Applications

Abstract

In this chapter we present several applications of the theory of GLT sequences. Special attention is devoted to the most important among them, that is, the computation of the singular value and eigenvalue distribution of matrix-sequences arising from the numerical discretization of differential equations.

In this chapter we present several applications of the theory of GLT sequences. Special attention is devoted to the most important among them, that is, the computation of the singular value and eigenvalue distribution of matrix-sequences arising from the numerical discretization of differential equations. In order to understand the content of this chapter, it is enough that the reader knows the summary of Chap. 9 and possesses the necessary prerequisites, most of which have been addressed in Chap. 2. Indeed, except for a few announced cases, our arguments/derivations in this chapter will never refer to Chaps. 3–8, i.e., they will only rely on the summary of Chap. 9, sometimes invoking also results from Chap. 2. For more applications than the ones presented herein, we refer the reader to Chap. 1, where specific pointers to the available literature are provided.

10.1 The Algebra Generated by Toeplitz Sequences

The algebra $\mathscr {T}$ generated  by Toeplitz sequences is the smallest subalgebra of the space of matrix-sequences

$$\begin{aligned} \mathscr {E}=\{\{A_n\}_n:\ \{A_n\}_n \text { is a matrix-sequence}\} \end{aligned}$$

containing the sequences of the form $\{T_n(f)\}_n$ with $f\in L^1([-\pi ,\pi ])$. Using the linearity of $T_n(\cdot )$, it is not difficult to see that

$$\begin{aligned} \mathscr {T}=\biggl \{\biggl \{\,\sum _{i=1}^r\prod _{j=1}^{q_i}T_n(f_{ij})\biggr \}_n:&\ \ r, q_1,\ldots , q_r\in \mathbb N,\ \ f_{ij}\in L^1([-\pi ,\pi ]) \text { for all }i, j\biggr \}. \end{aligned}$$

By GLT 3 it is clear that $\mathscr {T}$ is a subalgebra of the GLT algebra

$$\begin{aligned} \mathscr {G}=\{\{A_n\}_n:\ \{A_n\}_n\sim _\mathrm{GLT}\kappa \text { for some measurable }\kappa :[0, 1]\times [-\pi ,\pi ]\rightarrow \mathbb C\}. \end{aligned}$$

By GLT 3 and GLT 4 we have

$$\begin{aligned} \biggl \{\,\sum _{i=1}^r\prod _{j=1}^{q_i}T_n(f_{ij})\biggr \}_n\sim _\mathrm{GLT}\sum _{i=1}^r\prod _{j=1}^{q_i}\, 1\otimes f_{ij}=1\otimes \sum _{i=1}^r\prod _{j=1}^{q_i}f_{ij}. \end{aligned}$$

Taking into account the definition of singular value and spectral distribution of a matrix-sequence, GLT 1 yields

$$\begin{aligned} \biggl \{\,\sum _{i=1}^r\prod _{j=1}^{q_i}T_n(f_{ij})\biggr \}_n\sim _\sigma \sum _{i=1}^r\prod _{j=1}^{q_i}f_{ij}. \end{aligned}$$

(10.1)

If the matrices $\sum _{i=1}^r\prod _{j=1}^{q_i}T_n(f_{ij})$ are Hermitian, GLT 1 also yields

$$\begin{aligned} \biggl \{\,\sum _{i=1}^r\prod _{j=1}^{q_i}T_n(f_{ij})\biggr \}_n\sim _\lambda \sum _{i=1}^r\prod _{j=1}^{q_i}f_{ij}. \end{aligned}$$

(10.2)

The result (10.1) was originally obtained in [104]. In the case where the functions $f_{ij}$ belong to $L^\infty ([-\pi ,\pi ])$ for all i, j, both (10.1) and (10.2) already appeared in [28, Sect. 5.7]. Clearly, the distribution relations (10.1) and (10.2) extend T 4, that is, the $L^1$ version of the Avram–Parter theorem and the Szegő first limit theorem for Toeplitz sequences $\{T_n(f)\}_n$; see Sect. 6.5 for the history of these theorems.

Extension of the spectral distribution to the non-Hermitian case. The extension of the spectral distribution (10.2) to the case where the matrices $\sum _{i=1}^r\prod _{j=1}^{q_i}T_n(f_{ij})$ are not Hermitian has been the subject of recent researches [44, 110]. Note that if we remove the hypothesis of Hermitianess, then we necessarily have to add some additional assumption because (10.2) does not hold in general (see Exercise 6.4). The main hypothesis added in [44, 110] is the same topological assumption on the essential range of the function $\sum _{i=1}^r\prod _{j=1}^{q_i}f_{ij}$ as in T 5. This hypothesis was first employed by Tilli in [121] in order to prove T 5 and it is sometimes expressed by saying that $\sum _{i=1}^r\prod _{j=1}^{q_i}f_{ij}$ belongs to the Tilli class (see also the discussion at the end of Sect. 6.5).

Singular value and spectral distribution results beyond the algebra generated by Toeplitz sequences. It is clear that $\mathbf{GLT\, 1}\!-\!\mathbf{GLT\, 9}$ allows one to derive singular value and spectral distribution results for matrix-sequences obtained from more complicated operations on Toeplitz sequences than sums and products. For example, assume $f, g, u\in L^1([-\pi ,\pi ])$ are such that f is real a.e. and $g>0$ a.e., so that the matrices $T_n(f)$ are Hermitian by T 2 and the matrices $T_n(g)$ are HPD by T 2 and T 6. Then, by GLT 1 and $\mathbf{GLT\, 3}\!-\!\mathbf{GLT\, 6}$ we obtain

$$\begin{aligned}&\{(T_n(g))^{-1/2}\cos (T_n(f))(T_n(g))^{-1/2}\}_n\sim _\mathrm{GLT}1\otimes (g^{-1}\cos (f)),\\&\{(T_n(g))^{-1/2}\cos (T_n(f))(T_n(g))^{-1/2}\}_n\sim _{\sigma ,\,\lambda }g^{-1}\cos (f),\\&\{T_n(u)\mathrm{e}^{T_n(f)}(T_n(u))^*+\log (T_n(g))\}_n\sim _\mathrm{GLT}1\otimes (|u|^2\mathrm{e}^f+\log (g)),\\&\{T_n(u)\mathrm{e}^{T_n(f)}(T_n(u))^*+\log (T_n(g))\}_n\sim _{\sigma ,\,\lambda }|u|^2\mathrm{e}^f+\log (g), \end{aligned}$$

etc. (we could continue indefinitely...).

10.2 Variable-Coefficient Toeplitz Sequences

Let

$$\begin{aligned}&\mathfrak L_1=\bigl \{a:[0, 1]^2\times [-\pi ,\pi ]\rightarrow \mathbb C:\\&\qquad \qquad a(x, y,\cdot )\in L^1([-\pi ,\pi ]) \text { for all } (x, y)\in [0, 1]^2\bigr \}. \end{aligned}$$

For every $a\in \mathfrak L_1$ and every point $(x, y)\in [0, 1]^2$, the Fourier coefficients of the function $a(x, y,\cdot )\in L^1([-\pi ,\pi ])$ are given by

$$\begin{aligned} a_k(x, y)=\frac{1}{2\pi }\int _{-\pi }^\pi a(x, y,\theta )\,\mathrm{e}^{-Ik\theta }\mathrm{d}\theta ,\qquad k\in \mathbb Z. \end{aligned}$$

Every $a\in \mathfrak L_1$ can be formally represented by its Fourier series in the last variable, according to the formal equation

$$\begin{aligned} a(x, y,\theta )=\sum _{k\in \mathbb Z}a_k(x, y)\,\mathrm{e}^{Ik\theta }. \end{aligned}$$

The nth variable-coefficient  Toeplitz matrix associated with a is defined for $n\ge 2$ as

$$\begin{aligned} A_n(a)=\biggl [a_{i-j}\Bigl (\frac{i-1}{n-1},\frac{j-1}{n-1}\Bigr )\biggr ]_{i, j=1}^n. \end{aligned}$$

We call $\{A_n(a)\}_n$ the sequence of variable-coefficient Toeplitz matrices (or simply the variable-coefficient Toeplitz sequence) generated by a. The function a is referred to as the generating function of $\{A_n(a)\}_n$. Note that $A_n(a)=T_n(a)$ whenever a is independent of x and y. Note also that, for each fixed $n\ge 2$, the map

$$\begin{aligned} A_n(\cdot ):\mathfrak L_1\rightarrow \mathbb C^{n\times n},\qquad a\mapsto A_n(a), \end{aligned}$$

is linear:

$$\begin{aligned} A_n(\alpha a+\beta b)=\alpha A_n(a)+\beta A_n(b),\qquad \alpha ,\beta \in \mathbb C,\qquad a, b\in \mathfrak L_1. \end{aligned}$$

Variable-coefficient Toeplitz matrices are also known as generalized convolutions and appear in many different contexts. As testified by the literature, this kind of matrices has received a certain attention in the last years; see, e.g., [25, 26, 49, 85, 100, 115, 116, 127, 128]. We also refer the reader to [50, 81, 82] for a numerical-oriented literature about orthogonal polynomials with varying recurrence coefficients: the associated  Jacobi matrices can be interpreted as (approximated) symmetric Toeplitz matrices with variable coefficients.

Following the analysis in [63], in this section we show that, under suitable assumptions on a, $\{A_n(a)\}_n$ is a GLT sequence with symbol $a(x, x,\theta )$. This property, in combination with the theory of GLT sequences, allows one to derive a lot of singular value and eigenvalue distribution results for various matrix-sequences, including those obtained from algebraic and non-algebraic operations on variable-coefficient Toeplitz sequences. Let us formulate the main result of this section in a precise way. For any $\varepsilon >0$, define the strip

$$\begin{aligned} S_\varepsilon =\{(x, y)\in [0, 1]^2:\ |x-y|\le \varepsilon \} \end{aligned}$$

and set

$$\begin{aligned} \mathcal W=\Bigl \{a\in \mathfrak L_1:&\ \sum _{k\in \mathbb Z}\ \sup _{(x, y)\in [0, 1]^2}|a_k(x, y)|<\infty ,\ \ \text {for all } k\in \mathbb Z \text { there is}\, \varepsilon (k)>0 \nonumber \\&\ \text {such that }a_k(\cdot ,\cdot ) \text { is continuous on}\, S_{\varepsilon (k)}\Bigr \}. \end{aligned}$$

(10.3)

Note that $\mathcal W$ contains every continuous function $a\in C([0, 1]^2\times [-\pi ,\pi ])$ satisfying the so-called Wiener-type condition

$$\begin{aligned} \sum _{k\in \mathbb Z}\ \sup _{(x, y)\in [0, 1]^2}|a_k(x, y)|<\infty . \end{aligned}$$

(10.4)

The main result of this section is the following.

Theorem 10.1

If $a\in \mathcal W$ then $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x, x,\theta )$.

Proof of the main result. Some work is necessary to prove Theorem 10.1. Given $\alpha :[0, 1]^2\rightarrow \mathbb C$, $n\ge 2$ and $k\in \mathbb Z$, we define the $n\times n$ diagonal matrix

$$\begin{aligned} D_{n, k}(\alpha )=\mathop {\mathrm{diag}}_{h=1,\ldots , n}\alpha \Bigl (\frac{(h-1+k)\,\mathrm{mod}\, n}{n-1},\frac{h-1}{n-1}\Bigr ). \end{aligned}$$

Lemma 10.1

For every $a\in \mathfrak L_1$ and $n\ge 2$, we have

$$\begin{aligned} A_n(a)=\sum _{k=-(n-1)}^{n-1}T_n(\mathrm{e}^{Ik\theta })\, D_{n, k}(a_k). \end{aligned}$$

Proof

For $n\ge 2$ and $k\in \mathbb Z$, we have

$$\begin{aligned} (T_n(\mathrm{e}^{Ik\theta }))_{ij}=\delta _{i-j,\, k}=\left\{ \begin{array}{ll}1, &{}\ \ \mathrm{if}\ \ i-j=k,\\ 0, &{}\ \ \mathrm{otherwise}, \end{array}\right. \qquad i, j=1,\ldots , n. \end{aligned}$$

Hence, for all $i, j=1,\ldots , n$,

$$\begin{aligned} \biggl (\,\sum _{k=-(n-1)}^{n-1}T_n(\mathrm{e}^{Ik\theta })\, D_{n, k}(a_k)\biggr )_{ij}&=\sum _{k=-(n-1)}^{n-1}\delta _{i-j,\, k}\, a_k\Bigl (\frac{(j-1+k)\,\mathrm{mod}\, n}{n-1},\frac{j-1}{n-1}\Bigr )\\&=a_{i-j}\Bigl (\frac{i-1}{n-1},\frac{j-1}{n-1}\Bigr )=(A_n(a))_{ij}, \end{aligned}$$

and the lemma is proved. $\square$

Lemma 10.2

If $k\in \mathbb Z$ and $\alpha :[0, 1]^2\rightarrow \mathbb C$ is continuous on the strip $S_\varepsilon$ for some $\varepsilon >0$, then

$$\begin{aligned} \{D_{n, k}(\alpha )\}_n\sim _\mathrm{GLT}\alpha (x, x). \end{aligned}$$

Proof

Let k and $\alpha$ be as in the statement of the lemma. We show that, for all $n\ge 2$,

$$\begin{aligned} D_{n, k}(\alpha ) = D_n(\alpha (x, x))+R_n+N_n, \end{aligned}$$

(10.5)

where

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\mathrm{rank}(R_n)}{n}=\lim _{n\rightarrow \infty }\Vert N_n\Vert =0. \end{aligned}$$

(10.6)

This implies that the matrix-sequence $\{Z_n=R_n+N_n\}_n$ is zero-distributed by Z 1, and the thesis follows from (10.5) in combination with GLT 3 and GLT 4.

Let $\omega _{\alpha ,\varepsilon }(\cdot )$ be the modulus of continuity of $\alpha$ over the strip $S_\varepsilon$ with respect to the infinity norm, i.e.,

$$\begin{aligned} \omega _{\alpha ,\varepsilon }(\delta )\ =\max _{\begin{array}{c} (x, y),\,(x', y')\in S_\varepsilon \\ |(x-x',\, y-y')|_\infty \le \delta \end{array}}|\alpha (x, y)-\alpha (x', y')|,\qquad \delta >0. \end{aligned}$$

If $k\ge 0$ and $n>\frac{k}{\varepsilon }+1$, for $h=1,\ldots , n-k$ we have

$$\begin{aligned} \left| (D_{n, k}(\alpha ))_{hh}-(D_n(\alpha (x, x)))_{hh}\right|&=\left| \alpha \Bigl (\frac{h-1+k}{n-1},\frac{h-1}{n-1}\Bigr )-\alpha \Bigl (\frac{h}{n},\frac{h}{n}\Bigr )\right| \nonumber \\&\le \omega _{\alpha ,\varepsilon }\Bigl (\frac{k+1}{n-1}\Bigr ), \end{aligned}$$

(10.7)

which tends to 0 as $n\rightarrow \infty$. Write

$$\begin{aligned} D_{n, k}(\alpha )-D_n(\alpha (x, x)) = N_n+R_n, \end{aligned}$$

where $N_n$ (resp., $R_n$) is the matrix obtained from $D_{n, k}(\alpha )-D_n(\alpha (x, x))$ by setting to 0 the diagonal elements corresponding to indices $h>n-k$ (resp., $h\le n-k$). Then, the decomposition (10.5)–(10.6) follows from (10.7) and from the obvious inequality $\mathrm{rank}(R_n)\le k$.

If $k<0$ and $n>\frac{|k|}{\varepsilon }+1$, for $h=|k|+1,\ldots , n$ we have

$$\begin{aligned} \left| (D_{n, k}(\alpha ))_{hh}-(D_n(\alpha (x, x)))_{hh}\right|&=\left| \alpha \Bigl (\frac{h-1+k}{n-1},\frac{h-1}{n-1}\Bigr )-\alpha \Bigl (\frac{h}{n},\frac{h}{n}\Bigr )\right| \nonumber \\&\le \omega _{\alpha ,\varepsilon }\Bigl (\frac{|k|+1}{n-1}\Bigr ), \end{aligned}$$

(10.8)

which tends to 0 as $n\rightarrow \infty$. Write

$$\begin{aligned} D_{n, k}(\alpha )-D_n(\alpha (x, x)) = N_n+R_n, \end{aligned}$$

where $N_n$ (resp., $R_n$) is the matrix obtained from $D_{n, k}(\alpha )-D_n(\alpha (x, x))$ by setting to 0 the diagonal elements corresponding to indices $h<|k|+1$ (resp., $h\ge |k|+1$). Then, the decomposition (10.5)–(10.6) follows from (10.8) and from the (obvious) inequality $\mathrm{rank}(R_n)\le |k|$. $\square$

We are now ready to prove Theorem 10.1. For $n\ge 2$ and $m\in \mathbb N$, consider the matrix

$$\begin{aligned} A_{n, m}(a) = \sum _{k=-m}^mT_n(\mathrm{e}^{Ik\theta })\, D_{n, k}(a_k). \end{aligned}$$

Note that $A_{n, m}(a)$ ‘resembles’ $A_n(a)$ because, by Lemma 10.1, the only difference between these two matrices is the range of indices where k varies. By Lemma 10.1 we have $A_{n, m}(a)=A_n(\alpha _m)$ for $n>m$, where

$$\begin{aligned} \alpha _m(x, y,\theta )=\sum _{k=-m}^ma_k(x, y)\,\mathrm{e}^{Ik\theta } \end{aligned}$$

is the m th Fourier sum of the function $\theta \mapsto a(x, y,\theta )$. We are going to show that:

• (i) $\{A_{n, m}(a)\}_n\sim _\mathrm{GLT}\alpha _m(x, x,\theta )$ for every $m\in \mathbb N$;

• (ii) $\alpha _m(x, x,\theta )\rightarrow a(x, x,\theta )$ in measure over $[0, 1]\times [-\pi ,\pi ]$;

• (iii) $\{A_{n, m}(a)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n(a)\}_n$.

Once this is done, the thesis follows from GLT 7. By the hypothesis on a, each $a_k$ is continuous on $S_{\varepsilon (k)}$ for some $\varepsilon (k)>0$, hence $\{D_{n, k}(a_k)\}_n\sim _\mathrm{GLT}a_k(x, x)$ by Lemma 10.2. Thus, by GLT 3 and GLT 4,

$$\begin{aligned} \{A_{n, m}(a)\}_n\sim _\mathrm{GLT}\sum _{k=-m}^m\mathrm{e}^{Ik\theta }a_k(x, x)=\alpha _m(x, x,\theta ), \end{aligned}$$

and item (i) is proved. Considering that a satisfies the Wiener-type condition (10.4), $\alpha _m(x, y,\theta )\rightarrow a(x, y,\theta )$ uniformly on $[-\pi ,\pi ]$ for each fixed $(x, y)\in [0, 1]^2$. In particular, the sequence of continuous functions $\alpha _m(x, x,\theta )$ converges pointwise to $a(x, x,\theta )$ over $[0, 1]\times [-\pi ,\pi ]$. Since the pointwise convergence on a set of finite measure implies the convergence in measure (by Lemma 2.4), item (ii) is proved. Finally, by Lemma 10.1 and the equality

$$\begin{aligned} \Vert T_n(\mathrm{e}^{Ik\theta })\Vert =1,\qquad |k|<n, \end{aligned}$$

for every $n>m$ we have

$$\begin{aligned} \Vert A_n(a)-A_{n, m}(a)\Vert&=\biggl \Vert \,\sum _{n>|k|>m}T_n(\mathrm{e}^{Ik\theta })D_{n, k}(a_k)\biggr \Vert \le \sum _{n>|k|>m}\Vert T_n(\mathrm{e}^{Ik\theta })\Vert \,\Vert D_{n, k}(a_k)\Vert \nonumber \\&\le \sum _{n>|k|>m}\ \sup _{(x, y)\in [0, 1]^2}|a_k(x, y)|=\varepsilon (m, n). \end{aligned}$$

Recalling that a satisfies the Wiener-type condition (10.4), we have

$$\begin{aligned} \lim _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }\,\varepsilon (m, n)=0, \end{aligned}$$

and ACS 6 implies that $\{A_{n, m}(a)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n(a)\}_n$. We then conclude that item (iii) holds.

Consequences of the main result. Consequences of Theorem 10.1 (and of the theory of GLT sequences) are all the results that can be deduced from the list of properties $\mathbf{GLT\, 1}\!-\!\mathbf{GLT\, 9}$ in which GLT 3 is extended to include the result of Theorem 10.1. For the sake of clarity, we use a different notation for the extended version of GLT 3 and we give it the label $\overline{ \mathbf{ GLT 3 }}$. Property $\overline{ \mathbf{GLT 3 }}$ is then the following.

$\overline{ \mathbf{GLT 3 }}{} \mathbf .$ :

We have

• $\{T_n(f)\}_n\sim _\mathrm{GLT}\kappa (x,\theta )=f(\theta )$ if $f\in L^1([-\pi ,\pi ])$,

• $\{D_n(a)\}_n\sim _\mathrm{GLT}\kappa (x,\theta )=a(x)$ if $a:[0, 1]\rightarrow \mathbb C$ is continuous a.e.,

• $\{Z_n\}_n\sim _\mathrm{GLT}\kappa (x,\theta )=0$ if and only if $\{Z_n\}_n\sim _\sigma 0$,

• $\{A_n(a)\}_n\sim _\mathrm{GLT}\kappa (x,\theta )=a(x, x,\theta )$ if $a\in \mathcal W$.

In the following we briefly discuss some consequences of $\mathbf{GLT\, 1}\!-\!\mathbf{GLT\, 2}$, $\overline{ \mathbf{GLT 3 }}$, $\mathbf{GLT\, 4}\!-\!\mathbf{GLT\, 9}$.

Singular value and spectral distribution results on the algebra generated by variable-coefficient Toeplitz sequences. Let $\mathscr {C}$  be the smallest subalgebra of

$$\begin{aligned} \mathscr {E}=\{\{A_n\}_n:\ \{A_n\}_n \text { is a matrix-sequence}\} \end{aligned}$$

containing the variable-coefficient Toeplitz sequences $\{A_n(a)\}_n$ with $a\in \mathcal W$. It is not difficult to see that

$$\begin{aligned} \mathscr {C} = \biggl \{\biggl \{\,\sum _{i=1}^r\prod _{j=1}^{q_i}A_n(a_{ij})\biggr \}_n:\ \ r, q_1,\ldots , q_r\in \mathbb N,\ \ a_{ij}\in \mathcal W \text { for all }\, i, j\biggr \}. \end{aligned}$$

By $\overline{ \mathbf{GLT 3 }}$ and GLT 4, $\mathscr {C}$ is a subalgebra of the GLT algebra $\mathscr {G}$, and for the generic element of $\mathscr {C}$ we have

$$\begin{aligned} \biggl \{\,\sum _{i=1}^r\prod _{j=1}^{q_i}A_n(a_{ij})\biggr \}_n\sim _\mathrm{GLT}\sum _{i=1}^r\prod _{j=1}^{q_i}a_{ij}(x, x,\theta ). \end{aligned}$$

Hence, by  GLT 1,

$$\begin{aligned} \biggl \{\,\sum _{i=1}^r\prod _{j=1}^{q_i}A_n(a_{ij})\biggr \}_n\sim _\sigma \sum _{i=1}^r\prod _{j=1}^{q_i}a_{ij}(x, x,\theta ) \end{aligned}$$

(10.9)

and, if the matrices $\sum _{i=1}^r\prod _{j=1}^{q_i}A_n(a_{ij})$ are Hermitian,

$$\begin{aligned} \biggl \{\,\sum _{i=1}^r\prod _{j=1}^{q_i}A_n(a_{ij})\biggr \}_n\sim _\lambda \sum _{i=1}^r\prod _{j=1}^{q_i}a_{ij}(x, x,\theta ). \end{aligned}$$

(10.10)

A result analogous to (10.9)–(10.10) was obtained by Silbermann and Zabroda in [115, Theorem 7.2].

Singular value and spectral distribution results beyond the algebra generated by variable-coefficient Toeplitz sequences.  It is clear that the relations (10.9) and (10.10) are far from exhausting the singular value and eigenvalue distribution results that can be derived from $\mathbf{GLT\, 1}\!-\!\mathbf{GLT\, 2}$, $\overline{ \mathbf{GLT 3 }}$, $\mathbf{GLT\, 4}\!-\!\mathbf{GLT\, 9}$. In particular, GLT 1, $\overline{ \mathbf{GLT 3 }}$ and $\mathbf{GLT\, 4}\!-\!\mathbf{GLT\, 6}$ allow one to compute the singular value and eigenvalue distribution of matrix-sequences that are obtained not only through sums and products of variable-coefficient Toeplitz sequences, but also through more complex operations involving all the GLT sequences listed in $\overline{ \mathbf{GLT 3 }}$. For example, let $a\in \mathcal W$. If $a(x, x,\theta )\ne 0$ a.e., then GLT 1, $\overline{ \mathbf{GLT 3 }}$, GLT 5 yield

$$\begin{aligned}&\{A_n(a)^\dag \}_n\sim _\mathrm{GLT}\frac{1}{a(x, x,\theta )}, \\&\{A_n(a)^\dag \}_n\sim _\sigma \frac{1}{a(x, x,\theta )}. \end{aligned}$$

If in addition the matrices $A_n(a)$ are Hermitian for all n, then GLT 1, $\overline{ \mathbf{GLT 3 }}$, GLT 5 also yield

$$\begin{aligned}&\{A_n(a)^\dag \}_n\sim _\lambda \frac{1}{a(x, x,\theta )}. \end{aligned}$$

If the matrices $A_n(a)$ are Hermitian for all n, then GLT 1, $\overline{ \mathbf{GLT 3 }}$, GLT 6 give

$$\begin{aligned}&\{\sin (A_n(a))\}_n\sim _\mathrm{GLT}\sin (a(x, x,\theta )),\\&\{\sin (A_n(a))\}_n\sim _{\sigma ,\,\lambda }\sin (a(x, x,\theta )). \end{aligned}$$

If $a(x, x,\theta )\ne 0$ a.e. and the matrices $A_n(a)$ are Hermitian for all n, then GLT 1, $\overline{ \mathbf{GLT 3 }}$, $\mathbf{GLT\, 4}\!-\!\mathbf{GLT\, 6}$ (and T 1) yield

$$\begin{aligned}&\{T_n(|\theta |^{-1/2})\, A_n(a)^\dag \mathrm{e}^{A_n(a)}A_n(a)^\dag T_n(|\theta |^{-1/2})\\&\qquad +8\, D_n(\log x)\, A_n(a)^\dag D_n(\log x)\}_n\sim _\mathrm{GLT}\frac{\mathrm{e}^{a(x, x,\theta )}}{|\theta |\, a(x, x,\theta )^2}+\frac{8\log ^2x}{a(x, x,\theta )},\\&\{T_n(|\theta |^{-1/2})\, A_n(a)^\dag \mathrm{e}^{A_n(a)}A_n(a)^\dag T_n(|\theta |^{-1/2})\\&\qquad +8\, D_n(\log x)\, A_n(a)^\dag D_n(\log x)\}_n\sim _{\sigma ,\,\lambda }\frac{\mathrm{e}^{a(x, x,\theta )}}{|\theta |\, a(x, x,\theta )^2}+\frac{8\log ^2x}{a(x, x,\theta )}. \end{aligned}$$

We could continue with this game indefinitely...

Extensions of the main result. We conclude this section on variable-coefficient Toeplitz sequences by mentioning two possible ways to extend Theorem 10.1, i.e., to prove the GLT relation $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x, x,\theta )$ for a space of functions a larger than $\mathcal W$.

• 1. The first way originates from the observation that, when proving items (i)–(iii) under the assumption $a\in \mathcal W$, we actually proved more than necessary. In particular, when proving item (iii) we showed that

  $$\begin{aligned} \Vert A_n(a)-A_{n, m}(a)\Vert \le \varepsilon (m, n) \end{aligned}$$

  (10.11)

  for some $\varepsilon (m, n)$ satisfying

  $$\begin{aligned} \lim _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }\,\varepsilon (m, n)=0. \end{aligned}$$

  (10.12)

  On the other hand, it would have been enough to show that $\{A_{n, m}(a)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n(a)\}_n$, by proving for example that

  $$\begin{aligned} \Vert A_n(a)-A_{n, m}(a)\Vert _p\le \varepsilon (m, n)n^{1/p} \end{aligned}$$

  (10.13)

  for some $p\in [1,\infty )$ and some $\varepsilon (m, n)$ satisfying (10.12); see ACS 6. Note that (10.13) is a condition weaker than (10.11), because for all $p\in [1,\infty )$ we have

  $$\begin{aligned} \frac{\Vert A\Vert _p}{n^{1/p}}\le \Vert A\Vert , \end{aligned}$$

  with the equality holding if and only if all the singular values of A are equal. In view of these considerations, there is room to refine the arguments used for the proof of Theorem 10.1, so as to weaken the hypotheses on a and, consequently, to enlarge the space of functions a such that $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x, x,\theta )$.

• 2. Suppose that, for all functions a belonging to a certain class $\mathfrak C\subseteq \mathfrak L_1$ we can construct a sequence $\{\alpha _m\}_m\subset \mathfrak L_1$ such that

  • (i) the relation $\{A_n(\alpha _m)\}_n\sim _\mathrm{GLT}\alpha _m(x, x,\theta )$ holds for all m,

  • (ii) $\alpha _m(x, x,\theta )\rightarrow a(x, x,\theta )$ in measure,

  • (iii) $\{A_n(\alpha _m)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{A_n(a)\}_n$.

  Then $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x, x,\theta )$ for all $a\in \mathfrak C$ (by GLT 7). This technique is the second possible way to extend Theorem 10.1, and it was already used in the proof of Theorem 10.1 with $\mathfrak C=\mathcal W$. What is important to point out is that, now that we have proved Theorem 10.1, we are allowed to take any sequence of functions in $\mathcal W$ as the sequence $\{\alpha _m\}_m$, because item (i) is automatically satisfied and item (ii) is satisfied as well, provided we choose an $\alpha _m$ converging to a in a suitable way which ensures the convergence in measure.

Exercise 10.1

Let

$$\begin{aligned} \mathcal X=\biggl \{\,\sum _{r=1}^q\alpha _r(x, y)\beta _r(\theta ):&\ \ \alpha _r\in C([0, 1]^2) \text { and }\,\beta _r\in L^2([-\pi ,\pi ])\\&\ \ \text {for all }\, r=1,\ldots , q,\quad q\in \mathbb N\biggr \}. \end{aligned}$$

Show that $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x, x,\theta )$ for all $a\in \mathcal X$.

Exercise 10.2

Let

$$\begin{aligned} \mathcal Y=\biggl \{\,\sum _{r=1}^q\alpha _r(x)\beta _r(y)\gamma _r(\theta ):&\ \ \alpha _r,\beta _r\in C([0, 1]) \text { and }\,\gamma _r\in L^1([-\pi ,\pi ])\\&\ \ \text {for all }\, r=1,\ldots , q,\quad q\in \mathbb N\biggr \}. \end{aligned}$$

Show that $\{A_n(a)\}_n\sim _\mathrm{GLT}a(x, x,\theta )$ for all $a\in \mathcal Y$.

10.3 Geometric Means of Matrices

Everyone knows that the geometric mean of two positive numbers a, b is $G(a, b)=(ab)^{1/2}$. But what is the geometric mean G(A, B) of two HPD matrices $A, B\in \mathbb C^{n\times n}$? An appropriate definition was proposed in a remarkable paper by Ando, Li and Mathias [2]. The approach of these authors was axiomatic: denoting by  $\mathscr {P}_n$ the set of HPD matrices of size n, a function $G:\mathscr {P}_n\times \mathscr {P}_n\rightarrow \mathscr {P}_n$ is said to be a geometric mean if it satisfies a suitable list of properties that any geometric mean worthy of the name should satisfy. Ando, Li and Mathias proposed a list of ten properties, which are referred to as the ALM axioms. Let us mention three of them.

• 1. Permutation invariance: $G(A, B)=G(B, A)$ for all $A, B\in \mathscr {P}_n$.

• 2. Congruence invariance: $G(M^*AM, M^*BM)=M^*G(A, B)M$ for all $A, B\in \mathscr {P}_n$ and all invertible matrices $M\in \mathbb C^{n\times n}$.

• 3. Consistency with scalars: $G(A, B)=(AB)^{1/2}$ for all commuting   $A, B\in \mathscr {P}_n$ (note that $AB\in \mathscr {P}_n$ for all commuting $A, B\in \mathscr {P}_n$ because $(AB)^*=B^*A^*=BA=AB$ and AB is similar to the HPD matrix $A^{1/2}BA^{1/2}=A^{-1/2}(AB)A^{1/2}$).

It may be proved [14, Chap. 4] that the unique function $G:\mathscr {P}_n\times \mathscr {P}_n\rightarrow \mathscr {P}_n$ satisfying both consistency with scalars and congruence invariance is

$$\begin{aligned} G(A, B)=A^{1/2}\bigl (A^{-1/2} B A^{-1/2}\bigr )^{1/2}A^{1/2}, \end{aligned}$$

(10.14)

which, moreover, satisfies all the ALM axioms. In particular, using the permutation invariance we obtain the alternative expression

$$\begin{aligned} G(A, B)=B^{1/2}\bigl (B^{-1/2} A B^{-1/2}\bigr )^{1/2}B^{1/2}. \end{aligned}$$

(10.15)

Suppose now that $\{A_n\}_n\sim _\mathrm{GLT}\kappa$ and $\{B_n\}_n\sim _\mathrm{GLT}\xi$, where $A_n,\, B_n\in \mathscr {P}_{n}$. Due to the positive definiteness of $A_n, B_n$, GLT 1 and S 3 imply that $\kappa ,\xi \ge 0$ a.e. Assuming that at least one between $\kappa$ and $\xi$ is nonzero a.e., in Theorem 10.2 we show that the sequence of geometric means $\{G(A_n, B_n)\}_n$ is a GLT sequence whose symbol is given by the geometric mean of the symbols $\kappa ,\xi$.

Theorem 10.2

Suppose $\{A_n\}_n\sim _\mathrm{GLT}\kappa$ and $\{B_n\}_n\sim _\mathrm{GLT}\xi$, where $A_n, B_n\in \mathscr {P}_{n}$. Assume that at least one between $\kappa$ and $\xi$ is nonzero a.e. Then

$$\begin{aligned} \{G(A_n, B_n)\}_n\sim _\mathrm{GLT}(\kappa \xi )^{1/2} \end{aligned}$$

(10.16)

and

$$\begin{aligned} \{G(A_n, B_n)\}_n\sim _{\sigma ,\,\lambda }(\kappa \xi )^{1/2}. \end{aligned}$$

(10.17)

Proof

It suffices to prove (10.16) because (10.17) follows immediately from (10.16) and GLT 1, taking into account that $G(A_n, B_n)$ is HPD. Let $f:\mathbb C\rightarrow \mathbb C$ be any continuous function such that $f(x)=x^{1/2}$ for all $x\ge 0$ (take, for example, $f(z)=|z|^{1/2}$). Using the expression of $G(A_n, B_n)$ in (10.14) we see that

$$\begin{aligned} G(A_n, B_n)&=A_n^{1/2}\bigl (A_n^{-1/2}B_n\, A_n^{-1/2}\bigr )^{1/2}A_n^{1/2}\\&=f(A_n)f(f(A_n)^{-1}B_nf(A_n)^{-1})f(A_n). \end{aligned}$$

If $\kappa \ne 0$ a.e. (and hence $\kappa >0$ a.e. because $\kappa \ge 0$ a.e. by the previous discussion), then $f(\kappa )=\kappa ^{1/2}\ne 0$ a.e. and $\mathbf{GLT\, 4}\!-\!\mathbf{GLT\, 6}$ yield

$$\begin{aligned} \{G(A_n, B_n)\}_n\sim _\mathrm{GLT}f(\kappa )f({f(\kappa )}^{-1}\xi {f(\kappa )}^{-1})f(\kappa ) \end{aligned}$$

with

$$\begin{aligned} f(\kappa )f(f(\kappa )^{-1}\xi f(\kappa )^{-1})f(\kappa )=\kappa ^{1/2}\bigl (\kappa ^{-1/2}\xi \kappa ^{-1/2}\bigr )^{1/2}\kappa ^{1/2}=(\kappa \xi )^{1/2}\ \ \text {a.e.} \end{aligned}$$

Thus, (10.16) is proved. If $\xi \ne 0$ a.e., the proof is the same as in the case $\kappa \ne 0$ a.e., with the only difference that now we use the expression of $G(A_n, B_n)$ in (10.15) instead of the one in (10.14). $\square$

In Theorem 10.2, the hypothesis that at least one between $\kappa$ and $\xi$ is nonzero a.e. looks a bit ‘artificial’, in the sense that it seems to be necessary only for the argument used in the proof as both (10.16) and (10.17) make perfect sense even if this hypothesis is violated. In fact, we believe that this hypothesis can be removed and we therefore formulate the following conjecture.

Conjecture 10.1

Suppose $\{A_n\}_n\sim _\mathrm{GLT}\kappa$ and $\{B_n\}_n\sim _\mathrm{GLT}\xi$, where $A_n, B_n\in \mathscr {P}_{n}$. Then, both (10.16) and (10.17) hold.

While it is easy to generalize the concept of geometric mean to the case where the numbers to be averaged are $k>2$, the same is not true for HPD matrices. In particular, the axiomatic approach by Ando, Li and Mathias is not satisfactory for $k>2$, because the ten ALM axioms do not lead to a unique definition [19, 87]. The path to the right definition was different, involving a little bit of differential geometry. In fact, the geometric mean (or  Karcher mean) of k matrices $A^{(1)},\ldots , A^{(k)}\in \mathscr {P}_n$ was defined as the barycenter of the matrices with respect to a certain Riemannian distance; see [15] and [14, Chap. 6]. More precisely, the Karcher mean $G(A^{(1)},\ldots , A^{(k)})$ is the unique minimizer over $\mathscr {P}_n$ of the functional

$$\begin{aligned} D(\cdot \,;\, A^{(1)},\ldots , A^{(k)}):\mathscr {P}_n\rightarrow \mathbb R,\qquad D(X;\, A^{(1)},\ldots , A^{(k)})=\sum _{i=1}^k\bigl (\delta (X, A^{(i)})\bigr )^2, \end{aligned}$$

where $\delta (A, B)$ is the distance given by the Riemannian structure,

$$\begin{aligned} \delta (A, B)=\bigl \Vert \log (A^{-1/2}BA^{-1/2})\bigr \Vert _2=\biggl (\,\sum _{\ell =1}^n\log ^2(\lambda _\ell (A^{-1/2}BA^{-1/2}))\biggr )^{1/2}. \end{aligned}$$

It was proved with some effort [16, 72, 84] that the Karcher mean satisfies all the ALM axioms and some further properties, and thus now everyone agrees that the Karcher mean has the right to be called the geometric mean of matrices. Besides the mathematical interest, the Karcher mean has been used in several applications (see [78, Sect. 6] and the references in [18]), and suitable algorithms for its computation have been designed [18, 79].

Suppose now that $\{A_n^{(i)}\}_n\sim _\mathrm{GLT}\kappa _i$ for $i=1,\ldots , k$, where $A_n^{(1)},\ldots , A_n^{(k)}\in \mathscr {P}_n$. Due to the positive definiteness of the $A_n^{(i)}$, GLT 1 and S 3 imply that each $\kappa _i$ is nonnegative a.e. In this situation, we have reason to believe that the sequence of Karcher means $\{G(A_n^{(1)},\ldots , A_n^{(k)})\}_n$ is a GLT sequence whose symbol is $(\kappa _1\cdots \kappa _k)^{1/k}$, i.e., the geometric means of the symbols $\kappa _1,\ldots ,\kappa _k$.

Conjecture 10.2

Suppose $\{A_n^{(i)}\}_n\sim _\mathrm{GLT}\kappa _i$ for $i=1,\ldots , k$, where $A_n^{(1)},\ldots , A_n^{(k)}\in \mathscr {P}_n$. Then

$$\begin{aligned} \{G(A_n^{(1)},\ldots , A_n^{(k)})\}_n\sim _\mathrm{GLT}(\kappa _1 \cdots \kappa _k)^{1/k} \end{aligned}$$

(10.18)

and

$$\begin{aligned} \{G(A_n^{(1)},\ldots , A_n^{(k)})\}_n\sim _{\sigma ,\,\lambda }(\kappa _1 \cdots \kappa _k)^{1/k}. \end{aligned}$$

(10.19)

Note that only the GLT relation (10.18) needs to be proved as the singular value and eigenvalue distributions in (10.19) follow from GLT 1 and the Hermitianess of $G(A_n^{(1)},\ldots , A_n^{(k)})$. Note also that Conjecture 10.2 is an extension of Conjecture 10.1. The formal proof of Conjecture 10.2, which might be achieved via GLT 7, is certainly an interesting subject for future research, also considering that geometric means of Toeplitz matrices are of interest in practical applications. For example, in a  radar application one is interested in computing a geometric mean of HPD matrices which are Toeplitz or  block Toeplitz with Toeplitz blocks (i.e., 2-level Toeplitz according to a more recent terminology); see [7, 8, 83].

10.4 Discretization of Integral Equations

As shown in [1, 98], the discretization of an IE usually leads to a zero-distributed sequence, i.e., a GLT sequence with symbol 0 (see GLT 3). In this section we present a specific example from [98] which illustrates this phenomenon. For more on IE discretizations we refer the reader to [1, 98].

Consider the IE

$$\begin{aligned} f(x)-\int _0^1k(x, y)f(y)\mathrm{d}y=g(x), \end{aligned}$$

(10.20)

where $g:[0, 1]\rightarrow \mathbb R$ is a given function and the kernel $k:[0, 1]^2\rightarrow \mathbb R$ is defined as

$$\begin{aligned} k(x, y)=\left\{ \begin{array}{ll} b(x)c(y), &{}\ \ \text {if }\, x\le y,\\ b(y)c(x), &{}\ \ \text {if }\, y\le x. \end{array}\right. \end{aligned}$$

(10.21)

For the GLT analysis we are going to perform, it will be enough to assume that $b, c:[0, 1]\rightarrow \mathbb R$ are continuous a.e. and belong to $L^1([0, 1])$.

Discretization. We consider the discretization of (10.20) using the rectangle formula on a uniform grid. Let $h=\frac{1}{n}$ and set $x_i=ih$ for $i=1,\ldots , n$. The rectangle formula yields the approximation

$$\begin{aligned} \int _0^1k(x_i, y)f(y)\mathrm{d}y\approx h\sum _{j=1}^nk(x_i, x_j)f(x_j),\qquad i=1,\ldots , n. \end{aligned}$$

This means that the nodal values of the solution f of (10.20) satisfy (approximately) the following linear system:

$$\begin{aligned} f(x_i)-h\sum _{j=1}^nk(x_i, x_j)f(x_j)=g(x_i),\qquad i=1,\ldots , n. \end{aligned}$$

We then approximate the solution by the piecewise linear function that takes the value $f_i$ at $x_i$ for $i=1,\ldots , n$, where $\mathbf {f}=(f_1,\ldots , f_n)^T$ solves

$$\begin{aligned} f_i-h\sum _{j=1}^nk(x_i, x_j)f_j=g(x_i),\qquad i=1,\ldots , n. \end{aligned}$$

This linear system can be written in matrix form as

$$\begin{aligned} (I_n-hG_n)\mathbf {f}=\mathbf {g}, \end{aligned}$$

where $\mathbf {g}=(g(x_1),\ldots , g(x_n))^T$ and

$$\begin{aligned} G_n=\bigl [k(x_i, x_j)\bigr ]_{i, j=1}^n. \end{aligned}$$

Using the specific form of the kernel given in (10.21), we finally obtain

$$\begin{aligned} G_n=\left[ \begin{array}{cccccc} b_1 c_1 &{} \ b_1 c_2 &{} \ b_1 c_3 &{} \ \cdots &{} \ \cdots &{} \ b_1 c_n \\ b_1 c_2 &{} \ b_2 c_2 &{} \ b_2 c_3 &{} \ \cdots &{} \ \cdots &{} \ b_2 c_n \\ b_1 c_3 &{} \ b_2 c_3 &{} \ b_3 c_3 &{} \ \cdots &{} \ \cdots &{} \ b_3 c_n \\ \vdots &{} \ \vdots &{} \ \vdots &{} \ \ddots &{} \ &{} \ \vdots \\ \vdots &{} \ \vdots &{} \ \vdots &{} \ &{} \ \ddots &{} \ \vdots \\ b_1 c_n &{} \ b_2 c_n &{} \ b_3c_n &{} \ \cdots &{} \ \cdots &{} \ b_n c_n\end{array} \right] , \end{aligned}$$

(10.22)

where $b_i=b(x_i)$ and $c_i=c(x_i)$ for all $i=1,\ldots , n$. Since a matrix of the form

$$ [\gamma _{ij}]_{i, j=1}^n,\qquad \gamma _{ij}=\left\{ \begin{array}{ll} \alpha _i\beta _j, &{}\ \ \text {if }\, \ i\le j,\\ \alpha _j\beta _i, &{}\ \ \text {if }\, \ i>j, \end{array}\right. \qquad \alpha _i,\beta _i\in \mathbb R\backslash \{0\} \text { for all }\, i=1,\ldots , n, $$

is referred to as a Green  matrix [90], it is clear that $G_n$ is a Green matrix; this is the reason why we use the notation “$G_n$” instead of the traditional “$A_n$”.

GLT analysis of the IE discretization matrices.  The main result of this section is stated in the next theorem, which shows that the sequence of Green matrices $G_n$ is zero-distributed.

Theorem 10.3

Let $b, c\in L^1([0, 1])$ be continuous a.e. Then

$$\begin{aligned} \{G_n\}_n\sim _\mathrm{GLT}0 \end{aligned}$$

(10.23)

and

$$\begin{aligned} \{G_n\}_n\sim _{\sigma ,\,\lambda }0. \end{aligned}$$

(10.24)

Proof

Since $G_n$ is symmetric, (10.24) follows from (10.23) and GLT 1. It is therefore enough to prove (10.23). Consider the following  decomposition of $G_n$:

$$\begin{aligned} G_n=L_n+U_n-D_n, \end{aligned}$$

(10.25)

where $L_n$, $U_n$, $D_n$ are, respectively, the lower triangular, upper triangular and diagonal part of $G_n$, i.e.,

$$\begin{aligned} L_n&=\left[ \begin{array}{cccccc} b_1 c_1 &{} \ &{} \ &{} \ &{} \ &{} \ \\ b_1 c_2 &{} \ b_2 c_2 &{} \ &{} \ &{} \ &{} \ \\ b_1 c_3 &{} \ b_2 c_3 &{} \ b_3 c_3 &{} \ &{} \ &{} \ \\ \vdots &{} \ \vdots &{} \ \vdots &{} \ \ddots &{} \ &{} \ \\ \vdots &{} \ \vdots &{} \ \vdots &{} \ &{} \ \ddots &{} \ \\ b_1 c_n &{} \ b_2 c_n &{} \ b_3c_n &{} \ \cdots &{} \ \cdots &{} \ b_n c_n\end{array} \right] , \\ U_n&=\left[ \begin{array}{cccccc} b_1 c_1 &{} \ b_1 c_2 &{} \ b_1 c_3 &{} \ \cdots &{} \ \cdots &{} \ b_1 c_n \\ &{} \ b_2 c_2 &{} \ b_2 c_3 &{} \ \cdots &{} \ \cdots &{} \ b_2 c_n \\ &{} \ &{} \ b_3 c_3 &{} \ \cdots &{} \ \cdots &{} \ b_3 c_n \\ &{} \ &{} \ &{} \ \ddots &{} \ &{} \ \vdots \\ &{} \ &{} \ &{} \ &{} \ \ddots &{} \ \vdots \\ &{} \ &{} \ &{} \ &{} \ &{} \ b_n c_n\end{array} \right] ,\\ D_n&=\left[ \begin{array}{cccccc} b_1 c_1 &{} \ &{} \ &{} \ &{} \ &{} \ \\ &{} \ b_2 c_2 &{} \ &{} \ &{} \ &{} \ \\ &{} \ &{} \ b_3 c_3 &{} \ &{} \ &{} \ \\ &{} \ &{} \ &{} \ \ddots &{} \ &{} \ \\ &{} \ &{} \ &{} \ &{} \ \ddots &{} \ \\ &{} \ &{} \ &{} \ &{} \ &{} \ b_n c_n\end{array} \right] . \end{aligned}$$

The matrix $L_n$ can be written as

$$\begin{aligned} L_n&=\left[ \begin{array}{cccccc} c_1 &{} \ &{} \ &{} \ &{} \ &{} \ \\ &{} \ c_2 &{} \ &{} \ &{} \ &{} \ \\ &{} \ &{} \ c_3 &{} \ &{} \ &{} \ \\ &{} \ &{} \ &{} \ \ddots &{} \ &{} \ \\ &{} \ &{} \ &{} \ &{} \ \ddots &{} \ \\ &{} \ &{} \ &{} \ &{} \ &{} \ c_n\end{array} \right] \left[ \begin{array}{cccccc} 1 &{} \ \ &{} \ \ &{} \ &{} \ &{} \ \\ 1 &{} \ \ 1 &{} \ \ &{} \ &{} \ &{} \ \\ 1 &{} \ \ 1 &{} \ \ 1 &{} \ &{} \ &{} \ \\ \vdots &{} \ \ \vdots &{} \ \ \vdots &{} \ \ddots &{} \ &{} \ \\ \vdots &{} \ \ \vdots &{} \ \ \vdots &{} \ &{} \ \ddots &{} \ \\ 1 &{} \ \ 1 &{} \ \ 1 &{} \ \cdots &{} \ \cdots &{} \ 1\end{array} \right] \left[ \begin{array}{cccccc} b_1 &{} \ &{} \ &{} \ &{} \ &{} \ \\ &{} \ b_2 &{} \ &{} \ &{} \ &{} \ \\ &{} \ &{} \ b_3 &{} \ &{} \ &{} \ \\ &{} \ &{} \ &{} \ \ddots &{} \ &{} \ \\ &{} \ &{} \ &{} \ &{} \ \ddots &{} \ \\ &{} \ &{} \ &{} \ &{} \ &{} \ b_n\end{array} \right] \\&=D_n(c)\left[ \begin{array}{cccccc} 1 &{} \ \ &{} \ \ &{} \ &{} \ &{} \ \\ 1 &{} \ \ 1 &{} \ \ &{} \ &{} \ &{} \ \\ 1 &{} \ \ 1 &{} \ \ 1 &{} \ &{} \ &{} \ \\ \vdots &{} \ \ \vdots &{} \ \ \vdots &{} \ \ddots &{} \ &{} \ \\ \vdots &{} \ \ \vdots &{} \ \ \vdots &{} \ &{} \ \ddots &{} \ \\ 1 &{} \ \ 1 &{} \ \ 1 &{} \ \cdots &{} \ \cdots &{} \ 1\end{array} \right] D_n(b). \end{aligned}$$

Similarly,

$$ U_n=D_n(b)\left[ \begin{array}{cccccc} 1 &{} \ \ 1 &{} \ \ 1 &{} \ \cdots &{} \ \cdots &{} \ 1 \\ &{} \ \ 1 &{} \ \ 1 &{} \ \cdots &{} \ \cdots &{} \ 1 \\ &{} \ \ &{} \ \ 1 &{} \ \cdots &{} \ \cdots &{} \ 1 \\ &{} \ \ &{} \ \ &{} \ \ddots &{} \ &{} \ \vdots \\ &{} \ \ &{} \ \ &{} \ &{} \ \ddots &{} \ \vdots \\ &{} \ \ &{} \ \ &{} \ &{} \ &{} \ 1\end{array} \right] D_n(c), $$

and, of course,

$$\begin{aligned} D_n = D_n(b)D_n(c). \end{aligned}$$

The crucial observation is that

$$\begin{aligned} \left[ \begin{array}{cccccc} 1 &{} \ \ &{} \ &{} \ &{} \ &{} \ \\ 1 &{} \ \ 1 &{} \ &{} \ &{} \ &{} \ \\ 1 &{} \ \ 1 &{} \ 1 &{} \ &{} \ &{} \ \\ \vdots &{} \ \ \vdots &{} \ \vdots &{} \ \ddots &{} \ &{} \ \\ \vdots &{} \ \ \vdots &{} \ \vdots &{} \ &{} \ \ddots &{} \ \\ 1 &{} \ \ 1 &{} \ \ 1 &{} \ \cdots &{} \ \cdots &{} \ 1\end{array} \right] =\left[ \begin{array}{cccccc} 1 &{} \ \ &{} \ \ &{} \ &{} \ &{} \ \\ -1 &{} \ \ 1 &{} \ \ &{} \ &{} \ &{} \ \\ &{} \ \ -1 &{} \ \ 1 &{} \ &{} \ &{} \ \\ &{} \ \ &{} \ \ \ddots &{} \ \ddots &{} \ &{} \ \\ &{} \ \ &{} \ \ &{} \ \ddots &{} \ \ddots &{} \ \\ &{} \ \ &{} \ \ &{} \ &{} \ -1 &{} \ 1\end{array} \right] ^{-1}&=(T_n(1-\mathrm{e}^{I\theta }))^{-1},\\ \left[ \begin{array}{cccccc} 1 &{} \ \ 1 &{} \ \ 1 &{} \ \cdots &{} \ \cdots &{} \ 1 \\ &{} \ \ 1 &{} \ \ 1 &{} \ \cdots &{} \ \cdots &{} \ 1 \\ &{} \ \ &{} \ \ 1 &{} \ \cdots &{} \ \cdots &{} \ 1 \\ &{} \ \ &{} \ \ &{} \ \ddots &{} \ &{} \ \vdots \\ &{} \ \ &{} \ \ &{} \ &{} \ \ddots &{} \ \vdots \\ &{} \ \ &{} \ \ &{} \ &{} \ &{} \ 1\end{array} \right] =\left[ \begin{array}{cccccc} 1 &{} \ \ -1 &{} \ \ &{} \ &{} \ &{} \ \\ &{} \ \ 1 &{} \ \ -1 &{} \ &{} \ &{} \ \\ &{} \ \ &{} \ \ 1 &{} \ \ddots &{} \ &{} \ \\ &{} \ \ &{} \ \ &{} \ \ddots &{} \ \ddots &{} \ \\ &{} \ \ &{} \ \ &{} \ &{} \ \ddots &{} \ -1\\ &{} \ \ &{} \ \ &{} \ &{} \ &{} \ 1\end{array} \right] ^{-1}&=(T_n(1-\mathrm{e}^{-I\theta }))^{-1}. \end{aligned}$$

The decomposition (10.25) can then be rewritten as

$$\begin{aligned} G_n=D_n(c)(T_n(1-\mathrm{e}^{I\theta }))^{-1}D_n(b)+D_n(b)(T_n(1-\mathrm{e}^{-I\theta }))^{-1}D_n(c)-D_n(b)D_n(c), \end{aligned}$$

and from $\mathbf{GLT\, 3}\!-\!\mathbf{GLT\, 5}$ we obtain

$$\begin{aligned} \{G_n\}_n\sim _\mathrm{GLT}\frac{b(x)c(x)}{1-\mathrm{e}^{I\theta }}+\frac{b(x)c(x)}{1-\mathrm{e}^{-I\theta }}-b(x)c(x)=0, \end{aligned}$$

which proves the theorem. $\square$

10.5 Finite Difference Discretization of Differential Equations

The main application of the theory of GLT sequences was already described in Chap. 1. It consists in the computation of the spectral distribution of the sequences of discretization matrices arising from the approximation of DEs by numerical methods. In fact, these sequences are often GLT sequences. This section and the next ones present the GLT analysis of several DEs discretized by various numerical methods. We begin by considering FD discretizations, then we will move to FE discretizations, and finally we will focus on IgA discretizations.

Before starting, we prove here an auxiliary result, which is interesting also in itself. If $n\in \mathbb N$ and $a:[0, 1]\rightarrow \mathbb C$, the nth arrow-shaped sampling  matrix generated by a is denoted by $S_n(a)$  and is defined as the following symmetric matrix of size n:

$$\begin{aligned} (S_n(a))_{i, j}=(D_n(a))_{\min (i, j),\min (i, j)}=a\Bigl (\frac{\min (i, j)}{n}\Bigr ), \qquad i, j=1,\ldots , n, \end{aligned}$$

(10.26)

that is,

$$\begin{aligned} S_n(a)=\left[ \begin{array}{cccccc} a(\frac{1}{n}) &{} a(\frac{1}{n}) &{} a(\frac{1}{n}) &{} \cdots &{} \ \cdots &{} a(\frac{1}{n})\\ a(\frac{1}{n}) &{} a(\frac{2}{n}) &{} a(\frac{2}{n}) &{} \cdots &{} \ \cdots &{} a(\frac{2}{n})\\ a(\frac{1}{n}) &{} a(\frac{2}{n}) &{} a(\frac{3}{n}) &{} \cdots &{} \ \cdots &{} a(\frac{3}{n})\\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} &{} \vdots \\ \vdots &{} \vdots &{} \vdots &{} &{} \ \ddots &{} \vdots \\ a(\frac{1}{n}) &{} a(\frac{2}{n}) &{} a(\frac{3}{n}) &{} \cdots &{} \ \cdots &{} a(1) \end{array}\right] . \end{aligned}$$

The name is due to the fact that, if we imagine to color the matrix $S_n(a)$ by assigning the color i to the entries $a(\frac{i}{n})$, the resulting picture looks like a sort of arrow pointing toward the upper left corner.

Theorem 10.4

Let $a:[0, 1]\rightarrow \mathbb C$ be continuous and let f be a trigonometric polynomial of degree r. Then,

$$\begin{aligned} \Vert S_n(a)\circ T_n(f)-D_n(a)T_n(f)\Vert \le (2r+1)\Vert f\Vert _\infty \,\omega _a\Bigl (\frac{r}{n}\Bigr ) \end{aligned}$$

(10.27)

for every $n\in \mathbb N$,

$$\begin{aligned} \Vert S_n(a)\circ T_n(f)\Vert \le C \end{aligned}$$

(10.28)

for every $n\in \mathbb N$ and for some constant C independent of n, and

$$\begin{aligned} \{S_n(a)\circ T_n(f)\}_n\sim _\mathrm{GLT}a\otimes f. \end{aligned}$$

(10.29)

Proof

For all $i, j=1,\ldots , n$,

• if $|i-j|>r$, then the Fourier coefficient $f_{i-j}$ is zero and, consequently,

  $$\begin{aligned} (S_n(a)\circ T_n(f))_{ij}&=(S_n(a))_{ij}(T_n(f))_{ij}=a\Bigl (\frac{\min (i, j)}{n}\Bigr )f_{i-j}=0,\\ (D_n(a)T_n(f))_{ij}&=(D_n(a))_{ii}(T_n(f))_{ij}=a\Bigl (\frac{i}{n}\Bigr )f_{i-j}=0; \end{aligned}$$

• if $|i-j|\le r$, then, using (2.34) and T 3, we obtain

  $$\begin{aligned}&|(S_n(a)\circ T_n(f))_{ij}-(D_n(a)T_n(f))_{ij}|\\&\, =|(S_n(a))_{ij}(T_n(f))_{ij}-(D_n(a))_{ii}(T_n(f))_{ij}|\\&\, =|(S_n(a))_{ij}-(D_n(a))_{ii}|\,|(T_n(f))_{ij}|\\&\, \le \biggl |a\Bigl (\frac{\min (i, j)}{n}\Bigr )-a\Bigl (\frac{i}{n}\Bigr )\biggr |\,\Vert T_n(f)\Vert \\&\, \le \Vert f\Vert _\infty \,\omega _a\Bigl (\Bigl |\frac{\min (i, j)}{n}-\frac{i}{n}\Bigr |\Bigr ). \end{aligned}$$

  Since $|i-j|\le r$,

  $$\Bigl |\frac{\min (i, j)}{n}-\frac{i}{n}\Bigr |\le \frac{|j-i|}{n}\le \frac{r}{n},$$

  hence

  $$\begin{aligned} \bigl |(S_n(a)\circ T_n(f))_{ij}-(D_n(a)T_n(f))_{ij}\bigr |\le \Vert f\Vert _\infty \,\omega _a\Bigl (\frac{r}{n}\Bigr ). \end{aligned}$$

It follows from the first item that the nonzero entries in each row and column of $S_n(a)\circ T_n(f)-D_n(a)T_n(f)$ are at most $2r+1$. Hence, from the second item we infer that the 1-norm and the $\infty$-norm of $S_n(a)\circ T_n(f)-D_n(a)T_n(f)$ are bounded by $(2r+1)\Vert f\Vert _\infty \,\omega _a(\frac{r}{n})$. The application of (2.31) yields (10.27). Using (10.27) we obtain

$$\begin{aligned} \Vert S_n(a)\circ T_n(f)\Vert&\le \Vert S_n(a)\circ T_n(f)-D_n(a)T_n(f)\Vert +\Vert D_n(a)\Vert \,\Vert T_n(f)\Vert \\&\le (2r+1)\Vert f\Vert _\infty \,\omega _a\Bigl (\frac{r}{n}\Bigr )+\Vert a\Vert _\infty \Vert f\Vert _\infty , \end{aligned}$$

which implies (10.28). Finally, since $\omega _a(\frac{r}{n})\rightarrow 0$ as $n\rightarrow \infty$, the matrix-sequence $\{S_n(a)\circ T_n(f)-D_n(a)T_n(f)\}_n$ is zero-distributed by (10.27) and Z 1 (or Z 2). Thus, (10.29) follows from GLT 3 and GLT 4. $\square$

Exercise 10.3

Let $a:[0, 1]\rightarrow \mathbb C$ be a function such that $\{D_n(a)\}_n\sim _\mathrm{GLT}a\otimes 1$ and let f be a trigonometric polynomial. Show that the GLT relation (10.29) holds. In particular, (10.29) holds whenever f is a trigonometric polynomial and a is continuous a.e., because in this case we have $\{D_n(a)\}_n\sim _\mathrm{GLT}a\otimes 1$ by Theorem 8.7.

10.5.1 FD Discretization of Diffusion Equations

Consider the following second-order differential problem:

$$\begin{aligned} \left\{ \begin{array}{ll} -(a(x)u'(x))'=f(x), &{}\quad x\in (0, 1), \\ u(0)=\alpha ,\quad u(1)=\beta , \end{array}\right. \end{aligned}$$

(10.30)

where $a\in C([0, 1])$ and f is a given function. To ensure the well-posedness of this problem, further conditions on a and f should be imposed; for example, $f\in L^2([0, 1])$ and $a\in C^1([0, 1])$ with $a(x)>0$ for every $x\in [0, 1]$, so that problem (10.30) is elliptic (see Chap. 8 of [31], especially the Sturm-Liouville problem on p. 223). However, we will only assume that $a\in C([0, 1])$ as the GLT analysis presented herein does not require any other assumption.

FD discretization. We consider the discretization of (10.30) by the classical second-order central FD scheme on a uniform grid. In the case where a(x) is constant, this is also known as the $(-1, 2,-1)$ scheme. Let us describe it shortly; for more details on FD methods, we refer the reader to the available literature (see, e.g., [117] or any good book on FDs). Choose a discretization parameter $n\in \mathbb N$, set $h=\frac{1}{n+1}$ and $x_j=jh$ for all $j\in [0, n+1]$. For $j=1,\ldots , n$ we approximate $-(a(x)u'(x))'|_{x=x_j}$ by the classical second-order central FD formula:

$$\begin{aligned}&-(a(x)u'(x))'|_{x=x_j}\approx -\frac{a(x_{j+\frac{1}{2}})u'(x_{j+\frac{1}{2}})-a(x_{j-\frac{1}{2}})u'(x_{j-\frac{1}{2}})}{h}\nonumber \\&\quad \approx -\frac{a(x_{j+\frac{1}{2}})\dfrac{u(x_{j+1})-u(x_j)}{h}-a(x_{j-\frac{1}{2}})\dfrac{u(x_j)-u({x_{j-1}})}{h}}{h}\nonumber \\&\quad =\frac{-a(x_{j+\frac{1}{2}})u(x_{j+1})+\bigl (a(x_{j+\frac{1}{2}})+a(x_{j-\frac{1}{2}})\bigr )u(x_j)-a(x_{j-\frac{1}{2}})u(x_{j-1})}{h^2}. \end{aligned}$$

(10.31)

This means that the nodal values of the solution u satisfy (approximately) the following linear system:

$$\begin{aligned} \begin{aligned}&-a(x_{j+\frac{1}{2}})u(x_{j+1})+\bigl (a(x_{j+\frac{1}{2}})+a(x_{j-\frac{1}{2}})\bigr )u(x_j)-a(x_{j-\frac{1}{2}})u(x_{j-1})=h^2f(x_j),\\&j=1,\ldots , n. \end{aligned} \end{aligned}$$

We then approximate the solution by the piecewise linear function that takes the value $u_j$ in $x_j$ for $j=0,\ldots , n+1$, where $u_0=\alpha$, $u_{n+1}=\beta$, and $\mathbf {u}=(u_1,\ldots , u_n)^T$ solves

$$\begin{aligned} \begin{aligned}&-a(x_{j+\frac{1}{2}})u_{j+1}+\bigl (a(x_{j+\frac{1}{2}})+a(x_{j-\frac{1}{2}})\bigr )u_j-a(x_{j-\frac{1}{2}})u_{j-1}=h^2f(x_j),\\&j=1,\ldots , n. \end{aligned} \end{aligned}$$

(10.32)

The matrix of the linear system (10.32) is the $n\times n$ tridiagonal symmetric matrix given by

$$\begin{aligned} A_n = \left[ \begin{array}{ccccc} a_{\frac{1}{2}}+a_{\frac{3}{2}} &{} \ \ -a_{\frac{3}{2}} &{} \ \ \ \ &{} \ \ \ \ &{} \ \ \ \ \\ -a_{\frac{3}{2}} &{} \ \ a_{\frac{3}{2}}+a_{\frac{5}{2}} &{} \ \ \ \ -a_{\frac{5}{2}} &{} \ \ \ \ &{} \ \ \ \ \\ &{} \ \ -a_{\frac{5}{2}} &{} \ \ \ \ \ddots &{} \ \ \ \ \ddots &{} \ \ \ \ \\ &{} \ \ &{} \ \ \ \ \ddots &{} \ \ \ \ \ddots &{} \ \ \ \ -a_{n-\frac{1}{2}} \\ &{} \ \ &{} \ \ \ \ &{} \ \ \ \ -a_{n-\frac{1}{2}} &{} \ \ \ \ a_{n-\frac{1}{2}}+a_{n+\frac{1}{2}} \end{array}\right] , \end{aligned}$$

(10.33)

where $a_i=a(x_i)$ for all $i\in [0, n+1]$. This is the matrix we already encountered in Sect. 7.1, when explaining the notion of LT sequences.

GLT analysis of the FD discretization matrices.  We are going to see that the theory of GLT sequences allows one to compute the singular value and spectral distribution of the sequence of FD discretization matrices $\{A_n\}_n$. Actually, this is the fundamental example that led to the birth of the theory of LT sequences and, subsequently, of GLT sequences. Given the importance, we will compute the singular value and spectral distribution of $\{A_n\}_n$ by two different methods, both of them instructive.

Theorem 10.5

If $a\in C([0, 1])$ then

$$\begin{aligned} \{A_n\}_n\sim _\mathrm{GLT}a(x)(2-2\cos \theta ) \end{aligned}$$

(10.34)

and

$$\begin{aligned} \{A_n\}_n\sim _{\sigma ,\,\lambda }a(x)(2-2\cos \theta ). \end{aligned}$$

(10.35)

Proof

It suffices to prove (10.34) because (10.35) follows from (10.34) and GLT 1 as the matrices $A_n$ are symmetric. We will see two proofs of (10.34), both of them instructive. In order to fully understand the first proof, the reader should know something from Chap. 7 as the summary of Chap. 9 is not enough in this case. On the contrary, the second proof can be understood on the basis of the summary only.

First proof. Suppose first that a(x) is constant, say $a(x)=1$ identically. In this case,

$$ A_n=\left[ \begin{array}{ccccc} 2 &{} -1 &{} &{} &{} \\ -1 &{} 2 &{} -1 &{} &{} \\ &{} -1 &{} \ddots &{} \ddots &{} \\ &{} &{} \ddots &{} \ddots &{} -1\\ &{} &{} &{} -1 &{} 2 \end{array}\right] =T_n(2-2\cos \theta ), $$

i.e., $A_n$ is simply the nth Toeplitz matrix generated by the function $2-2\cos \theta$. Hence, (10.34) follows from GLT 3. Now let us turn to the case where a(x) is not constant. In this case, the expression of $A_n$ is given by (10.33) and the Toeplitzness seems to be completely lost. In reality, we find it again ‘in an approximated sense’ and ‘at a local scale’. Indeed, we note that a(x) varies smoothly from a(0) to a(1), because it is uniformly continuous. Therefore, assuming that n is large with respect to k, any $k\times k$  leading principal submatrix of $A_n$ shows an approximate Toeplitz structure. Let us be more quantitative. Fix a large $m\in \mathbb N$ and assume $n>m$. Then, n is large with respect to $\lfloor n/m\rfloor$, and so, according to the previous reasoning, any $\lfloor n/m\rfloor \times \lfloor n/m\rfloor$ leading principal submatrix of $A_n$ shows an approximate Toeplitz structure. In fact, the evaluations of a(x) appearing in the first $\lfloor n/m\rfloor \times \lfloor n/m\rfloor$ leading principal submatrix are approximately equal to $a(\frac{1}{m})$; the evaluations of a(x) appearing in the second $\lfloor n/m\rfloor \times \lfloor n/m\rfloor$ leading principal submatrix are approximately equal to $a(\frac{2}{m})$; and so on until the evaluations of a(x) appearing in the mth $\lfloor n/m\rfloor \times \lfloor n/m\rfloor$ leading principal submatrix, which are approximately equal to a(1). If, for all $i=1,\ldots , m$, we replace by $a(\frac{i}{m})$ the evaluations of a(x) in the ith $\lfloor n/m\rfloor \times \lfloor n/m\rfloor$ leading principal submatrix, this submatrix becomes $a(\frac{i}{m})T_{\lfloor n/m\rfloor }(2-2\cos \theta )$. In conclusion, the matrix $A_n$ is approximated by the locally Toeplitz operator

$$ LT_n^m(a(x), 2-2\cos \theta )=\mathop {\mathrm{diag}}_{i=1,\ldots , m}\biggl [a\Bigl (\frac{i}{m}\Bigr )T_{\left\lfloor n/m\right\rfloor }(2-2\cos \theta )\biggr ]\,\oplus \, O_{n\,\mathrm{mod}\, m}. $$

In fact, we have $\{LT_n^m(a(x), 2-2\cos \theta )\}_n\mathop {\longrightarrow }\limits ^{\mathrm{a.c.s.}}\{A_n\}_n$, because, using the suggestions in Remark 7.1, it can be shown that

$$\begin{aligned}\begin{gathered} A_n=LT_n^m(a(x), 2-2\cos \theta )+R_{n, m}+N_{n, m}\\ \mathrm{rank}(R_{n, m})\le 3m,\qquad \Vert N_{n, m}\Vert \le \omega _a\Bigl (\frac{1}{m}+\frac{m+1}{n+1}\Bigr ). \end{gathered}\end{aligned}$$

Thus, $\{A_n\}_n\sim _\mathrm{LT}a(x)(2-2\cos \theta )$ by definition and (10.34) is proved.

Second proof. As already pointed out, the example we are dealing with led to the birth of the theory of GLT sequences. In particular, the procedure followed in the first proof to obtain (10.34) (and hence (10.35)) motivated the definition of locally Toeplitz sequences, as well as the introduction of the locally Toeplitz operator in Sect. 7.1. However, now that we have developed the theory of GLT sequences, we should say that the procedure followed in the first proof is not the most effective way to obtain (10.34) and (10.35) as the method we are going to see now is definitely more powerful. Consider the matrix

$$\begin{aligned} D_n(a)T_n(2-2\cos \theta )=\begin{bmatrix} 2a(\frac{1}{n})&-a(\frac{1}{n})&\,&\,&\, \\ -a(\frac{2}{n})&2a(\frac{2}{n})&-a(\frac{2}{n})&\,&\, \\ \,&-a(\frac{3}{n})&\ddots&\ddots&\, \\ \,&\,&\ddots&\ddots&-a(\frac{n-1}{n}) \\ \,&\,&\,&-a(1)&2a(1) \end{bmatrix}. \end{aligned}$$

(10.36)

In view of the inequalities $\bigl |x_j-\frac{j}{n}\bigr |\le \frac{1}{n+1}=h,\ j=1,\ldots , n$, a direct comparison between (10.36) and (10.33) shows that the modulus of each diagonal entry of the matrix $A_n-D_n(a)T_n(2-2\cos \theta )$ is bounded by $2\,\omega _a(3h/2)$, and the modulus of each off-diagonal entry of $A_n-D_n(a)T_n(2-2\cos \theta )$ is bounded by $\omega _a(3h/2)$. Therefore, the 1-norm and the $\infty$-norm of $A_n-D_n(a)T_n(2-2\cos \theta )$ are bounded by $4\,\omega _a(3h/2)$, and so, by (2.31),

$$\begin{aligned} \Vert A_n-D_n(a)T_n(2-2\cos \theta )\Vert \le 4\,\omega _a(3h/2)\rightarrow 0 \ \text { as }\, n\rightarrow \infty . \end{aligned}$$

Setting $Z_n=A_n-D_n(a)T_n(2-2\cos \theta )$, we have $\{Z_n\}_n\sim _\sigma 0$ by Z 1 (or Z 2). Since

$$\begin{aligned} A_n=D_n(a)T_n(2-2\cos \theta )+Z_n, \end{aligned}$$

GLT 3 and GLT 4 yield (10.34). $\square$

Remark 10.1

(formal structure of the symbol) From a formal viewpoint (i.e., disregarding the regularity of a(x) and u(x)), problem (10.30) can be rewritten in the form

$$\begin{aligned} \left\{ \begin{array}{ll} -a(x)u''(x)-a'(x)u'(x)=f(x), &{}\quad x\in (0, 1),\\ u(0)=\alpha ,\quad u(1)=\beta . \end{array}\right. \end{aligned}$$

From this reformulation, it appears more clearly that the symbol $a(x)(2-2\cos \theta )$ consists of the following two ‘ingredients’.

• The coefficient of the higher-order differential operator, namely a(x), in the physical variable x. To make a parallelism with Hörmander’s theory [73], the higher-order  differential operator $-a(x)u''(x)$ is the so-called principal  symbol of the complete differential operator $-a(x)u''(x)-a'(x)u'(x)$ and a(x) is then the coefficient of the principal symbol.

• The trigonometric polynomial associated with the FD formula $(-1, 2,-1)$ used to approximate the higher-order derivative $-u''(x)$, namely $2-2\cos \theta =-\mathrm{e}^{I\theta }+2-\mathrm{e}^{-I\theta }$, in the Fourier variable $\theta$. To see that $(-1, 2,-1)$ is precisely the FD formula used to approximate $-u''(x)$, simply imagine $a(x)=1$ and note that in this case the FD scheme (10.31) becomes

  $$\begin{aligned} -u''(x_j)\approx \frac{-u(x_{j+1})+2u(x_j)-u(x_{j-1})}{h^2}, \end{aligned}$$

  i.e., the FD formula $(-1, 2,-1)$ to approximate $-u''(x_j)$.

We observe that the term $-a'(x)u'(x)$, which only depends on lower-order derivatives of u(x), does not enter the expression of the symbol.

Remark 10.2

(nonnegativity and order of the zero at $\theta =0$) The trigonometric  polynomial $2-2\cos \theta$ is nonnegative on $[-\pi ,\pi ]$ and it has a unique zero of order 2 at $\theta =0$, because

$$\begin{aligned} \lim _{\theta \rightarrow 0}\frac{2-2\cos \theta }{\theta ^2}=1. \end{aligned}$$

This reflects the fact that the associated FD formula $(-1, 2,-1)$ approximates $-u''(x)$, which is a differential operator of order 2 (it is also nonnegative on the space of functions $v\in C^2([0, 1])$ such that $v(0)=v(1)=0$, in the sense that $\int _0^1-v''(x)v(x)\mathrm{d}x=\int _0^1(v'(x))^2\mathrm{d}x\ge 0$ for all such v).

Example 10.1

Since the symbol $a(x)(2-2\cos \theta )$ is symmetric with respect to the Fourier variable $\theta$, it is clear from the definition of singular value and spectral distribution that the relations (10.35) continue to hold even if we consider $[0, 1]\times [0,\pi ]$ as the domain of $a(x)(2-2\cos \theta )$ instead of $[0, 1]\times [-\pi ,\pi ]$. Indeed, the function $a(x)(2-2\cos \theta ):[0, 1]\times [0,\pi ]\rightarrow \mathbb R$ is a rearranged version of $a(x)(2-2\cos \theta ):[0, 1]\times [-\pi ,\pi ]\rightarrow \mathbb R$ (see Sect. 3.2). According to the informal meaning of the spectral distribution (see Remark 3.2), if $n=\ell ^2$ is large enough, then, assuming we have no outliers, the eigenvalues

$$\begin{aligned} \lambda _k(A_n),\qquad k=1,\ldots , n, \end{aligned}$$

(10.37)

are approximately equal to the uniform samples

$$\begin{aligned} a\Bigl (\frac{i}{\ell }\Bigr )\Bigl (2-2\cos \frac{j\pi }{\ell }\Bigr ),\qquad i, j=1,\ldots ,\ell . \end{aligned}$$

(10.38)

This is confirmed by Fig. 10.1 and Table 10.1 for the case $a(x)=2+\cos (3x)$. In Fig. 10.1 we plotted the eigenvalues (10.37) and the samples (10.38), both of them arranged in non-decreasing order. We observe an excellent agreement between the eigenvalues and the samples (which also indicates the absence of outliers in this case). In Table 10.1 we computed, for increasing values of $n=\ell ^2$, the $\infty$-norm of the difference $\mathbf {s}_n-\mathbf {e}_n$, where

• $\mathbf {e}_n$ is the vector of the eigenvalues (10.37) sorted in non-decreasing order,

• $\mathbf {s}_n$ is the vector of the samples (10.38) sorted in non-decreasing order.

As shown in Table 10.1, the norm $\Vert \mathbf {s}_n-\mathbf {e}_n\Vert _\infty$ apparently converges to 0 as $n\rightarrow \infty$, though the convergence is very slow.

Fig. 10.1

Comparison between the spectrum of $A_n$ and the symbol $a(x)(2-2\cos \theta )$ for $n=400$ and $a(x)=2+\cos (3x)$

Table 10.1 Computation of $\Vert \mathbf {s}_n-\mathbf {e}_n\Vert _\infty$ for $a(x)=2+\cos (3x)$ and for increasing values of n

Example 10.2

Another way to check the agreement between the eigenvalues of $A_n$ and the symbol $a(x)(2-2\cos \theta )$ consists in comparing the eigenvalues of $A_n$ with a good rearranged version of the symbol instead of the symbol itself. The rearranged version we used in Fig. 10.1 is not so good, because it depends on two variables $x,\theta$; in fact, it is essentially the same as the symbol since it was obtained by a mere restriction to $[0, 1]\times [0,\pi ]$. A good rearranged version, which has the very nice properties of being univariate and non-decreasing, can be obtained from the construction of Sect. 3.2, which we repeat here for the reader’s convenience. Instead of applying the construction to the symbol, we apply it to the rearranged version $a(x)(2-2\cos \theta ):[0, 1]\times [0,\pi ]\rightarrow \mathbb R$ (clearly, a rearranged version of a rearranged version of a function is another rearranged version of that function). For each $r\in \mathbb N$, compute the uniform samples

$$\begin{aligned} a\Bigl (\frac{i}{r}\Bigr )\Bigl (2-2\cos \frac{j\pi }{r}\Bigr ),\qquad i, j=1,\ldots , r, \end{aligned}$$

sort them in non-decreasing order and put them in a vector $(s_1, s_2,\ldots , s_{r^2})$. Let $g_r:[0, 1]\rightarrow \mathbb R$ be the piecewise linear non-decreasing function that interpolates the samples $(s_0=s_1, s_1, s_2,\ldots , s_{r^2})$ over the nodes $(0,\frac{1}{r^2},\frac{2}{r^2},\ldots , 1)$. When increasing r, one can graphically verify that the function $g_r$ converges pointwise to a limit function $g:[0, 1]\rightarrow \mathbb R$. This is the desired rearranged version. In Fig. 10.2 we fixed $a(x)=x\mathrm{e}^{-x}$ and we plotted the graph of g and the eigenvalues of $A_n$ for $n=50$. The graph of g has been obtained by plotting the graph of $g_r$ corresponding to a large value of r ($r=1000$). The eigenvalues of $A_n$ have been sorted in non-decreasing order and placed at the points $(\frac{k}{n},\lambda _k(A_n)),\ k=1,\ldots , n$. Figure 10.2 is then a comparison between the eigenvalues

$$\begin{aligned} \lambda _k(A_n),\qquad k=1,\ldots , n, \end{aligned}$$

(10.39)

and the uniform samples

$$\begin{aligned} g\Bigl (\frac{k}{n}\Bigr ),\qquad k=1,\ldots , n. \end{aligned}$$

(10.40)

We clearly see an excellent agreement, which apparently becomes perfect in the limit of mesh refinement $n\rightarrow \infty$, as shown in Table 10.2. Indeed, it is clear from the table that $\Vert \mathbf {s}_n-\mathbf {e}_n\Vert _\infty \rightarrow 0$ as $n\rightarrow \infty$, where

• $\mathbf {e}_n$ is the vector of the eigenvalues (10.39) sorted in non-decreasing order,

• $\mathbf {s}_n$ is the vector of the samples (10.40) sorted in non-decreasing order.

We point out that in Table 10.2 we approximated g by $g_r$ with $r=5000$ instead of $r=1000$, so as to obtain more accurate values for the samples (10.40).

Fig. 10.2

Comparison between the spectrum of $A_n$ and the rearranged version g of the symbol $a(x)(2-2\cos \theta )$ for $n=50$ and $a(x)=x\mathrm{e}^{-x}$

Table 10.2 Computation of $\Vert \mathbf {s}_n-\mathbf {e}_n\Vert _\infty$ for $a(x)=x\mathrm{e}^{-x}$ and for increasing values of n

10.5.2 FD Discretization of Convection-Diffusion-Reaction Equations

1st Part

Suppose we add to the diffusion equation (10.30) a convection and a reaction term. In this way, we obtain the following convection-diffusion-reaction equation in divergence form with Dirichlet boundary conditions:

$$\begin{aligned} \left\{ \begin{array}{ll} -(a(x)u'(x))'+b(x)u'(x)+c(x)u(x)=f(x), &{}\quad x\in (0, 1),\\ u(0)=\alpha ,\quad u(1)=\beta , \end{array}\right. \end{aligned}$$

(10.41)

where $a:[0, 1]\rightarrow \mathbb R$ is continuous as before and we assume that $b, c:[0, 1]\rightarrow \mathbb R$ are bounded. Based on Remark 10.1, we expect that the term $b(x)u'(x)+c(x)u(x)$, which only involves lower-order derivatives of u(x), does not enter the expression of the symbol. In other words, if we discretize the higher-order term $-(a(x)u'(x))'$ as in (10.31), the symbol of the resulting FD discretization matrices $B_n$ should be again $a(x)(2-2\cos \theta )$. We are going to show that this is in fact the case.

FD discretization. Let $n\in \mathbb N$, set $h=\frac{1}{n+1}$ and $x_j=jh$ for all $j\in [0, n+1]$. Consider the discretization of (10.41) by the FD scheme defined as follows.

• To approximate the higher-order (diffusion)  term $-(a(x)u'(x))'$, use again the FD formula (10.31), i.e.,

  $$\begin{aligned}&-(a(x)u'(x))'|_{x=x_j}\nonumber \\&\quad \approx \frac{-a(x_{j+\frac{1}{2}})u(x_{j+1})+\bigl (a(x_{j+\frac{1}{2}})+a(x_{j-\frac{1}{2}})\bigr )u(x_j)-a(x_{j-\frac{1}{2}})u(x_{j-1})}{h^2}. \end{aligned}$$

  (10.42)

• To approximate the convection term $b(x)u'(x)$, use any (consistent) FD formula; to fix the ideas, here we use the second-order central formula

  $$\begin{aligned} b(x)u'(x)|_{x=x_j}\approx b(x_j)\frac{u(x_{j+1})-u(x_{j-1})}{2h}. \end{aligned}$$

  (10.43)

• To approximate the reaction term c(x)u(x), use the obvious equation

  $$\begin{aligned} c(x)u(x)|_{x=x_j}=c(x_j)u(x_j). \end{aligned}$$

  (10.44)

The resulting FD discretization matrix $B_n$ admits a natural decomposition as

$$\begin{aligned} B_n=A_n+Z_n, \end{aligned}$$

(10.45)

where $A_n$ is the matrix coming from the discretization of the higher-order (diffusion) term $-(a(x)u'(x))$, while $Z_n$ is the matrix coming from the discretization of the lower-order (convection and reaction) terms $b(x)u'(x)$ and c(x)u(x). Note that $A_n$ is given by (10.33) and $Z_n$ is given by

$$\begin{aligned} Z_n&=\frac{h}{2}\begin{bmatrix} 0&b_1&\,&\,&\, \\ -b_2&0&b_2&\,&\, \\ \,&\ddots&\ddots&\ddots&\, \\ \,&\,&-b_{n-1}&0&b_{n-1}\\ \,&\,&\,&-b_n&0 \end{bmatrix}+h^2\begin{bmatrix} c_1&\,&\,&\,&\, \\ \,&c_2&\,&\,&\, \\ \,&\,&\ddots&\,&\, \\ \,&\,&\,&c_{n-1}&\, \\ \,&\,&\,&\,&c_n \end{bmatrix}, \end{aligned}$$

(10.46)

where $b_i=b(x_i)$ and $c_i=c(x_i)$ for all $i=1,\ldots , n$.

GLT analysis of the FD discretization matrices.  We now prove that Theorem 10.5 holds unchanged with $B_n$ in place of $A_n$. This highlights a general aspect: lower-order terms such as $b(x)u'(x)+c(x)u(x)$ do not enter the expression of the symbol and do not affect in any way the asymptotic singular value and spectral distribution of DE discretization matrices.

Theorem 10.6

If $a\in C([0, 1])$ and $b, c:[0, 1]\rightarrow \mathbb R$ are bounded then

$$\begin{aligned} \{B_n\}_n\sim _\mathrm{GLT}a(x)(2-2\cos \theta ) \end{aligned}$$

(10.47)

and

$$\begin{aligned} \{B_n\}_n\sim _{\sigma ,\,\lambda }a(x)(2-2\cos \theta ). \end{aligned}$$

(10.48)

Proof

By (2.31), the matrix $Z_n$ in (10.46) satisfies

$$\begin{aligned} \Vert Z_n\Vert \le h\Vert b\Vert _\infty +h^2\Vert c\Vert _\infty \le C/n \end{aligned}$$

(10.49)

for some constant C independent of n. As a consequence, $\{Z_n\}_n$ is zero-distributed by Z 1 (or Z 2), hence $\{Z_n\}_n\sim _\mathrm{GLT}0$ by GLT 3. Since $\{A_n\}_n\sim _\mathrm{GLT}a(x)(2-2\cos \theta )$ by Theorem 10.5, the decomposition (10.45) and GLT 4 imply (10.47).

Now, if the convection term is not present, i.e. $b(x)=0$ identically, then $B_n$ is symmetric and (10.48) follows from (10.47) and GLT 1. If b(x) is not identically 0, then $B_n$ is not symmetric in general and so (10.47) and GLT 1 only imply the singular value distribution $\{B_n\}_n\sim _\sigma a(x)(2-2\cos \theta )$. Nevertheless, in view of the decomposition (10.45), since $A_n$ is symmetric and $\{A_n\}_n\sim _\lambda a(x)(2-2\cos \theta )$ by Theorem 10.5, since $\Vert Z_n\Vert _1=O(1)$ by the inequalities (10.49) and (2.42), and since $\Vert A_n\Vert \le 4\Vert a\Vert _\infty$ by (2.31), the spectral distribution $\{B_n\}_n\sim _\lambda a(x)(2-2\cos \theta )$ holds (by GLT 2) even if b(x) is an arbitrary bounded function. $\square$

Fig. 10.3

Comparison between the spectrum of $B_n$ and the rearranged version g of the symbol $a(x)(2-2\cos \theta )$ for $n=50$ and $a(x)=x\mathrm{e}^{-x},\ b(x)=0,\ c(x)=100$

Table 10.3 Computation of $\Vert \mathbf {s}_n-\mathbf {e}_n\Vert _\infty$ for $a(x)=x\mathrm{e}^{-x}$, $b(x)=0$, $c(x)=c=1, 10, 100$, and for increasing values of n

Example 10.3

Let $a(x)=x\mathrm{e}^{-x},\ b(x)=0,\ c(x)=100$. Figure 10.3 is the same as Fig. 10.2 with the eigenvalues of $B_n$ in place of the eigenvalues of $A_n$. We see from Fig. 10.3 that the approximation of the spectrum provided by  g is not as good as in Fig. 10.2. This is due to the presence of a  reaction term c which is quite large with respect to $1/h^2$, being $h^2$ the coefficient appearing in front of the reaction matrix (see (10.46)). Actually, c is almost of the same order as $1/h^2=(n+1)^2=2601$. The reduced accuracy in the symbol-to-spectrum approximation could be expected, because the symbol does not depend on c and therefore it is unlikely that it yields a good approximation of the spectrum for any c and n. However, as soon as $h^2c$ becomes small, the approximation becomes accurate. This is shown in Table 10.3, in which we fixed $a(x)=x\mathrm{e}^{-x}$ and $b(x)=0$, and we computed, for $c(x)=c=1, 10, 100$ and for increasing values of n, the $\infty$-norm of the difference $\mathbf {s}_n-\mathbf {e}_n$, where

• $\mathbf {e}_n$ is the vector of the eigenvalues of $B_n$ sorted in non-decreasing order,

• $\mathbf {s}_n$ is the vector of the samples $g(\frac{k}{n})$, $k=1,\ldots , n$, sorted in non-decreasing order.

Table 10.3 shows that the norm $\Vert \mathbf {s}_n-\mathbf {e}_n\Vert _\infty$ converges to 0 as $n\rightarrow \infty$ with the same asymptotic speed as in the absence of the reaction term; cf. Tables 10.2 and 10.3. The presence of a large reaction term such as $c=100$ affects the norm $\Vert \mathbf {s}_n-\mathbf {e}_n\Vert _\infty$ only when n is small (more precisely, when the quantity $h^2c$ is not negligible), whereas for large n the value of $\Vert \mathbf {s}_n-\mathbf {e}_n\Vert _\infty$ provided by Table 10.3 is essentially the same as the corresponding value provided by Table 10.2.

Remark 10.3

In Example 10.3 we addressed the case of a nonzero reaction term c(x). Similar considerations also hold in the presence of a nonzero convection term b(x). In particular, a bad symbol-to-spectrum approximation may be expected if b is large with respect to h, being h the coefficient appearing in front of the convection matrix in (10.46). However, as soon as the quantity hb becomes negligible, the approximation of $\varLambda (B_n)$ provided by the symbol becomes accurate. In other words, the influence of the convection and reaction terms disappears when n is large enough, because in the limit where $n\rightarrow \infty$ we have $\Vert Z_n\Vert \approx 0,\ B_n\approx A_n$ by (10.45), and, consequently, $\varLambda (B_n)\approx \varLambda (A_n)$. In this respect, we recall that the notion of spectral distribution is asymptotic: for small n, the spectral symbol might be far from approximating the spectrum.

2nd Part

So far, we only considered differential equations with Dirichlet boundary conditions. A natural question is the following: if we change the boundary conditions in (10.41), does the expression of the symbol change? The answer is ‘no’: boundary conditions do not affect the singular value and eigenvalue distribution because they only produce a small-rank perturbation in the resulting discretization matrices. To understand better this point, we consider problem (10.41) with  Neumann boundary conditions:

$$\begin{aligned} \left\{ \begin{array}{ll} -(a(x)u'(x))'+b(x)u'(x)+c(x)u(x)=f(x), &{}\quad x\in (0, 1),\\ u'(0)=\alpha ,\quad u'(1)=\beta . \end{array}\right. \end{aligned}$$

(10.50)

FD discretization. We discretize (10.50) by the same FD scheme considered in the 1st part, which is defined by the FD formulas (10.42)–(10.44). In this way, we arrive at the linear system

$$\begin{aligned}&-a(x_{j+\frac{1}{2}})u_{j+1}+\bigl (a(x_{j+\frac{1}{2}})+a(x_{j-\frac{1}{2}})\bigr )u_j-a(x_{j-\frac{1}{2}})u_{j-1}\nonumber \\&\quad +\frac{h}{2}\bigl (b(x_j)u_{j+1}-b(x_j)u_{j-1}\bigr )+h^2c(x_j)u_j=h^2f(x_j),\qquad j=1,\ldots , n, \end{aligned}$$

(10.51)

which is formed by n equations in the $n+2$ unknowns $u_0, u_1,\ldots , u_n, u_{n+1}$. Note that $u_0$ and $u_{n+1}$ should now be considered as unknowns, because they are not specified by the Dirichlet boundary conditions. However, as it is common in the FD context, $u_0$ and $u_{n+1}$ are expressed in terms of $u_1,\ldots , u_n$ by exploiting the Neumann boundary conditions. The simplest choice is to express $u_0$ and $u_{n+1}$ as a function of $u_1$ and $u_n$, respectively, by imposing the conditions

$$\begin{aligned} \frac{u_1-u_0}{h}=\alpha ,\qquad \frac{u_{n+1}-u_n}{h}=\beta , \end{aligned}$$

(10.52)

which yield $u_0=u_1-\alpha h$ and $u_{n+1}=u_n+\beta h$. Substituting into (10.51), we obtain a linear system with n equations and n unknowns $u_1,\ldots , u_n$. Setting ${a_i=a(x_i)}$, ${b_i=b(x_i)}$, ${c_i=c(x_i)}$ for all $i\in [0, n+1]$, the matrix of this system is

$$\begin{aligned} C_n=B_n+R_n=A_n+Z_n+R_n, \end{aligned}$$

(10.53)

where $A_n$, $B_n$, $Z_n$ are given by (10.33), (10.45), (10.46), and

$$R_n=\begin{bmatrix} -a_{\frac{1}{2}}-\dfrac{h}{2}b_1&&\\&&\\&&\\&&-a_{n+\frac{1}{}2}+\dfrac{h}{2}b_n \end{bmatrix}$$

is a small-rank correction coming from the discretization (10.52) of the boundary conditions.

GLT analysis of the FD discretization matrices.  We prove that Theorems 10.5 and 10.6 hold unchanged with $C_n$ in place of $A_n$ and $B_n$, respectively.

Theorem 10.7

If $a\in C([0, 1])$ and $b, c:[0, 1]\rightarrow \mathbb R$ are bounded then

$$\begin{aligned} \{C_n\}_n\sim _\mathrm{GLT}a(x)(2-2\cos \theta ) \end{aligned}$$

(10.54)

and

$$\begin{aligned} \{C_n\}_n\sim _{\sigma ,\,\lambda }a(x)(2-2\cos \theta ). \end{aligned}$$

(10.55)

Proof

Throughout this proof, the letter C will denote a generic constant independent of n. It is clear that $\Vert R_n\Vert \le \Vert a\Vert _\infty +(h/2)\Vert b\Vert _\infty \le C$. Moreover, since $\Vert R_n\Vert _1\le \mathrm{rank}(R_n)\Vert R_n\Vert \le C$, the matrix-sequence $\{R_n\}_n$ is zero-distributed by Z 2. Note that $\{Z_n\}_n$ is zero-distributed as well because $\Vert Z_n\Vert \le C/n$ by (10.49). In view of the decomposition (10.53), Theorem 10.5 and $\mathbf{GLT\, 3}\!-\!\mathbf{GLT\, 4}$ imply (10.54).

If the matrices $C_n$ are symmetric (this happens if $b(x)=0$), from (10.54) and GLT 1 we immediately obtain (10.55). If the matrices $C_n$ are not symmetric, from (10.54) and GLT 1 we only obtain the singular value distribution in (10.55). However, in view of (10.53), since $\Vert R_n+Z_n\Vert _1=o(n)$ and $\Vert R_n+Z_n\Vert ,\,\Vert A_n\Vert \le C$, the spectral distribution in (10.55) holds (by GLT 2) even if the matrices $C_n$ are not symmetric. $\square$

3rd Part

Consider the following convection-diffusion-reaction problem:

$$\begin{aligned} \left\{ \begin{array}{ll} -a(x)u''(x)+b(x)u'(x)+c(x)u(x)=f(x), &{}\quad x\in (0, 1),\\ u(0)=\alpha ,\quad u(1)=\beta , \end{array}\right. \end{aligned}$$

(10.56)

where $a:[0, 1]\rightarrow \mathbb R$ is continuous and $b, c:[0, 1]\rightarrow \mathbb R$ are bounded. The difference with respect to problem (10.41) is that the higher-order differential operator now appears in non-divergence form, i.e., we have $-a(x)u''(x)$ instead of $-(a(x)u'(x))'$. Neverthelss, based on Remark 10.1, if we use again the FD formula $(-1, 2,-1)$ to discretize the second derivative $-u''(x)$, the symbol of the resulting FD discretization matrices should be again $a(x)(2-2\cos \theta )$. We are going to show that this is in fact the case.

FD discretization. Let $n\in \mathbb N$, set $h=\frac{1}{n+1}$ and $x_j=jh$ for all $j=0,\ldots , n+1$. We discretize again (10.56) by the central second-order FD scheme, which in this case is defined by the following formulas:

$$\begin{aligned} -a(x)u''(x)|_{x=x_j}&\approx a(x_j)\frac{-u(x_{j+1})+2u(x_j)-u(x_{j-1})}{h^2},\qquad j=1,\ldots , n,\\ b(x)u'(x)|_{x=x_j}&\approx b(x_j)\frac{u(x_{j+1})-u(x_{j-1})}{2h},\qquad j=1,\ldots , n,\\ c(x)u(x)|_{x=x_j}&=c(x_j)u(x_j),\qquad j=1,\ldots , n. \end{aligned}$$

Then, we approximate the solution of (10.56) by the piecewise linear function that takes the value $u_j$ in $x_j$ for $j=0,\ldots , n+1$, where $u_0=\alpha$, $u_{n+1}=\beta$, and $\mathbf {u}=(u_1,\ldots , u_n)^T$ solves the linear system

$$\begin{aligned}&a(x_j)(-u_{j+1}+2u_j-u_{j-1})+\frac{h}{2}b(x_j)(u_{j+1}-u_{j-1})+h^2c(x_j)u_j=h^2f(x_j),\\&j=1,\ldots , n. \end{aligned}$$

The matrix $E_n$ of this linear system can be decomposed according to the diffusion, convection and reaction term, as follows:

$$\begin{aligned} E_n=K_n+Z_n, \end{aligned}$$

(10.57)

where $Z_n$ is the sum of the convection and reaction matrix and is given by (10.46), while

$$\begin{aligned} K_n&=\begin{bmatrix} 2a_1&\, -a_1&\,&\,&\, \\ -a_2&\, 2a_2&\, -a_2&\,&\, \\&\, \ddots&\, \ddots&\, \ddots&\, \\&\,&\, -a_{n-1}&\, 2a_{n-1}&\, -a_{n-1}\\&\,&\,&\, -a_n&\, 2a_n \end{bmatrix} \end{aligned}$$

(10.58)

is the diffusion matrix ($a_i=a(x_i)$ for all $i=1,\ldots , n$).

GLT analysis of the FD discretization matrices.  Despite the nonsymmetry of the diffusion matrix, which is due to the non-divergence form of the higher-order (diffusion) operator $-a(x)u''(x)$, we will prove that Theorems 10.5–10.7 hold unchanged with $E_n$ in place of $A_n$, $B_n$, $C_n$.

Theorem 10.8

If $a\in C([0, 1])$ and $b, c:[0, 1]\rightarrow \mathbb R$ are bounded then

$$\begin{aligned} \{E_n\}_n\sim _\mathrm{GLT}a(x)(2-2\cos \theta ) \end{aligned}$$

(10.59)

and

$$\begin{aligned} \{E_n\}_n\sim _{\sigma ,\,\lambda }a(x)(2-2\cos \theta ). \end{aligned}$$

(10.60)

Proof

Throughout this proof, the letter C will denote a generic constant independent of n. By (10.49),

$$\begin{aligned} \Vert Z_n\Vert \le C/n, \end{aligned}$$

hence $\{Z_n\}_n$ is zero-distributed. We prove that

$$\begin{aligned} \{K_n\}_n\sim _\mathrm{GLT}a(x)(2-2\cos \theta ), \end{aligned}$$

(10.61)

after which (10.59) will follow from $\mathbf{GLT\, 3}\!-\!\mathbf{GLT\, 4}$ and the decomposition (10.57). It is clear from (10.58) that

$$\begin{aligned} K_n=\mathop {\mathrm{diag}}_{i=1,\ldots , n}(a_i)\,\, T_n(2-2\cos \theta ). \end{aligned}$$

By T 3 we obtain

$$\begin{aligned} \Vert K_n-D_n(a)T_n(2-2\cos \theta )\Vert&\le \Bigl \Vert \,\mathop {\mathrm{diag}}_{i=1,\ldots , n}(a_i)-D_n(a)\Bigr \Vert \,\Vert T_n(2-2\cos \theta )\Vert \\&\le \omega _a(h)\,\Vert 2-2\cos \theta \Vert _\infty =4\,\omega _a(h), \end{aligned}$$

which tends to 0 as $n\rightarrow \infty$. We conclude that $\{K_n-D_n(a)T_n(2-2\cos \theta )\}_n$ is zero-distributed, and so (10.61) follows from $\mathbf{GLT\, 3}\!-\!\mathbf{GLT\, 4}$.

From (10.59) and GLT 1 we obtain the singular value distribution in (10.60). To obtain the spectral distribution, the idea is to exploit the fact that $K_n$ is ‘almost’ symmetric, because a(x) varies continuously when x ranges in [0, 1], and so $a(x_j)\approx a(x_{j+1})$ for all $j=1,\ldots , n-1$ (when n is large enough). Therefore, by replacing $K_n$ with one of its  symmetric approximations $\tilde{K}_n$, we can write

$$\begin{aligned} E_n=\tilde{K}_n+(K_n-\tilde{K}_n)+Z_n, \end{aligned}$$

(10.62)

and then we will want to obtain the spectral distribution in (10.60) from GLT 2 applied with $X_n=\tilde{K}_n$ and $Y_n=(K_n-\tilde{K}_n)+Z_n$. Let

$$\begin{aligned} \tilde{K}_n=\begin{bmatrix} 2a_1&\, -a_1&\,&\,&\, \\ -a_1&\, 2a_2&\, -a_2&\,&\, \\&\, \ddots&\, \ddots&\, \ddots&\, \\&\,&\, -a_{n-2}&\, 2a_{n-1}&\, -a_{n-1}\\&\,&\,&\, -a_{n-1}&\, 2a_n \end{bmatrix}. \end{aligned}$$

(10.63)

Since

$$\begin{aligned}&\Vert K_n-\tilde{K}_n\Vert \le \sqrt{|K_n-\tilde{K}_n|_1\,|K_n-\tilde{K}_n|_\infty }\le \max _{i=1,\ldots , n-1}|a_{i+1}-a_i|\le \omega _a(h)\rightarrow 0,\\&\Vert K_n\Vert \le \sqrt{|K_n|_1|K_n|_\infty }\le 4\Vert a\Vert _\infty \le C,\\&\Vert Z_n\Vert \rightarrow 0, \end{aligned}$$

it follows from GLT 2 that $\{E_n\}_n\sim _\lambda a(x)(2-2\cos \theta )$. $\square$

Remark 10.4

In the proof of Theorem 10.8 we could also choose

$$\tilde{K}_n=S_n(a)\circ T_n(2-2\cos \theta )= \begin{bmatrix} 2\tilde{a}_1&-\tilde{a}_1&\,&\,&\, \\ -\tilde{a}_1&2\tilde{a}_2&-\tilde{a}_2&\,&\, \\ \,&\ddots&\ddots&\ddots&\, \\ \,&\,&-\tilde{a}_{n-2}&2\tilde{a}_{n-1}&-\tilde{a}_{n-1}\\ \,&\,&\,&-\tilde{a}_{n-1}&2\tilde{a}_n \end{bmatrix},$$

where $\tilde{a}_i=a(\frac{i}{n})$ for all $i=1,\ldots , n$ and $S_n(a)$ is the arrow-shaped sampling matrix defined in (10.26). With this choice of $\tilde{K}_n$, nothing changes in the proof of Theorem 10.8 except for the bound of $\Vert K_n-\tilde{K}_n\Vert$, which becomes $\Vert K_n-\tilde{K}_n\Vert \le 4\,\omega _a(h)$.

4th Part

Based on Remark 10.1, if we change the FD scheme to discretize the differential problem (10.56), the symbol should become $a(x)p(\theta )$, where $p(\theta )$ is the trigonometric polynomial associated with the new FD formula used to approximate the second derivative $-u''(x)$ (the higher-order differential operator). We are going to show through an example that this is indeed the case.

FD discretization. Consider the convection-diffusion-reaction problem (10.56). Instead of the second-order central FD scheme $(-1, 2,-1)$, this time we use the fourth-order central FD scheme $\frac{1}{12}(1,-16, 30,-16, 1)$ to approximate the second derivative $-u''(x)$. In other words, for $j=2,\ldots , n-1$ we approximate the higher-order term $-a(x)u''(x)$ by the FD formula

$$-a(x)u''(x)|_{x=x_j}\approx a(x_j)\frac{u(x_{j+2})-16u(x_{j+1})+30u(x_j)-16u(x_{j-1})+u(x_{j-2})}{12h^2},$$

while for $j=1, n$ we use again the FD scheme $(-1, 2,-1)$,

$$-a(x)u''(x)|_{x=x_j}\approx a(x_j)\frac{-u(x_{j+1})+2u(x_j)-u(x_{j-1})}{h^2}.$$

From a numerical viewpoint, this is not a good choice because the FD formula $\frac{1}{12}(1,-16, 30,-16, 1)$ is a very accurate fourth-order formula, and in order not to destroy the accuracy one would gain from this formula, one should use a fourth-order scheme also for $j=1, n$ instead of the classical $(-1, 2,-1)$. However, in this book we are not concerned with this kind of issues and we use the classical $(-1, 2,-1)$ because it is simpler and allows us to better illustrate the GLT analysis without introducing useless technicalities. As already observed before, the FD schemes used to approximate the lower-order terms $b(x)u'(x)$ and c(x)u(x) do not affect the symbol, as well as the singular value and eigenvalue distribution, of the resulting sequence of discretization matrices. To illustrate once again this point, in this example we assume to approximate $b(x)u'(x)$ and c(x)u(x) by the following ‘strange’ FD formulas: for $j=1,\ldots , n$,

$$\begin{aligned} b(x)u'(x)|_{x=x_j}&\approx b(x_j)\frac{u(x_j)-u(x_{j-1})}{h},\\ c(x)u(x)|_{x=x_j}&\approx c(x_j)\frac{u(x_{j+1})+u(x_j)+u(x_{j-1})}{3}. \end{aligned}$$

Setting $a_i=a(x_i),\ b_i=b(x_i),\ c_i=c(x_i)$ for all $i=1,\ldots , n$, the resulting FD discretization matrix $P_n$ can be decomposed according to the diffusion, convection and reaction term, as follows:

$$\begin{aligned} P_n=K_n+Z_n, \end{aligned}$$

where $Z_n$ is the sum of the convection and reaction matrix,

$$Z_n=h\begin{bmatrix} b_1&\,&\,&\,&\, \\ -b_2&\, b_2&\,&\,&\, \\&\, \ddots&\, \ddots&\,&\, \\&\,&\, -b_{n-1}&\, b_{n-1}\\&\,&\,&\, -b_n&\ b_n \end{bmatrix}+ \frac{h^2}{3}\begin{bmatrix} c_1&\, c_1&\,&\,&\, \\ c_2&\, c_2&\, c_2&\,&\, \\&\, \ddots&\, \ddots&\, \ddots&\ \\&\,&\, c_{n-1}&\, c_{n-1}&\, c_{n-1}\\&\,&\,&\, c_n&\, c_n \end{bmatrix},$$

while $K_n$ is the diffusion matrix,

$$\begin{aligned} K_n&=\frac{1}{12}\begin{bmatrix} 24a_1&\, -12a_1&\,&\,&\,&\,&\, \\ -16a_2&\, 30a_2&\, -16a_2&\, a_2&\,&\,&\, \\ a_3&\, -16a_3&\, 30a_3&\, -16a_3&\, a_3&\,&\, \\&\, \ddots&\, \ddots&\, \ddots&\, \ddots&\, \ddots&\, \\&\,&\ a_{n-2}&\ -16a_{n-2}&\ 30a_{n-2}&\ -16a_{n-2}&\ a_{n-2} \\&\,&\,&\, a_{n-1}&\ -16a_{n-1}&\ 30a_{n-1}&\ -16a_{n-1} \\&\,&\,&\,&\,&\ -12a_n&24a_n \end{bmatrix}. \end{aligned}$$

GLT analysis of the FD discretization matrices.  The trigonometric polynomial associated with the FD formula $\frac{1}{12}(1,-16, 30,-16, 1)$ used to approximate the second derivative $-u''(x)$ is

$$\begin{aligned} p(\theta )=\frac{1}{12}(\mathrm{e}^{-2I\theta }-16\mathrm{e}^{-I\theta }+30-16\mathrm{e}^{I\theta }+\mathrm{e}^{2I\theta })=\frac{1}{12}(30-32\cos \theta +2\cos (2\theta )). \end{aligned}$$

Based on Remark 10.1, the following result is not unexpected.

Theorem 10.9

If $a\in C([0, 1])$ and $b, c:[0, 1]\rightarrow \mathbb R$ are bounded then

$$\begin{aligned} \{P_n\}_n\sim _\mathrm{GLT}a(x)p(\theta ) \end{aligned}$$

(10.64)

and

$$\begin{aligned} \{P_n\}_n\sim _{\sigma ,\,\lambda }a(x)p(\theta ). \end{aligned}$$

(10.65)

Proof

Throughout this proof, the letter C will denote a generic constant independent of n. To simultaneously obtain (10.64) and (10.65), we consider the following decomposition of $P_n$:

$$P_n=\tilde{K}_n+(K_n-\tilde{K}_n)+Z_n,$$

where $\tilde{K}_n$ is the  symmetric approximation of $K_n$ given by

$$\begin{aligned} \tilde{K}_n&=S_n(a)\circ T_n(p)\\&=\frac{1}{12} \begin{bmatrix} 30\tilde{a}_1&\, -16\tilde{a}_1&\, \tilde{a}_1&\,&\,&\,&\, \\ -16\tilde{a}_1&\, 30\tilde{a}_2&\, -16\tilde{a}_2&\, \tilde{a}_2&\,&\,&\, \\ \tilde{a}_1&\, -16\tilde{a}_2&\, 30\tilde{a}_3&\, -16\tilde{a}_3&\, \tilde{a}_3&\,&\, \\&\, \ddots&\, \ddots&\, \ddots&\, \ddots&\, \ddots&\, \\&\,&\, \tilde{a}_{n-4}&\, -16\tilde{a}_{n-3}&\, 30\tilde{a}_{n-2}&\, -16\tilde{a}_{n-2}&\, \tilde{a}_{n-2} \\&\,&\,&\, \tilde{a}_{n-3}&\, -16\tilde{a}_{n-2}&\, 30\tilde{a}_{n-1}&\, -16\tilde{a}_{n-1} \\&\,&\,&\,&\, \tilde{a}_{n-2}&\, -16\tilde{a}_{n-1}&30\tilde{a}_n \end{bmatrix} \end{aligned}$$

($\tilde{a}_i=a(\frac{i}{n})$ for all $i=1,\ldots , n$). We show that:

• (a) $\{\tilde{K}_n\}_n\sim _\mathrm{GLT}a(x)p(\theta )$;

• (b) $\Vert K_n\Vert ,\,\Vert \tilde{K}_n\Vert \le C$ and $\Vert Z_n\Vert \rightarrow 0$;

• (c) $\Vert K_n-\tilde{K}_n\Vert _1=o(n)$.

Note that (b)–(c) imply that $\{(K_n-\tilde{K}_n)+Z_n\}_n\sim _\sigma 0$ by Z 2. Once we have proved (a)–(c), the GLT relation (10.64) follows from GLT 4, the singular value distribution in (10.65) follows from (10.64) and GLT 1, and the spectral distribution in (10.65) follows from GLT 2 applied with $X_n=\tilde{K}_n$ and $Y_n=(K_n-\tilde{K}_n)+Z_n$.

Proof of (a). See Theorem 10.4.

Proof of (b). We have

$$\begin{aligned}&\Vert Z_n\Vert \le \sqrt{|Z_n|_1\,|Z_n|_\infty }\le 2h\Vert b\Vert _\infty +h^2\Vert c\Vert _\infty \rightarrow 0,\\&\Vert K_n\Vert \le \sqrt{|K_n|_1\,|K_n|_\infty }\le \frac{64}{12}\Vert a\Vert _\infty , \\&\Vert \tilde{K}_n\Vert \le \sqrt{|\tilde{K}_n|_1\,|\tilde{K}_n|_\infty }\le \frac{64}{12}\Vert a\Vert _\infty . \end{aligned}$$

Note that the uniform boundedness of $\Vert \tilde{K}_n\Vert$ with respect to n was already known from Theorem 10.4.

Proof of (c). A direct comparison between $K_n$ and $\tilde{K}_n$ shows that

$$K_n=\tilde{K}_n+R_n+N_n,$$

where $N_n=K_n-\tilde{K}_n-R_n$ and $R_n$ is the matrix whose rows are all zeros except for the first and the last one, which are given by

$$\begin{aligned} \left[ 24a_1-30\tilde{a}_1\qquad -12a_1+16\tilde{a}_1\qquad -\tilde{a}_1\qquad 0\qquad \cdots \qquad 0\right] \end{aligned}$$

and

$$\begin{aligned} \left[ 0\qquad \cdots \qquad 0\qquad -\tilde{a}_{n-2}\qquad -12a_n+16\tilde{a}_{n-1}\qquad 24a_n-30\tilde{a}_n\right] , \end{aligned}$$

respectively. We have

$$\begin{aligned} \Vert R_n\Vert \le \frac{83}{12}\Vert a\Vert _\infty ,\qquad \mathrm{rank}(R_n)\le 2,\qquad \Vert N_n\Vert \le \frac{64}{12}\,\omega _a\Bigl (\frac{2}{n}\Bigr ) \end{aligned}$$

and

$$\begin{aligned} \Vert K_n-\tilde{K}_n\Vert _1\le \Vert R_n\Vert _1+\Vert N\Vert _1\le \mathrm{rank}(R_n)\Vert R_n\Vert +n\Vert N_n\Vert \end{aligned}$$

hence $\Vert K_n-\tilde{K}_n\Vert _1=o(n)$. $\square$

Remark 10.5

(nonnegativity and order of the zero at $\theta =0$) Despite we  have changed the FD scheme to approximate the second derivative $-u''(x)$, the resulting trigonometric polynomial $p(\theta )$ retains some properties of $2-2\cos \theta$. In particular, $p(\theta )$ is nonnegative over $[-\pi ,\pi ]$ and it has a unique zero of order 2 at $\theta =0$, because

$$\lim _{\theta \rightarrow 0}\frac{p(\theta )}{\theta ^2}=1=\lim _{\theta \rightarrow 0}\frac{2-2\cos \theta }{\theta ^2}.$$

This reflects the fact the the associated FD formula $\frac{1}{12}(1,-16, 30,-16, 1)$ approximates $-u''(x)$, which is a differential operator of order 2 and it is also nonnegative on $\{v\in C^2([0, 1]):\ v(0)=v(1)=0\}$; cf. Remark 10.2.

10.5.3 FD Discretization of Higher-Order Equations

So far we only considered the FD discretization of second-order differential equations. In order to show that the GLT analysis is not limited to second-order equations, in this section we deal with an higher-order problem. For simplicity, we focus on the following fourth-order problem with homogeneous Dirichlet–Neumann boundary conditions:

$$\begin{aligned} \left\{ \begin{array}{ll} a(x)u^{(4)}(x)=f(x), &{}\quad x\in (0, 1),\\ u(0)=0,\quad u(1)=0,\\ u'(0)=0,\quad u'(1)=0, \end{array}\right. \end{aligned}$$

(10.66)

where $a\in C([0, 1])$ and f is a given function. We do not consider more complicated boundary conditions, and we do not include terms with lower-order derivatives, because we know from Remark 10.1 and the experience gained from the previous section that both these ingredients only serve to complicate things, but ultimately they do not affect the symbol, as well as the singular value and eigenvalue distribution, of the resulting discretization matrices. Based on Remark 10.1, the symbol of the matrix-sequence arising from the FD discretization of (10.66) should be $a(x)q(\theta )$, where $q(\theta )$ is the trigonometric polynomial associated with the FD formula used to discretize $u^{(4)}(x)$. We will see that this is in fact the case.

FD discretization. To approximate the fourth derivative $u^{(4)}(x)$, we use the second-order central FD scheme $(1,-4, 6,-4, 1)$, which yields the approximation

$$\begin{aligned} a(x)u^{(4)}(x)|_{x=x_j}\approx a(x_j)\frac{u(x_{j+2})-4u(x_{j+1})+6u(x_j)-4u(x_{j-1})+u(x_{j-2})}{h^4}, \end{aligned}$$

for all $j=2,\ldots , n+1$; here, $h=\frac{1}{n+3}$ and $x_j=jh$ for $j=0,\ldots , n+3$. Taking into account the homogeneous boundary conditions, we approximate the solution of (10.66) by the piecewise linear function that takes the value $u_j$ in $x_j$ for $j=0,\ldots , n+3$, where $u_0=u_1=u_{n+2}=u_{n+3}=0$ and $\mathbf {u}=(u_2,\ldots , u_{n+1})^T$ is the solution of the linear system

$$a(x_j)(u_{j+2}-4u_{j+1}+6u_j-4u_{j-1}+u_{j-2})=h^4f(x_j),\qquad j=2,\ldots , n+1.$$

The matrix $A_n$ of this linear system is given by

$$A_n=\begin{bmatrix} 6a_2&-4a_2&a_2&\,&\,&\,&\, \\ -4a_3&6a_3&-4a_3&a_3&\,&\,&\, \\ a_4&-4a_4&6a_4&-4a_4&a_4&\,&\, \\ \,&\ddots&\ddots&\ddots&\ddots&\ddots&\, \\ \,&\,&a_{n-1}&-4a_{n-1}&6a_{n-1}&-4a_{n-1}&a_{n-1}\\ \,&\,&\,&a_n&-4a_n&6a_n&-4a_n\\ \,&\,&\,&\,&a_{n+1}&-4a_{n+1}&6a_{n+1} \end{bmatrix},$$

where $a_i=a(x_i)$ for all $i=2,\ldots , n+1$.

GLT analysis of the FD discretization matrices. Let  $q(\theta )$ be the trigonometric polynomial associated with the FD formula $(1,-4, 6,-4, 1)$, i.e.,

$$\begin{aligned} q(\theta )=\mathrm{e}^{-2I\theta }-4\mathrm{e}^{-I\theta }+6-4\mathrm{e}^{I\theta }+\mathrm{e}^{2I\theta }=6-8\cos \theta +2\cos (2\theta ). \end{aligned}$$

Theorem 10.10

If $a\in C([0, 1])$ then

$$\begin{aligned} \{A_n\}_n\sim _\mathrm{GLT}a(x)q(\theta ) \end{aligned}$$

(10.67)

and

$$\begin{aligned} \{A_n\}_n\sim _{\sigma ,\,\lambda }a(x)q(\theta ), \end{aligned}$$

(10.68)

Proof

We show that

$$\begin{aligned} \Vert A_n-S_n(a)\circ T_n(q)\Vert \rightarrow 0. \end{aligned}$$

(10.69)

Once this is proved, since $\{S_n(a)\circ T_n(q)\}_n\sim _\mathrm{GLT}a(x)q(\theta )$ and $\Vert S_n(a)\circ T_n(q)\Vert$ is uniformly bounded with respect to n (by Theorem 10.4), and since $\Vert A_n\Vert \le 16\Vert a\Vert _\infty$ by (2.31), the relations (10.67) and (10.68) follow from the decomposition

$$\begin{aligned} A_n=S_n(a)\circ T_n(q)+(A_n-S_n(a)\circ T_n(q)) \end{aligned}$$

and from $\mathbf{GLT\, 1}\!-\!\mathbf{GLT\, 4}$, taking into account that $S_n(a)\circ T_n(p)$ is symmetric and $\{A_n-S_n(a)\circ T_n(q)\}_n$ is zero-distributed by (10.69) and Z 1 (or Z 2). Let us then prove (10.69). The matrices $A_n$ and $S_n(a)\circ T_n(q)$ are banded (pentadiagonal) and, for all $i, j=1,\ldots , n$ with $|i-j|\le 2$, a crude estimates gives

$$\begin{aligned} \bigl |(A_n)_{ij}-(S_n(a)\circ T_n(q))_{ij}\bigr |&=\Bigl |a_{i+1}(T_n(q))_{ij}-a\Bigl (\frac{\min (i, j)}{n}\Bigr )(T_n(q))_{ij}\Bigr |\\&=\Bigl |a\Bigl (\frac{i+1}{n+3}\Bigr )-a\Bigl (\frac{\min (i, j)}{n}\Bigl )\Bigr |\,|(T_n(q))_{ij}|\\&\le 6\,\omega _a\Bigl (\frac{6}{n}\Bigr ). \end{aligned}$$

Hence, by (2.31), $\Vert A_n-S_n(a)\circ T_n(q)\Vert \le 5\cdot 6\,\omega _a(\frac{6}{n})\rightarrow 0$. $\square$

Remark 10.6

(nonnegativity and order of the zero at $\theta =0$) The polynomial  $q(\theta )$ is nonnegative over $[-\pi ,\pi ]$ and has a unique zero of order 4 at $\theta =0$, because

$$\lim _{\theta \rightarrow 0}\frac{q(\theta )}{\theta ^4}=1.$$

This reflects the fact that the FD  formula $(1,-4, 6,-4, 1)$ associated with $q(\theta )$ approximates the fourth derivative $u^{(4)}(x)$, which is a differential operator of order 4 (it is also nonnegative on the space of functions $v\in C^4([0, 1])$ such that $v(0)=v(1)=0$ and $v'(0)=v'(1)=0$, in the sense that $\int _0^1v^{(4)}(x)v(x)\mathrm{d}x=\int _0^1(v''(x))^2\mathrm{d}x\ge 0$ for all such v); see also Remarks 10.2 and 10.5.

10.5.4 Non-uniform FD Discretizations

All the FD discretizations considered in the previous sections are based on uniform grids. It is natural to ask whether the theory of GLT sequences finds applications also in the context of non-uniform FD discretizations. The answer to this question is affirmative, at least in the case where the non-uniform grid is obtained as the mapping of a uniform grid through a fixed function G, independent of the mesh size. In this section we illustrate this claim by means of a simple example.

FD discretization. Consider the diffusion equation (10.30) with $a \in C([0, 1])$. Take a discretization parameter $n\in \mathbb N$, fix a set of grid points $0=x_0<x_1< \ldots < x_{n+1}=1$ and define the corresponding stepsizes $h_j=x_j-x_{j-1}, j=1,\ldots , n+1$. For each $j=1,\ldots , n$, we approximate $-(a(x)u'(x))'|_{x=x_j}$ by the FD formula

$$\begin{aligned} -(a(x)u'(x))'|_{x=x_j}&\approx -\frac{a(x_j+\frac{h_{j+1}}{2^{}})u'(x_j+\frac{h_{j+1}}{2^{}})-a(x_j-\frac{h_j}{2^{}})u'(x_j-\frac{h_j}{2^{}})}{\frac{h_{j+1}}{2^{}}+\frac{h_j}{2^{}}}\\&\approx -\frac{a(x_j+\frac{h_{j+1}}{2^{}})\dfrac{u(x_{j+1})-u(x_j)}{h_{j+1}}-a(x_j-\frac{h_j}{2^{}})\dfrac{u(x_j)-u(x_{j-1})}{h_j}}{\frac{h_{j+1}}{2^{}}+\frac{h_j}{2^{}}} \end{aligned}$$

which is equal to $\dfrac{2}{h_j+h_{j+1}}$ times

$$ -\frac{a(x_j-\frac{h_j}{2^{}})}{h_j}u(x_{j-1})+\biggl (\frac{a(x_j-\frac{h_j}{2^{}})}{h_j} +\frac{a(x_j+\frac{h_{j+1}}{2^{}})}{h_{j+1}}\biggr )u(x_j)-\frac{a(x_j+\frac{h_{j+1}}{2^{}})}{h_{j+1}}u(x_{j+1}). $$

This means that the nodal values of the solution u satisfy (approximately) the following linear system:

$$\begin{aligned}&-\frac{a(x_j-\frac{h_j}{2^{}})}{h_j}u(x_{j-1})+\biggl (\frac{a(x_j-\frac{h_j}{2^{}})}{h_j}+\frac{a(x_j+\frac{h_{j+1}}{2^{}})}{h_{j+1}}\biggr )u(x_j)-\frac{a(x_j+\frac{h_{j+1}}{2^{}})}{h_{j+1}}u(x_{j+1})\\&\quad =\frac{h_j+h_{j+1}}{2}f(x_j),\quad \ j=1,\ldots , n. \end{aligned}$$

We then approximate the solution by the piecewise linear function that takes the value $u_j$ in $x_j$ for $j=0,\ldots , n+1$, where $u_0=\alpha$, $u_{n+1}=\beta$, and $\mathbf u=(u_1,\ldots , u_n)^T$ solves

$$\begin{aligned}&-\frac{a(x_j-\frac{h_j}{2^{}})}{h_j}u_{j-1}+\biggl (\frac{a(x_j-\frac{h_j}{2^{}})}{h_j}+\frac{a(x_j+\frac{h_{j+1}}{2^{}})}{h_{j+1}}\biggr )u_j-\frac{a(x_j+\frac{h_{j+1}}{2^{}})}{h_{j+1}}u_{j+1}\nonumber \\&\quad =\frac{h_j+h_{j+1}}{2}f(x_j),\quad \ j=1,\ldots , n. \end{aligned}$$

The matrix of this linear system is the $n\times n$ tridiagonal symmetric matrix given by

$$\begin{aligned} \mathrm{tridiag}_n\biggl [-\frac{a(x_j-\frac{h_j}{2^{}})}{h_j},\ \frac{a(x_j-\frac{h_j}{2^{}})}{h_j}+\frac{a(x_j+\frac{h_{j+1}}{2^{}})}{h_{j+1}},\ -\frac{a(x_j+\frac{h_{j+1}}{2^{}})}{h_{j+1}}\biggr ]. \end{aligned}$$

(10.70)

GLT analysis of the FD discretization matrices. Let  $h=\frac{1}{n+1}$ and $\hat{x}_j=jh$, $j=0,\ldots , n+1$. In the following, we assume that the set of points $\{x_0, x_1,\ldots , x_{n+1}\}$ is obtained as the mapping of the uniform grid $\{\hat{x}_0,\hat{x}_1,\ldots ,\hat{x}_{n+1}\}$ through a fixed function G, i.e., $x_j=G(\hat{x}_j)$ for $j=0,\ldots , n+1$, where $G:[0, 1]\rightarrow [0, 1]$ is an increasing and bijective map, independent of the mesh parameter n. The resulting FD discretization matrix (10.70) will be denoted by $A_{G, n}$ in order to emphasize its dependence on G. In formulas,

$$\begin{aligned}&A_{G, n}=\\&\mathrm{tridiag}_n\biggl [-\frac{a(G(\hat{x}_j)-\frac{h_j}{2^{}})}{h_j},\ \frac{a(G(\hat{x}_j)-\frac{h_j}{2^{}})}{h_j}+\frac{a(G(\hat{x}_j)+\frac{h_{j+1}}{2^{}})}{h_{j+1}},\ -\frac{a(G(\hat{x}_j)+\frac{h_{j+1}}{2^{}})}{h_{j+1}}\biggr ]\nonumber \end{aligned}$$

(10.71)

with

$$\begin{aligned} h_j=G(\hat{x}_j)-G(\hat{x}_{j-1}),\qquad j=1,\ldots , n. \end{aligned}$$

Theorem 10.11

Let $a\in C([0, 1])$. Suppose $G:[0, 1]\rightarrow [0, 1]$ is an increasing bijective map in $C^1([0, 1])$ and there exist at most finitely many points $\hat{x}$ such that $G'(\hat{x})=0$. Then

$$\begin{aligned} \Bigl \{\frac{1}{n+1}A_{G, n}\Bigr \}_n\sim _\mathrm{GLT}\frac{a(G(\hat{x}))}{G'(\hat{x})}(2-2\cos \theta ) \end{aligned}$$

(10.72)

and

$$\begin{aligned} \Bigl \{\frac{1}{n+1}A_{G, n}\Bigr \}_n\sim _{\sigma ,\,\lambda }\frac{a(G(\hat{x}))}{G'(\hat{x})}(2-2\cos \theta ). \end{aligned}$$

(10.73)

Proof

We only prove (10.72) because (10.73) follows immediately from (10.72) and GLT 1 as the matrices $A_{G, n}$ are symmetric. Since $G\in C^1([0, 1])$, for every $j=1,\ldots , n$ there exist $\alpha _j\in [\hat{x}_{j-1},\hat{x}_j]$ and $\beta _j\in [\hat{x}_j,\hat{x}_{j+1}]$ such that

$$\begin{aligned} h_j&=G(\hat{x}_j)-G(\hat{x}_{j-1})=G'(\alpha _j)h=(G'(\hat{x}_j)+\delta _j)h,\end{aligned}$$

(10.74)

$$\begin{aligned} h_{j+1}&=G(\hat{x}_{j+1})-G(\hat{x}_j)=G'(\beta _j)h=(G'(\hat{x}_j)+\varepsilon _j)h, \end{aligned}$$

(10.75)

where

$$\begin{aligned} \delta _j&= G'(\alpha _j)-G'(\hat{x}_j),\\ \varepsilon _j&= G'(\beta _j)-G'(\hat{x}_j). \end{aligned}$$

Note that

$$\begin{aligned} |\delta _j|,|\varepsilon _j|\le \omega _{G'}(h),\qquad j=1,\ldots , n, \end{aligned}$$

where $\omega _{G'}$ is the modulus of continuity of $G'$. In view of (10.74) and (10.75), we have, for each $j=1,\ldots , n$,

$$\begin{aligned} a\Bigl (G(\hat{x}_j)-\frac{h_j}{2}\Bigr )&=a\Bigl (G(\hat{x}_j)-\frac{h}{2}(G'(\hat{x}_j)+\delta _j)\Bigr )=a(G(\hat{x}_j))+\mu _j,\end{aligned}$$

(10.76)

$$\begin{aligned} a\Bigl (G(\hat{x}_j)+\frac{h_{j+1}}{2}\Bigr )&=a\Bigl (G(\hat{x}_j)+\frac{h}{2}(G'(\hat{x}_j)+\varepsilon _j)\Bigr )=a(G(\hat{x}_j))+\eta _j, \end{aligned}$$

(10.77)

where

$$\begin{aligned} \mu _j&=a\Bigl (G(\hat{x}_j)-\frac{h}{2}(G'(\hat{x}_j)+\delta _j)\Bigr )-a(G(\hat{x}_j)),\\ \eta _j&=a\Bigl (G(\hat{x}_j)+\frac{h}{2}(G'(\hat{x}_j)+\epsilon _j)\Bigr )-a(G(\hat{x}_j)). \end{aligned}$$

This time

$$\begin{aligned} |\mu _j|,|\eta _j|\le C_G\omega _a(h),\qquad j=1,\ldots , n, \end{aligned}$$

where $\omega _a$ is the modulus of continuity of a and $C_G$ is a constant depending only on G. Substituting (10.74)–(10.77) in (10.71), we obtain

$$\begin{aligned}&\frac{1}{n+1}A_{G, n}=h\, A_{G, n}=\\&tridiag _n\biggl [-\frac{a(G(\hat{x}_j))+\mu _j}{G'(\hat{x}_j)+\delta _j},\; \frac{a(G(\hat{x}_j))+\mu _j}{G'(\hat{x}_j)+\delta _j}+\frac{a(G(\hat{x}_j))+\eta _j}{G'(\hat{x}_j)+\varepsilon _j},\;-\frac{a(G(\hat{x}_j))+\eta _j}{G'(\hat{x}_j)+\varepsilon _j}\biggr ].\nonumber \end{aligned}$$

(10.78)

Consider the matrix

$$\begin{aligned} D_n\Bigl (\frac{a(G(\hat{x}))}{G'(\hat{x})}\Bigr )T_n(2-2\cos \theta ) = tridiag _n\biggl [-\frac{a(G(\hat{x}_j))}{G'(\hat{x}_j)},\;2\,\frac{a(G(\hat{x}_j))}{G'(\hat{x}_j)},\;-\frac{a(G(\hat{x}_j))}{G'(\hat{x}_j)}\biggr ]. \end{aligned}$$

(10.79)

Note that this matrix seems to be an ‘approximation’ of $\frac{1}{n+1}A_{G, n}$; cf. (10.78) and (10.79). Since the function $a(G(\hat{x}))/G'(\hat{x})$ is continuous a.e., GLT 3 and GLT 4 yield

$$\begin{aligned} \biggl \{D_n\Bigl (\frac{a(G(\hat{x}))}{G'(\hat{x})}\Bigr )T_n(2-2\cos \theta )\biggr \}_n\sim _\mathrm{GLT}\frac{a(G(\hat{x}))}{G'(\hat{x})}(2-2\cos \theta ). \end{aligned}$$

We are going to show that

$$\begin{aligned} \Bigl \{D_n\Bigl (\frac{a(G(\hat{x}))}{G'(\hat{x})}\Bigr )T_n(2-2\cos \theta )\Bigr \}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\Bigl \{\frac{1}{n+1}A_{G, n}\Bigr \}_n. \end{aligned}$$

(10.80)

Once this is proved, (10.72) follows immediately from GLT 7. Incidentally, we note that the convergence statement (10.80) is equivalent to the equality

$$\begin{aligned} d_\mathrm{a.c.s.}\biggl (\Bigl \{D_n\Bigl (\frac{a(G(\hat{x}))}{G'(\hat{x})}\Bigr )T_n(2-2\cos \theta )\Bigr \}_n,\Bigl \{\frac{1}{n+1}A_{G, n}\Bigr \}_n\biggr )=0, \end{aligned}$$

which in turn is equivalent to saying that $\{Z_n\}_n\sim _\sigma 0$, where

$$\begin{aligned} Z_n = \frac{1}{n+1}A_{G, n}-D_n\Bigl (\frac{a(G(\hat{x}))}{G'(\hat{x})}\Bigr )T_n(2-2\cos \theta ). \end{aligned}$$

(10.81)

Therefore, once (10.80) is proved, we could also derive (10.72) from GLT 3 and GLT 4 without invoking GLT 7.

We first prove (10.80) in the case where $G'(\hat{x})$ does not vanish over [0, 1], so that

$$\begin{aligned} m_{G'}=\min _{\hat{x}\in [0, 1]}G'(\hat{x})>0. \end{aligned}$$

In this case, we actually show directly that $\{Z_n\}_n\sim _\sigma 0$. The matrix $Z_n$ in (10.81) is tridiagonal and a straightforward computation based on (10.78) and (10.79) shows that all its components are bounded in modulus by a quantity that depends only on n, G, a and that converges to 0 as $n\rightarrow \infty$. For example, if $j=2,\ldots , n$, then

$$\begin{aligned} |(Z_n)_{j, j-1}|&=\left| \frac{a(G(\hat{x}_j))+\mu _j}{G'(\hat{x}_j)+\delta _j}-\frac{a(G(\hat{x}_j))}{G'(\hat{x}_j)}\right| \nonumber \\&\le \left| \frac{a(G(\hat{x}_j))+\mu _j}{G'(\hat{x}_j)+\delta _j}-\frac{a(G(\hat{x}_j))}{G'(\hat{x}_j)+\delta _j}\right| +\left| \frac{a(G(\hat{x}_j))}{G'(\hat{x}_j)+\delta _j}-\frac{a(G(\hat{x}_j))}{G'(\hat{x}_j)}\right| \nonumber \\&=\left| \frac{\mu _j}{G'(\hat{x}_j)+\delta _j}\right| +\left| \frac{a(G(\hat{x}_j))\delta _j}{G'(\hat{x}_j)(G'(\hat{x}_j)+\delta _j)}\right| \nonumber \\&\le \frac{C_G\omega _a(h)}{m_{G'}-\omega _{G'}(h)}+\frac{\Vert a\Vert _\infty \omega _{G'}(h)}{m_{G'}(m_{G'}-\omega _{G'}(h))}, \end{aligned}$$

(10.82)

which tends to 0 as $n\rightarrow \infty$. Thus, $\Vert Z_n\Vert \rightarrow 0$ as $n\rightarrow \infty$ and $\{Z_n\}_n\sim _\sigma 0$.

Now we consider the case where G has a finite number of points $\hat{x}$ where ${G'(\hat{x})=0}$. In this case, the previous argument does not work because $m_{G'}=0$. However, we can still prove (10.80) in the following way. Let $\hat{x}^{(1)},\ldots ,\hat{x}^{(s)}$ be the points where $G'$ vanishes, and consider the balls (intervals) $B(\hat{x}^{(k)},\frac{1}{m})=\{\hat{x}\in [0, 1]:\, |\hat{x}-\hat{x}^{(k)}|<\frac{1}{m}\}$. The function $G'$ is continuous and positive on the complement of the union $\bigcup _{k=1}^sB(\hat{x}^{(k)},\frac{1}{m})$, so

$$\begin{aligned} m_{G',\, m}=\min _{\hat{x}\in [0, 1]\backslash \bigcup _{k=1}^sB(\hat{x}^{(k)},\frac{1}{m})}G'(\hat{x})>0. \end{aligned}$$

For all indices $j=1,\ldots , n$ such that $\hat{x}_j\in [0, 1]\backslash \bigcup _{k=1}^sB(\hat{x}^{(k)},\frac{1}{m})$, the components in the jth row of the matrix (10.81) are bounded in modulus by a quantity that depends only on n, m, G, a and that converges to 0 as $n\rightarrow \infty$. This becomes immediately clear if we note that, for such indices j, the inequality (10.82) holds unchanged with $m_{G'}$ replaced by $m_{G',\, m}$. The number of remaining rows of $Z_n$ (the rows corresponding to indices j such that $\hat{x}_j\in \bigcup _{k=1}^sB(\hat{x}^{(k)},\frac{1}{m})$) is at most $2s(n+1)/m+s$. Indeed, each interval $B(\hat{x}^{(k)},\frac{1}{m})$ has length 2 / m (at most) and can contain at most $2(n+1)/m+1$ grid points $\hat{x}_j$. Thus, for every n, m we can split the matrix $Z_n$ into the sum of two terms, i.e.,

$$\begin{aligned} Z_n = R_{n, m} + N_{n, m}, \end{aligned}$$

where $N_{n, m}$ is obtained from $Z_n$ by setting to zero all the rows corresponding to indices j such that $\hat{x}_j\in \bigcup _{k=1}^sB(\hat{x}^{(k)},\frac{1}{m})$ and $R_{n, m}=Z_n-N_{n, m}$ is obtained from $Z_n$ by setting to zero all the rows corresponding to indices j such that $\hat{x}_j\in [0, 1]\backslash \bigcup _{k=1}^sB(\hat{x}^{(k)},\frac{1}{m})$. From the above discussion we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert N_{n, m}\Vert =0 \end{aligned}$$

for all m, and

$$\begin{aligned} \mathrm{rank}(R_{n, m})\le \frac{2s(n+1)}{m}+s \end{aligned}$$

for all m, n. In particular, for each m we can choose $n_m$ such that, for $n\ge n_m$, $\mathrm{rank}(R_{n, m})\le 3sn/m$ and $\Vert N_{n, m}\Vert \le 1/m$. The convergence (10.80) now follows from the definition of a.c.s. $\square$

Remark 10.7

An increasing bijective map $G:[0, 1]\rightarrow [0, 1]$ in $C^1([0, 1])$ is said to be  regular if $G'(\hat{x})\ne 0$ for all $\hat{x}\in [0, 1]$ and is said to be singular otherwise, i.e., if $G'(\hat{x})=0$ for some $\hat{x}\in [0, 1]$. If G is singular, any point $\hat{x}\in [0, 1]$ such that $G'(\hat{x})=0$ is referred to as a singularity point (or simply a singularity) of G. We note that, if G is singular, the symbol in (10.72) is unbounded (except in some rare cases where $a(G(\hat{x}))$ and $G'(\hat{x})$ vanish simultaneously).

Remark 10.8

(Why should one be interested in using a singular map?) The choice of a map G with one or more singularity points corresponds to adopting a local  refinement strategy, according to which the grid points $x_j$ rapidly accumulate at the G-images of the singularities as n increases. For example, if

$$\begin{aligned} G(\hat{x})=\hat{x}^q, \qquad q>1, \end{aligned}$$

(10.83)

then 0 is a singularity of G (because $G'(0)=0$) and the grid points

$$\begin{aligned} x_j=G(\hat{x}_j)=\Bigl (\frac{j}{n+1}\Bigr )^q,\quad \ j=0,\ldots , n+1, \end{aligned}$$

rapidly accumulate at $G(0)=0$ as $n\rightarrow \infty$. But why should one be interested in discretizing the diffusion equation (10.30) with a grid that rapidly accumulates at a point? The answer is that this local refinement is necessary in some situations where the coefficient a(x) is strongly anisotropic. In the subregions of the domain where a(x) is, say, ‘not well-behaved’ (e.g., remarkably oscillatory), the solution u(x) is expected to have a wild behavior as well, and a fine grid should be used to approximate it adequately. If a uniform discretization were used, the associated discretization step should be chosen very small, and this would result in a linear system with extremely large size: the computational cost to solve it would be unsustainable. For this reason, one adopts a local refinement, so that a coarse grid is used in the subregions of the domain where a(x) is sufficiently smooth, and a finer grid is used only in the subregions where a(x) is not well-behaved. For instance, the map G in (10.83) is a better way to approximate the solution u(x) in a neighborhood of 0, and this is necessary if a(x) has a ‘wild behavior’ near 0 (like, e.g., the function $\frac{1}{x+\varepsilon }\sin (\frac{1}{x+\varepsilon })$ with $\varepsilon \approx 0$). We refer the reader to [21, Sect. 3] for a nice diffusion model in which a local grid refinement is advisable.

10.6 Finite Element Discretization of Differential Equations

10.6.1 FE Discretization of Convection-Diffusion-Reaction Equations

Consider the following convection-diffusion-reaction problem in divergence form with Dirichlet boundary conditions:

$$\begin{aligned} \left\{ \begin{array}{l} -(a(x)u'(x))'+b(x)u'(x)+c(x)u(x) = f(x),\quad x\in (0, 1),\\ u(0)=u(1)=0, \end{array}\right. \end{aligned}$$

(10.84)

where $f\in L^2([0, 1])$ and the coefficients a, b, c are only assumed to be in $L^\infty ([0, 1])$. These sole assumptions are enough to perform the GLT analysis of the matrices arising from the FE discretization of (10.84). In this sense, we are going to see that the theory of GLT sequences allows one to derive the singular value and spectral distribution of DE discretization matrices under very weak hypotheses on the DE coefficients.

FE discretization. We consider the approximation of (10.84) by classical linear FEs on a uniform mesh in [0, 1] with stepsize $h=\frac{1}{n+1}$. We briefly describe here this approximation technique and for more details we refer the reader to [91, Chap. 4] or to any other good book on FEs. We first recall from [31, Chap. 8] that, if ${\varOmega \subset \mathbb R}$ is a bounded interval whose endpoints are, say, $\alpha$ and $\beta$,  $H^1(\varOmega )$ denotes the (Sobolev)  space of functions $v\in L^2(\varOmega )$ possessing a weak (Sobolev) derivative in $L^2(\varOmega )$. We also recall that each $v\in H^1(\varOmega )$ coincides a.e. with a continuous function in $C(\overline{\varOmega })$, and $H^1(\varOmega )$ can also be defined as the following subspace of $C(\overline{\varOmega })$:

$$\begin{aligned} H^1(\varOmega )=\biggl \{v\in C(\overline{\varOmega }):&\ \ v \text { is differentiable a.e. with }\ v'\in L^2(\varOmega ),\\&\ \ v(x)=v(\alpha )+\int _\alpha ^xv'(y)\mathrm{d}y\quad \text{ for } \text{ all }\quad x\in \overline{\varOmega }\biggr \}. \end{aligned}$$

Fig. 10.4

Graph of the hat-functions $\varphi _1,\ldots ,\varphi _n$ for $n=9$

In  this definition, the weak derivative of a $v\in H^1(\varOmega )$ is just the classical derivative $v'$ (which exists a.e.). Let

$$\begin{aligned} H^1_0(\varOmega )=\{v\in H^1(\varOmega ):\ v(\alpha )=v(\beta )=0\}. \end{aligned}$$

The weak form of (10.84) reads as follows [31, Chap. 8]: find $u\in H^1_0([0, 1])$ such that

$$\begin{aligned} a (u, w)=f (w),\qquad \forall \, w\in H^1_0([0, 1]), \end{aligned}$$

where

$$\begin{aligned} a (u, w)&=\int _0^1a(x)u'(x)w'(x)\mathrm{d}x+\int _0^1b(x)u'(x)w(x)\mathrm{d}x+\int _0^1c(x)u(x)w(x)\mathrm{d}x,\\ f (w)&=\int _0^1f(x)w(x)\mathrm{d}x. \end{aligned}$$

Let $h=\frac{1}{n+1}$ and $x_i=ih,\ i=0,\ldots , n+1$. In the linear FE approach based on the uniform mesh $\{x_0,\ldots , x_{n+1}\}$, we fix the subspace $\mathscr {W}_n=span (\varphi _1,\ldots ,\varphi _n)\subset H^1_0([0, 1])$, where $\varphi _1,\ldots ,\varphi _n$ are the so-called hat-functions:

$$\begin{aligned} \varphi _i(x)&=\frac{x-x_{i-1}}{x_i-x_{i-1}}\chi _{[x_{i-1},\, x_i)}(x)+\frac{x_{i+1}-x}{x_{i+1}-x_i}\chi _{[x_i,\, x_{i+1})}(x),\qquad i=1,\ldots , n; \end{aligned}$$

(10.85)

see Fig. 10.4. Note that $\mathscr {W}_n$ is the space of piecewise linear functions corresponding to the sequence of points $0=x_0<x_1<\ldots <x_{n+1}=1$ and vanishing on the boundary of the domain [0, 1]. In formulas,

$$\begin{aligned} \mathscr {W}_n=\bigl \{s:[0, 1]\rightarrow \mathbb R:\ s_{\left| \left[ \frac{i}{n+1^{}},\frac{i+1}{n+1^{}}\right) \right. }\in \mathbb P_1,\ \ i=0,\ldots , n,\ \ s(0)=s(1)=0\bigr \}, \end{aligned}$$

where $\mathbb P_1$ is the space of polynomials of degree less than or equal to 1. Then, we look for an approximation $u_{\mathscr {W}_n}$ of u by solving the following (Galerkin) problem: find $u_{\mathscr {W}_n}\in \mathscr {W}_n$ such that

$$\begin{aligned} a (u_{\mathscr {W}_n}, w)=f (w),\qquad \forall \, w\in \mathscr {W}_n. \end{aligned}$$

Since $\{\varphi _1,\ldots ,\varphi _n\}$ is a basis of $\mathscr {W}_n$, we can write $u_{\mathscr {W}_n}=\sum _{j=1}^nu_j\varphi _j$ for a unique vector $\mathbf {u}=(u_1,\ldots , u_n)^T$. By linearity, the computation of $u_{\mathscr {W}_n}$ (i.e., of $\mathbf {u}$) reduces to solving the linear system

$$\begin{aligned} A_n\mathbf {u}=\mathbf {f}, \end{aligned}$$

where $\mathbf {f}=(f (\varphi _1),\ldots , f (\varphi _n))^T$ and $A_n$ is the stiffness matrix,

$$\begin{aligned} A_n=[a (\varphi _j,\varphi _i)]_{i, j=1}^n. \end{aligned}$$

Note that $A_n$ admits the following decomposition:

$$\begin{aligned} A_n=K_n+Z_n, \end{aligned}$$

(10.86)

where

$$\begin{aligned} K_n=\left[ \int _0^1a(x)\varphi _j'(x)\varphi _i'(x)\mathrm{d}x\right] _{i, j=1}^n \end{aligned}$$

(10.87)

is the (symmetric) diffusion matrix and

$$\begin{aligned} Z_n=\left[ \int _0^1b(x)\varphi _j'(x)\varphi _i(x)\mathrm{d}x\right] _{i, j=1}^n+\left[ \int _0^1c(x)\varphi _j(x)\varphi _i(x)\mathrm{d}x\right] _{i, j=1}^n \end{aligned}$$

(10.88)

is the sum of the convection and reaction matrix.

GLT analysis of the FE discretization matrices.  Using the theory of GLT sequences we now derive the spectral and singular value distribution of the sequence of normalized stiffness matrices $\{\frac{1}{n+1}A_n\}_n$.

Theorem 10.12

If $a, b, c\in L^\infty ([0, 1])$ then

$$\begin{aligned} \Bigl \{\frac{1}{n+1}A_n\Bigr \}_n\sim _\mathrm{GLT}a(x)(2-2\cos \theta ) \end{aligned}$$

(10.89)

and

$$\begin{aligned} \Bigl \{\frac{1}{n+1}A_n\Bigr \}_n\sim _{\sigma ,\,\lambda }a(x)(2-2\cos \theta ). \end{aligned}$$

(10.90)

Proof

The proof consists of the following steps. Throughout the proof, the letter C will denote a generic constant independent of n.

Step 1. We show that

$$\begin{aligned} \Bigl \Vert \frac{1}{n+1}K_n\Bigr \Vert \le C \end{aligned}$$

(10.91)

and

$$\begin{aligned} \Bigl \Vert \frac{1}{n+1}Z_n\Bigr \Vert \le C/n. \end{aligned}$$

(10.92)

To prove (10.91), we note that $K_n$ is a banded (tridiagonal) matrix, due to the local support property $supp (\varphi _i)=[x_{i-1}, x_{i+1}]$, $i=1,\ldots , n$. Moreover, by the inequality $|\varphi _i'(x)|\le n+1$, for all $i, j=1,\ldots , n$ we have

$$\begin{aligned} |(K_n)_{ij}|&=\left| \int _0^1a(x)\varphi _j'(x)\varphi _i'(x)\mathrm{d}x\right| =\left| \int _{x_{i-1}}^{x_{i+1}}a(x)\varphi _j'(x)\varphi _i'(x)\mathrm{d}x\right| \\&\le (n+1)^2\Vert a\Vert _{L^\infty }\int _{x_{i-1}}^{x_{i+1}}\mathrm{d}x=2(n+1)\Vert a\Vert _{L^\infty }. \end{aligned}$$

Thus, the components of the tridiagonal matrix $\frac{1}{n+1}K_n$ are bounded (in modulus) by $2\Vert a\Vert _{L^\infty }$, and (10.91) follows from (2.31).

To prove (10.92), we follow the same argument as for the proof of (10.91). Due to the local support property of the hat-functions, $Z_n$ is tridiagonal. Moreover, by the inequalities $|\varphi _i(x)|\le 1$ and $|\varphi _i'(x)|\le n+1$, for all $i, j=1,\ldots , n$ we have

$$\begin{aligned} |(Z_n)_{ij}|&= \left| \int _{x_{i-1}}^{x_{i+1}}b(x)\varphi _j'(x)\varphi _i(x)\mathrm{d}x+\int _{x_{i-1}}^{x_{i+1}}c(x)\varphi _j(x)\varphi _i(x)\mathrm{d}x\right| \\&\le 2\Vert b\Vert _{L^\infty }+\frac{2\Vert c\Vert _{L^\infty }}{n+1}, \end{aligned}$$

and (10.92) follows from (2.31).

Step 2. Consider the linear operator $K_n(\cdot ):L^1([0, 1])\rightarrow \mathbb R^{n\times n}$,

$$\begin{aligned} K_n(g)=\left[ \int _0^1g(x)\varphi _j'(x)\varphi _i'(x)\mathrm{d}x\right] _{i, j=1}^n. \end{aligned}$$

By (10.87), we have $K_n=K_n(a)$. The next three steps are devoted to show that

$$\begin{aligned} \Bigl \{\frac{1}{n+1}K_n(g)\Bigr \}_n\sim _\mathrm{GLT}g(x)(2-2\cos \theta ),\qquad \forall \, g\in L^1([0, 1]). \end{aligned}$$

(10.93)

Once this is done, the theorem is proved. Indeed, by applying (10.93) with ${g=a}$ we immediately get $\{\frac{1}{n+1}K_n\}_n\sim _\mathrm{GLT}a(x)(2-2\cos \theta )$. Since $\{\frac{1}{n+1}Z_n\}_n$ is zero-distributed by Step 1, (10.89) follows from the decomposition

$$\begin{aligned} \frac{1}{n+1}A_n=\frac{1}{n+1}K_n+\frac{1}{n+1}Z_n \end{aligned}$$

(10.94)

and from $\mathbf{GLT\, 3}\!-\!\mathbf{GLT\, 4}$; and the singular value distribution in (10.90) follows from GLT 1. If $b(x)=0$ identically, then $\frac{1}{n+1}A_n$ is symmetric and also the spectral distribution in (10.90) follows from GLT 1. If b(x) is not identically 0, the spectral distribution in (10.90) follows from GLT 2 applied to the decomposition (10.94), taking into account what we have proved in Step 1.

Step 3. We first prove (10.93) in the constant-coefficient case where $g=1$ identically. In this case, a direct computation based on (10.85) shows that

$$\begin{aligned} K_n(1)=\left[ \int _0^1\varphi _j'(x)\varphi _i'(x)\mathrm{d}x\right] _{i, j=1}^n=\frac{1}{h}\begin{bmatrix} 2&-1&\, \\ -1&2&-1&\, \\ \,&\ddots&\ddots&\ddots \\ \,&\,&-1&2&-1\\ \,&\,&\,&-1&2 \end{bmatrix}=\frac{1}{h}\, T_n(2-2\cos \theta ), \end{aligned}$$

and the desired relation $\{\frac{1}{n+1}K_n(1)\}_n\sim _\mathrm{GLT}2-2\cos \theta$ follows from GLT 3. Note that it is precisely the analysis of the constant-coefficient case considered in this step that allows one to realize what is the correct normalization factor. In our case, this is $\frac{1}{n+1}$, which removes the $\frac{1}{h}$ from $K_n(1)$ and yields a normalized matrix $\frac{1}{n+1}K_n(1)=T_n(2-2\cos \theta )$, whose components are bounded away from 0 and $\infty$ (actually, in the present case they are even constant).

Step 4. Now we prove (10.93) in the case where $g\in C([0, 1])$. We first illustrate the idea, and then we go into the details. The proof is based on the fact that the hat-functions (10.85) are ‘locally supported’. Indeed, the support $[x_{i-1}, x_{i+1}]$ of the ith hat-function $\varphi _i(x)$ is located near the point $\frac{i}{n}\in [x_i, x_{i+1}]$, and the amplitude of the support tends to 0 as $n\rightarrow \infty$. In this sense, the linear FE method considered herein belongs to the family of the so-called ‘local’ methods. Since g(x) varies continuously over [0, 1], the (i, j) entry of $K_n(g)$ can be approximated as follows, for every $i, j=1,\ldots , n$:

$$\begin{aligned}&(K_n(g))_{ij}=\int _0^1g(x)\varphi _j'(x)\varphi _i'(x)\mathrm{d}x=\int _{x_{i-1}}^{x_{i+1}}g(x)\varphi _j'(x)\varphi _i'(x)\mathrm{d}x\\&\approx g\Bigl (\frac{i}{n}\Bigr )\int _{x_{i-1}}^{x_{i+1}}\varphi _j'(x)\varphi _i'(x)\mathrm{d}x=g\Bigl (\frac{i}{n}\Bigr )\int _0^1\varphi _j'(x)\varphi _i'(x)\mathrm{d}x=g\Bigl (\frac{i}{n}\Bigr )(K_n(1))_{ij}. \end{aligned}$$

This approximation can be rewritten in matrix form as

$$\begin{aligned} K_n(g)\approx D_n(g)K_n(1). \end{aligned}$$

(10.95)

We will see that (10.95) implies that $\{\frac{1}{n+1}K_n(g)-\frac{1}{n+1}D_n(g)K_n(1)\}_n\sim _\sigma 0$, and (10.93) will then follow from Step 3 and $\mathbf{GLT\, 3}\!-\!\mathbf{GLT\, 4}$.

Let us now go into the details. Since $\mathrm{supp}(\varphi _i)=[x_{i-1}, x_{i+1}]$ and $|\varphi _i'(x)|\le n+1$, for all $i, j=1,\ldots , n$ we have

$$\begin{aligned}&\left| (K_n(g))_{ij}-(D_n(g)K_n(1))_{ij}\right| =\left| \int _0^1\Bigl [g(x)-g\Bigl (\frac{i}{n}\Bigr )\Bigr ]\varphi _j'(x)\varphi _i'(x)\mathrm{d}x\right| \\&\le (n+1)^2\int _{x_{i-1}}^{x_{i+1}}\Bigl |g(x)-g\Bigl (\frac{i}{n}\Bigr )\Bigr |\mathrm{d}x\le 2(n+1)\,\omega _g\Bigl (\frac{2}{n+1}\Bigr ). \end{aligned}$$

It follows that each entry of the matrix $Z_n=\frac{1}{n+1}K_n(g)-\frac{1}{n+1}D_n(g)K_n(1)$ is bounded in modulus by $2\,\omega _g(\frac{2}{n+1})$. Moreover, $Z_n$ is banded (tridiagonal), because of the local support property of the hat-functions. Thus, both the 1-norm and the $\infty$-norm of $Z_n$ are bounded by $C\,\omega _g(\frac{2}{n+1})$, and (2.31) yields $\Vert Z_n\Vert \le C\,\omega _g(\frac{2}{n+1})\rightarrow 0$ as $n\rightarrow \infty$. Hence, $\{Z_n\}_n\sim _\sigma 0$, which implies (10.93) by Step 3 and $\mathbf{GLT\, 3}\!-\!\mathbf{GLT\, 4}$.

Step 5. Finally, we prove (10.93) in the general case where $g\in L^1([0, 1])$. By the density of C([0, 1]) in $L^1([0, 1])$, there exist continuous functions $g_m\in C([0, 1])$ such that $g_m\rightarrow g$ in $L^1([0, 1])$. By Step 4,

$$\begin{aligned} \Bigl \{\frac{1}{n+1}K_n(g_m)\Bigr \}_n\sim _\mathrm{GLT}g_m(x)(2-2\cos \theta ). \end{aligned}$$

(10.96)

Moreover,

$$\begin{aligned} g_m(x)(2-2\cos \theta )\rightarrow g(x)(2-2\cos \theta ) \ \text {in measure}. \end{aligned}$$

(10.97)

We show that

$$\begin{aligned} \Bigl \{\frac{1}{n+1}K_n(g_m)\Bigr \}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\Bigl \{\frac{1}{n+1}K_n(g)\Bigr \}_n. \end{aligned}$$

(10.98)

Since $\sum _{i=1}^n|\varphi _i'(x)|\le 2(n+1)$ for all $x\in [0, 1]$, by (2.43) we obtain

$$\begin{aligned} \Vert K_n(g)-K_n(g_m)\Vert _1&\le \sum _{i, j=1}^n|(K_n(g))_{ij}-(K_n(g_m))_{ij}|\\&= \sum _{i, j=1}^n\left| \int _0^1\bigl [g(x)-g_m(x)\bigr ]\varphi _j'(x)\varphi _i'(x)\mathrm{d}x\right| \\&\le \int _0^1\bigl |g(x)-g_m(x)\bigr |\sum _{i, j=1}^n|\varphi _j'(x)|\,|\varphi _i'(x)|\mathrm{d}x\\&\le 4(n+1)^2\Vert g-g_m\Vert _{L^1} \end{aligned}$$

and

$$\begin{aligned} \Bigl \Vert \frac{1}{n+1}K_n(g)-\frac{1}{n+1}K_n(g_m)\Bigr \Vert _1\le Cn\Vert g-g_m\Vert _{L^1}. \end{aligned}$$

Thus, $\{\frac{1}{n+1}K_n(g_m)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{\frac{1}{n+1}K_n(g)\}_n$ by ACS 6. In view of (10.96)–(10.98), the relation (10.93) follows from GLT 7. $\square$

Remark 10.9

(formal structure of the symbol) Problem (10.84) can be formally rewritten as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} -a(x)u''(x)+(b(x)-a'(x))u'(x)+c(x)u(x)=f(x), &{}\quad x\in (0, 1),\\ u(0)=u(1)=0. \end{array}\right. \end{aligned}$$

(10.99)

It is then clear that the symbol $a(x)(2-2\cos \theta )$ has the same formal structure of the higher-order  differential operator $-a(x)u''(x)$ associated with (10.99) (as in the FD case; see Remark 10.1). The formal analogy becomes even more evident if we note that $2-2\cos \theta$ is the trigonometric polynomial in the Fourier variable coming from the FE discretization of the (negative) second derivative $-u''(x)$. Indeed, as we have seen in Step 3 of the proof of Theorem 10.12, $2-2\cos \theta$ is the symbol of the sequence of FE diffusion matrices $\{\frac{1}{n+1}K_n(1)\}_n$, which arises from the FE approximation of the Poisson problem

$$\begin{aligned} \left\{ \begin{array}{ll} -u''(x)=f(x), &{}\quad x\in (0, 1),\\ u(0)=u(1)=0, \end{array}\right. \end{aligned}$$

that is, problem (10.84) in the case where $a(x)=1$ and $b(x)=c(x)=0$ identically.

Fig. 10.5

Comparison between the spectrum of $\frac{1}{n+1}A_n$ and the rearranged version g of the symbol $a(x)(2-2\cos \theta )$ for $n=50$ and $a(x)=\mathrm{e}^x\sin (4x),\ b(x)=c(x)=0$

Example 10.4

Let $a(x)=\mathrm{e}^x\sin (4x),\ b(x)=c(x)=0$. Let g be the rearranged version of the symbol $a(x)(2-2\cos \theta )$ obtained as the pointwise limit of the functions $g_r$, in the same way as explained in Example 10.2. In Fig. 10.5 we plotted the graph of g and the eigenvalues of $\frac{1}{n+1}A_n$ for $n=50$. The graph of g has been obtained by plotting the graph of $g_r$ corresponding to a large r ($r=1000$). The eigenvalues of $\frac{1}{n+1}A_n$ have been sorted in non-decreasing order and placed at the points $(\frac{k}{n},\lambda _k(\frac{1}{n+1}A_n)),\ k=1,\ldots , n$. We see from the figure that the symbol-to-spectrum approximation is quite accurate, which also indicates the absence of outliers.

Exercise 10.4

Suppose the convection term is not present, i.e., $b(x)=0$ identically. Show that (10.89) and (10.90)  continue to hold even if the diffusion and reaction coefficients a(x) and c(x) are only assumed to be in $L^1([0, 1])$. This is an extension of Theorem 10.12, which demonstrates once again that the theory of GLT sequences allows one to derive the singular value and spectral distribution of DE discretization matrices under very weak hypotheses on the DE coefficients.

10.6.2 FE Discretization of a System of Equations

In this section we consider the linear FE approximation of a system of differential equations, namely

$$\begin{aligned} \left\{ \begin{aligned} -(a(x)u'(x))'+v'(x)&=f(x),&\quad x\in (0, 1),\\ -u'(x)-\rho v(x)&=g(x),&\quad x\in (0, 1),\\ u(0)=0,\quad u(1)=0,&\\ v(0)=0,\quad v(1)=0,&\end{aligned}\right. \end{aligned}$$

(10.100)

where $\rho$ is a constant and a is only assumed to be in $L^1([0, 1])$. As we shall see, the resulting discretization matrices appear in the so-called saddle point form [12, p. 3], and we will illustrate the way to compute the asymptotic spectral and singular value distribution of their Schur complements using the theory of GLT sequences. It is worth noting that the Schur complement is a key tool for the numerical treatment of the related linear systems [12, Sect. 5]. The analysis of this section is similar to the analysis in [47, Sect. 2], but the discretization technique considered herein is a pure FE approximation, whereas in [47, Sect. 2] the authors adopted a mixed FD/FE technique.

FE discretization. We consider the approximation of (10.100) by linear FEs on a uniform mesh in [0, 1] with stepsize $h=\frac{1}{n+1}$. Let us describe it shortly. The weak form of (10.100) reads as follows¹: find $u, v\in H^1_0([0, 1])$ such that, for all $w\in H^1_0([0, 1])$,

$$\begin{aligned} \left\{ \begin{aligned} \textstyle {\int _0^1a(x)u'(x)w'(x)\mathrm{d}x + \int _0^1v'(x)w(x)\mathrm{d}x}&= \textstyle {\int _0^1f(x)w(x)\mathrm{d}x,}\\ \textstyle {-\int _0^1u'(x)w(x)\mathrm{d}x - \rho \int _0^1v(x)w(x)\mathrm{d}x}&= \textstyle {\int _0^1g(x)w(x)\mathrm{d}x.} \end{aligned}\right. \end{aligned}$$

(10.101)

Let $h=\frac{1}{n+1}$ and $x_i=ih,\ i=0,\ldots , n+1$. In the linear FE approach based on the mesh $\{x_0,\ldots , x_{n+1}\}$, we fix the subspace $\mathscr {W}_n=span (\varphi _1,\ldots ,\varphi _n)\subset H^1_0([0, 1])$, where $\varphi _1,\ldots ,\varphi _n$ are the hat-functions in (10.85) (see also Fig. 10.4). Then, we look for approximations $u_{\mathscr {W}_n}, v_{\mathscr {W}_n}$ of u, v by solving the following (Galerkin) problem: find $u_{\mathscr {W}_n}, v_{\mathscr {W}_n}\in \mathscr {W}_n$ such that, for all $w\in \mathscr {W}_n$,

$$\begin{aligned} \left\{ \begin{aligned} \textstyle {\int _0^1a(x)u_{\mathscr {W}_n}'(x)w'(x)\mathrm{d}x + \int _0^1v_{\mathscr {W}_n}'(x)w(x)\mathrm{d}x}&= \textstyle {\int _0^1f(x)w(x)\mathrm{d}x,}\\ \textstyle {-\int _0^1u_{\mathscr {W}_n}'(x)w(x)\mathrm{d}x - \rho \int _0^1v_{\mathscr {W}_n}(x)w(x)\mathrm{d}x}&= \textstyle {\int _0^1g(x)w(x)\mathrm{d}x.} \end{aligned}\right. \end{aligned}$$

Since $\{\varphi _1,\ldots ,\varphi _n\}$ is a basis of $\mathscr {W}_n$, we can write $u_{\mathscr {W}_n}=\sum _{j=1}^nu_j\varphi _j$ and $v_{\mathscr {W}_n}=\sum _{j=1}^nv_j\varphi _j$ for unique vectors $\mathbf {u}=(u_1,\ldots , u_n)^T$ and $\mathbf {v}=(v_1,\ldots , v_n)^T$. By linearity, the computation of $u_{\mathscr {W}_n}, v_{\mathscr {W}_n}$ (i.e., of $\mathbf {u},\mathbf {v}$) reduces to solving the linear system

$$\begin{aligned} A_{2n}\begin{bmatrix}\mathbf {u}\\\mathbf {v}\end{bmatrix}=\begin{bmatrix}\mathbf {f}\\\mathbf {g}\end{bmatrix}, \end{aligned}$$

where $\mathbf {f}=\bigl [\int _0^1f(x)\varphi _i(x)\mathrm{d}x\bigr ]_{i=1}^n,\ \mathbf {g}=\bigl [\int _0^1g(x)\varphi _i(x)\mathrm{d}x\bigr ]_{i=1}^n$ and $A_{2n}$ is the stiffness matrix, which possesses the following  saddle point structure:

$$A_{2n} =\begin{bmatrix} K_n&H_n\\ H_n^T&-\rho M_n \end{bmatrix}.$$

Here, the blocks $K_n, H_n, M_n$ are square matrices of size n, and precisely

$$\begin{aligned} K_n&=\left[ \int _0^1a(x)\varphi _j'(x)\varphi _i'(x)\mathrm{d}x\right] _{i, j=1}^n,\\ H_n&=\left[ \int _0^1\varphi _j'(x)\varphi _i(x)\mathrm{d}x\right] _{i, j=1}^n=\frac{1}{2}\begin{bmatrix} 0&1&\,&\,&\, \\ -1&0&1&\,&\, \\ \,&\ddots&\ddots&\ddots&\,\\ \,&\,&-1&0&1 \\ \,&\,&\,&-1&0 \end{bmatrix}=-I\, T_n(\sin \theta ),\\ M_n&=\left[ \int _0^1\varphi _j(x)\varphi _i(x)\mathrm{d}x\right] _{i, j=1}^n=\frac{h}{6}\begin{bmatrix} 4&1&\, \\ 1&4&1&\, \\ \,&\ddots&\ddots&\ddots \\ \,&\,&1&4&1\\ \,&\,&\,&1&4 \end{bmatrix}=\frac{h}{3}\, T_n(2+\cos \theta ). \end{aligned}$$

Note that $K_n$ is exactly the matrix appearing in (10.87). Note also that the matrices $K_n,\, M_n$ are symmetric, while $H_n$ is skew-symmetric: $H_n^T=-H_n=I\, T_n(\sin \theta )$.

GLT analysis of the Schur complements of the FE discretization matrices.  Assume that the matrices $K_n$ are invertible. This is satisfied, for example, if $a>0$ a.e., in which case the matrices $K_n$ are positive definite. The (negative)  Schur complement of $A_{2n}$ is the symmetric matrix given by

$$\begin{aligned} S_n=\rho M_n+H_n^T\, K_n^{-1}\, H_n=\frac{\rho h}{3}\, T_n(2+\cos \theta )+T_n(\sin \theta )\, K_n^{-1}\, T_n(\sin \theta ). \end{aligned}$$

(10.102)

In the following, we perform the GLT analysis of the sequence of normalized Schur complements $\{(n+1)S_n\}_n$, and we compute its asymptotic spectral and singular value distribution under the additional necessary assumption that $a\ne 0$ a.e.

Theorem 10.13

Let $\rho \in \mathbb R$ and $a\in L^1([0, 1])$. Suppose that the matrices $K_n$ are invertible and that $a\ne 0$ a.e. Then

$$\begin{aligned} \{(n+1)S_n\}_n\sim _\mathrm{GLT}\varsigma (x,\theta ) \end{aligned}$$

(10.103)

and

$$\begin{aligned} \{(n+1)S_n\}_n\sim _{\sigma ,\,\lambda }\varsigma (x,\theta ), \end{aligned}$$

(10.104)

where

$$\begin{aligned} \varsigma (x,\theta )=\frac{\rho }{3}(2+\cos \theta )+\frac{\sin ^2\theta }{a(x)(2-2\cos \theta )}. \end{aligned}$$

Proof

In view of (10.102), we have

$$\begin{aligned} (n+1)S_n=\frac{\rho }{3}\, T_n(2+\cos \theta )+T_n(\sin \theta )\,\Bigl (\frac{1}{n+1}K_n\Bigr )^{-1}\, T_n(\sin \theta ). \end{aligned}$$

Moreover, by (10.93),

$$\begin{aligned} \Bigl \{\frac{1}{n+1}K_n\Bigr \}_n=\Bigl \{\frac{1}{n+1}K_n(a)\Bigr \}_n\sim _\mathrm{GLT}a(x)(2-2\cos \theta ). \end{aligned}$$

Therefore, under the assumption that $a\ne 0$ a.e., the GLT relation (10.103) follows from $\mathbf{GLT\, 3}\!-\!\mathbf{GLT\, 5}$. The singular value and spectral distributions in (10.104) follow from (10.103) and GLT 1 as the Schur complements $S_n$ are symmetric. $\square$

Fig. 10.6

Comparison between the spectrum of $(n+1)S_n$ and the symbol $\varsigma (x,\theta )=\varsigma (\theta )$ for $n=50$ and $a(x)=1,\ \rho =1.2$

Example 10.5

Consider the constant-coefficient case $a(x)=1$. In this case, the symbol

$$\begin{aligned} \varsigma (x,\theta )=\frac{\rho }{3}(2+\cos \theta )+\frac{\sin ^2\theta }{2-2\cos \theta } \end{aligned}$$

does not depend on the physical variable x and, moreover, it is symmetric with respect to 0 in the Fourier variable $\theta$. This means that one of its  rearranged versions is simply its restriction to $[0,\pi ]$. In Fig. 10.6 we fixed $\rho =1.2$ and we plotted the graph of $\varsigma$ over $[0,\pi ]$ and the eigenvalues of $(n+1)S_n$ for $n=50$. The eigenvalues have been sorted so as to match as much as possible the graph of $\varsigma$ (i.e., in decreasing order) and have been placed at the points $(\frac{k\pi }{n},\lambda _k((n+1)S_n)),\ k=1,\ldots , n$. The figure shows an excellent agreement between the spectrum and the symbol, in accordance with the informal meaning behind the eigenvalue distribution $\{(n+1)S_n\}_n\sim _\lambda \varsigma$; see Remark 3.2. In particular, no outliers are observed.

Example 10.6

In the variable-coefficient case, i.e., when a(x) is not constant, the symbol $\varsigma (x,\theta )$ depends on the physical variable x but it is still symmetric with respect to 0 in the Fourier variable $\theta$, so one of its rearranged versions is its restriction to $[0, 1]\times [0,\pi ]$. However, instead of comparing the eigenvalues of $(n+1)S_n$ with either $\varsigma (x,\theta )$ or its restriction to $[0, 1]\times [0,\pi ]$, which are both bivariate and hence ‘complicated’, we compare the eigenvalues of $(n+1)S_n$ with the univariate non-decreasing rearranged version g obtained from the construction of Sect. 3.2 applied to the restriction $\varsigma (x,\theta ):[0, 1]\times [0,\pi ]\rightarrow \mathbb R$. In Fig. 10.7 we fixed $a(x)=1+\sqrt{x}$ and $\rho =3.7$ and we plotted the graph of g and the eigenvalues of $(n+1)S_n$ for $n=100$. The eigenvalues of $(n+1)S_n$ have been sorted in non-decreasing order and placed at the points $(\frac{k}{n},\lambda _k((n+1)S_n)),\ k=1,\ldots , n$. We note an excellent agreement, which further numerical experiments reveal to become perfect in the limit of mesh refinement $n\rightarrow \infty$.

Fig. 10.7

Comparison between the spectrum of $(n+1)S_n$ and the rearranged version g of the symbol $\varsigma (x,\theta )$ for $n=100$ and $a(x)=1+\sqrt{x},\ \rho =3.7$

10.7 Isogeometric Analysis Discretization of Differential Equations

Isogeometric analysis (IgA) is a modern and successful paradigm introduced in [34, 74] for analyzing problems governed by DEs. Its goal is to improve the connection between numerical simulation and computer-aided design (CAD) systems. The main idea in IgA is to use directly the geometry provided by CAD systems and to approximate the solutions of DEs by the same type of functions (usually, B-splines or NURBS). In this way, it is possible to save about 80% of the CPU time, which is normally employed in the translation between two different languages (e.g., between FEs and CAD or between FDs and CAD). In its original formulation [34, 74], IgA employs Galerkin discretizations, which are typical of the FE approach. In the Galerkin framework an efficient implementation requires special numerical  quadrature rules when constructing the resulting system of equations; see, e.g., [77]. To avoid this issue, isogeometric  collocation methods have been recently introduced in [3, 4]. Detailed comparisons with IgA Galerkin have shown the advantages of IgA collocation in terms of accuracy versus computational cost, in particular when higher-order approximation degrees are adopted [99]. Within the framework of IgA collocation, many applications have been successfully tackled, showing its potential and flexibility. Interested readers are referred to the recent review [93] and references therein. Section 10.7.1 is devoted to the isogeometric collocation approach, whereas the more traditional isogoemetric Galerkin methods will be addressed in Sects. 10.7.2–10.7.3.

10.7.1 B-Spline IgA Collocation Discretization of Convection-Diffusion-Reaction Equations

Consider the convection-diffusion-reaction problem

$$\begin{aligned} \left\{ \begin{array}{ll} -(a(x)u'(x))'+b(x)u'(x)+c(x)u(x) = f(x), &{}\quad x\in \varOmega ,\\ u(x)=0, &{}\quad x\in \partial \varOmega , \end{array}\right. \end{aligned}$$

(10.105)

where $\varOmega$ is a bounded open interval of $\mathbb R,\ a:\overline{\varOmega }\rightarrow \mathbb R$ is a function in $C^1(\overline{\varOmega })$ and $b, c, f:\overline{\varOmega }\rightarrow \mathbb R$ are functions in $C(\overline{\varOmega })$. We consider the isogeometric collocation approximation of (10.105) based on uniform B-splines of degree $p\ge 2$. Since this approximation technique is not as known as FDs or FEs, we describe it below in some detail. For more on IgA collocation methods, see [3, 4, 93].

Isogeometric collocation approximation.  Problem (10.105) can be reformulated as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} -a(x)u''(x)+s(x)u'(x)+c(x)u(x)=f(x), &{}\quad x\in \varOmega ,\\ u(x)=0, &{}\quad x\in \partial \varOmega , \end{array}\right. \end{aligned}$$

(10.106)

where $s(x)=b(x)-a'(x)$. In the standard collocation method, we choose a finite dimensional vector space $\mathscr {W}$, consisting of sufficiently smooth functions defined on $\overline{\varOmega }$ and vanishing on the boundary $\partial \varOmega$; we call $\mathscr {W}$ the approximation space. Then, we introduce a set of $N=\dim \mathscr {W}$  collocation points $\{\tau _1,\ldots ,\tau _N\}\subset \varOmega$ and we look for a function $u_\mathscr {W}\in \mathscr {W}$ satisfying the differential equation (10.106) at the points $\tau _i$, i.e.,

$$\begin{aligned} -a(\tau _i)u_{\mathscr {W}}''(\tau _i)+s(\tau _i)u_{\mathscr {W}}'(\tau _i)+c(\tau _i)u_{\mathscr {W}}(\tau _i)=f(\tau _i),\qquad i=1,\ldots , N. \end{aligned}$$

The function $u_\mathscr {W}$ is taken as an approximation to the solution u of (10.106). If $\{\varphi _1,\ldots ,\varphi _N\}$ is a basis of $\mathscr {W}$, then we have $u_\mathscr {W}=\sum _{j=1}^Nu_j\varphi _j$ for a unique vector $\mathbf {u}=(u_1,\ldots , u_N)^T$, and, by linearity, the computation of $u_\mathscr {W}$ (i.e., of $\mathbf {u}$) reduces to solving the linear system

$$\begin{aligned} A\mathbf {u}=\mathbf {f}, \end{aligned}$$

where $\mathbf {f}=\bigl [f(\tau _i)\bigr ]_{i=1}^N$ and

$$\begin{aligned} A&=\bigl [-a(\tau _i)\varphi _j''(\tau _i)+s(\tau _i)\varphi _j'(\tau _i)+c(\tau _i)\varphi _j(\tau _i)\bigr ]_{i, j=1}^N\nonumber \\&=\Bigl (\,\mathop {\mathrm{diag}}_{i=1,\ldots , N}a(\tau _i)\Bigr )\bigl [-\varphi _j''(\tau _i)\bigr ]_{i, j=1}^N+\Bigl (\,\mathop {\mathrm{diag}}_{i=1,\ldots , N}s(\tau _i)\Bigr )\bigl [\varphi _j'(\tau _i)\bigr ]_{i, j=1}^N\nonumber \\&\qquad +\Bigl (\,\mathop {\mathrm{diag}}_{i=1,\ldots , N}c(\tau _i)\Bigr )\bigl [\varphi _j(\tau _i)\bigr ]_{i, j=1}^N \end{aligned}$$

(10.107)

is the collocation matrix.

Now, suppose that the physical domain $\varOmega$ can be described by a global  geometry function $G:[0, 1]\rightarrow \overline{\varOmega }$, which is invertible and satisfies $G(\partial ([0, 1]))=\partial \overline{\varOmega }$. Let

$$\begin{aligned} \{\hat{\varphi }_1,\ldots ,\hat{\varphi }_N\} \end{aligned}$$

(10.108)

be a set of basis functions defined on the parametric (or reference) domain [0, 1] and vanishing on the boundary $\partial ([0, 1])$. Let

$$\begin{aligned} \{\hat{\tau }_1,\ldots ,\hat{\tau }_N\} \end{aligned}$$

(10.109)

be a set of N  collocation points in (0, 1). In the isogeometric collocation approach, we find an approximation $u_\mathscr {W}$ of u by using the standard collocation method described above, in which

• the approximation space is chosen as $\mathscr {W}=span (\varphi _1,\ldots \varphi _N)$, with

  $$\begin{aligned} \varphi _i(x)=\hat{\varphi }_i(G^{-1}(x))=\hat{\varphi }_i(\hat{x}), \qquad x=G(\hat{x}),\qquad i=1,\ldots , N, \end{aligned}$$

  (10.110)

• the collocation points in the physical domain $\varOmega$ are defined as

  $$\begin{aligned} \tau _i=G(\hat{\tau }_i),\qquad i=1,\ldots , N. \end{aligned}$$

  (10.111)

The resulting collocation matrix A is given by (10.107), with the basis functions $\varphi _i$ and the collocation points $\tau _i$ defined as in (10.110) and (10.111).

Assuming that G and $\hat{\varphi }_i$, $i=1,\ldots , N$, are sufficiently regular, we can apply  standard differential calculus to express A in terms of G and $\hat{\varphi }_i$, $\hat{\tau }_i$, $i=1,\ldots , N$. Let us work out this expression. For any $u:\overline{\varOmega }\rightarrow \mathbb R$, consider the corresponding function $\hat{u}:[0, 1]\rightarrow \mathbb R$, which is defined on the parametric domain by

$$\begin{aligned} \hat{u}(\hat{x})=u(x),\qquad x=G(\hat{x}). \end{aligned}$$

(10.112)

In other words, $\hat{u}(\hat{x})=u(G(\hat{x}))$.² Then, u satisfies (10.106) if and only if $\hat{u}$ satisfies the corresponding transformed problem

$$\begin{aligned} \left\{ \begin{array}{ll} -a_G(\hat{x})\hat{u}''(\hat{x})+s_G(\hat{x})\hat{u}'(\hat{x})+c_G(\hat{x})\hat{u}(\hat{x})=f(G(\hat{x})), &{}\quad \hat{x}\in (0, 1),\\ \hat{u}(\hat{x})=0, &{}\quad \hat{x}\in \partial ((0, 1)), \end{array}\right. \end{aligned}$$

(10.113)

where $a_G$, $s_G$, $c_G$ are, respectively, the transformed diffusion, convection,  reaction coefficient. They are given by

$$\begin{aligned} a_G(\hat{x})&=\frac{a(G(\hat{x}))}{(G'(\hat{x}))^2},\end{aligned}$$

(10.114)

$$\begin{aligned} s_G(\hat{x})&=\dfrac{a(G(\hat{x}))G''(\hat{x})}{(G'(\hat{x}))^3}+\dfrac{s(G(\hat{x}))}{G'(\hat{x})},\end{aligned}$$

(10.115)

$$\begin{aligned} c_G(\hat{x})&=c(G(\hat{x})), \end{aligned}$$

(10.116)

for $\hat{x}\in [0, 1]$. The collocation matrix A in (10.107) can be expressed in terms of G and $\hat{\varphi }_i$, $\hat{\tau }_i$, $i=1,\ldots , N$, as follows:

$$\begin{aligned} A&=\bigl [-a_G(\hat{\tau }_i)\hat{\varphi }_j''(\hat{\tau }_i)+s_G(\hat{\tau }_i)\hat{\varphi }_j'(\hat{\tau }_i)+c_G(\hat{\tau }_i)\hat{\varphi }_j(\hat{\tau }_i)\bigr ]_{i, j=1}^N\nonumber \\&=\Bigl (\,\mathop {\mathrm{diag}}_{i=1,\ldots , N}a_G(\hat{\tau }_i)\Bigr )\bigl [-\hat{\varphi }_j''(\hat{\tau }_i)\bigr ]_{i, j=1}^N+\Bigl (\,\mathop {\mathrm{diag}}_{i=1,\ldots , N}s_G(\hat{\tau }_i)\Bigr )\bigl [\hat{\varphi }_j'(\hat{\tau }_i)\bigr ]_{i, j=1}^N\nonumber \\&\qquad +\Bigl (\,\mathop {\mathrm{diag}}_{i=1,\ldots , N}c_G(\hat{\tau }_i)\Bigr )\bigl [\hat{\varphi }_j(\hat{\tau }_i)\bigr ]_{i, j=1}^N. \end{aligned}$$

(10.117)

In the IgA context, the geometry map G is expressed in terms of the functions $\hat{\varphi }_i$, in accordance with the isoparametric approach [34, Sect. 3.1]. Moreover, the functions $\hat{\varphi }_i$ themselves are usually B-splines or their rational versions, the so-called NURBS. In this section, the role of the $\hat{\varphi }_i$ will be played by B-splines over uniform knot sequences. Furthermore, we do not limit ourselves to the isoparametric approach, but we allow the geometry map G to be any sufficiently regular function from [0, 1] to $\overline{\varOmega }$, not necessarily expressed in terms of B-splines. Finally, following [3], the collocation points $\hat{\tau }_i$ will be chosen as the Greville abscissae corresponding to the B-splines $\hat{\varphi }_i$.

B-splines and Greville abscissae.  For $p, n\ge 1$, consider the uniform knot sequence

$$\begin{aligned} t_1=\cdots =t_{p+1}=0<t_{p+2}<\cdots<t_{p+n}<1=t_{p+n+1}=\cdots =t_{2p+n+1}, \end{aligned}$$

(10.118)

where

$$\begin{aligned} t_{i+p+1}=\frac{i}{n},\qquad i=0,\ldots , n. \end{aligned}$$

(10.119)

The B-splines of degree p on this knot sequence are denoted by

$$\begin{aligned} N_{i,[p]}:[0, 1]\rightarrow \mathbb R,\qquad i=1,\ldots , n+p, \end{aligned}$$

(10.120)

and are defined recursively as follows [37]: for $1\le i\le n+2p$,

$$\begin{aligned} N_{i,[0]}(t) = \chi _{[t_i,\, t_{i+1})}(t),\qquad t\in [0, 1]; \end{aligned}$$

(10.121)

for $1\le k\le p$ and $1\le i\le n+2p-k$,

$$\begin{aligned} N_{i,[k]}(t) = \frac{t-t_i}{t_{i+k}-t_i}\, N_{i,[k-1]}(t)+\frac{t_{i+k+1}-t}{t_{i+k+1}-t_{i+1}}\, N_{i+1,[k-1]}(t),\qquad t\in [0, 1], \end{aligned}$$

(10.122)

where we assume that a fraction with zero denominator is zero. The Greville abscissa $\xi _{i,[p]}$ associated with the B-spline $N_{i,[p]}$ is defined by

$$\begin{aligned} \xi _{i,[p]}=\frac{t_{i+1}+t_{i+2}+\ldots +t_{i+p}}{p},\qquad i=1,\ldots , n+p. \end{aligned}$$

(10.123)

We know from [37] that the functions $N_{1,[p]},\ldots , N_{n+p,[p]}$ belong to $C^{p-1}([0, 1])$ and form a basis for the spline space

$$\Bigl \{s\in C^{p-1}([0, 1]):\ \ s_{\left| \left[ \frac{i}{n},\frac{i+1}{n}\right) \right. }\in \mathbb P_p,\ \ i=0,\ldots , n-1\Bigr \},$$

where $\mathbb P_p$ is the space of polynomials of degree less than or equal to p. Moreover, $N_{1,[p]},\ldots , N_{n+p,[p]}$ possess the following properties [37].

• Local support property:

  $$\begin{aligned} supp (N_{i,[p]})=[t_i, t_{i+p+1}],\qquad i=1,\ldots , n+p. \end{aligned}$$

  (10.124)

• Vanishment on the boundary:

  $$\begin{aligned} N_{i,[p]}(0)=N_{i,[p]}(1)=0,\qquad i=2,\ldots , n+p-1. \end{aligned}$$

  (10.125)

• Nonnegative partition of unity:

  $$\begin{aligned} N_{i,[p]}(t)&\ge 0,\qquad t\in [0, 1],\qquad i=1,\ldots , n+p,\end{aligned}$$

  (10.126)
  $$\begin{aligned} \sum _{i=1}^{n+p}N_{i,[p]}(t)&=1,\qquad t\in [0, 1]. \end{aligned}$$

  (10.127)

• Bounds for derivatives:

  $$\begin{aligned} \sum _{i=1}^{n+p}|N_{i,[p]}'(t)|&\le 2pn,\qquad t\in [0, 1],\end{aligned}$$

  (10.128)
  $$\begin{aligned} \sum _{i=1}^{n+p}|N_{i,[p]}''(t)|&\le 4p(p-1)n^2,\qquad t\in [0, 1]. \end{aligned}$$

  (10.129)

  Note that the derivatives $N_{1,[p]}'(t),\ldots , N_{n+p,[p]}'(t)$ (resp., $N_{1,[p]}''(t),\ldots , N_{n+p,[p]}''(t)$) may not be defined at some of the points $\frac{1}{n},\ldots ,\frac{n-1}{n}$ when $p=1$ (resp., $p=1, 2$). In the summations (10.128) and (10.129), it is understood that the undefined values are counted as 0.

Let $\phi _{[q]}$ be the cardinal B-spline of degree $q\ge 0$ over the uniform knot sequence $\{0, 1,\ldots , q+1\}$, which is defined recursively as follows [37]:

$$\begin{aligned} \phi _{[0]}(t)&=\chi _{[0, 1)}(t),\qquad t\in \mathbb R, \end{aligned}$$

(10.130)

$$\begin{aligned} \phi _{[q]}(t)&=\frac{t}{q}\phi _{[q-1]}(t)+\frac{q+1-t}{q}\phi _{[q-1]}(t-1),\qquad t\in \mathbb R,\qquad q\ge 1. \end{aligned}$$

(10.131)

It is known from [32, 37] that $\phi _{[q]}\in C^{q-1}(\mathbb R)$ and

$$\begin{aligned} supp (\phi _{[q]})=[0, q+1]. \end{aligned}$$

(10.132)

Moreover, the following symmetry property holds by [56, Lemma 3] (see also [32, p. 86]):

$$\begin{aligned} \phi _{[q]}^{(r)}\Bigl (\frac{q+1}{2}+t\Bigr )=(-1)^r\phi _{[q]}^{(r)}\Bigl (\frac{q+1}{2}-t\Bigr ),\qquad t\in \mathbb R,\qquad r, q\ge 0, \end{aligned}$$

(10.133)

where $\phi _{[q]}^{(r)}$ is the rth derivative of $\phi _{[q]}$. Note that $\phi _{[q]}^{(r)}(t)$ is defined for all $t\in \mathbb R$ if $r<q$, and for all $t\in \mathbb R\backslash \{0, 1,\ldots , q+1\}$ if $r\ge q$. Nevertheless, (10.133) holds for all $t\in \mathbb R$, because when the left-hand side is not defined, the right-hand side is not defined as well. Concerning the $L^2$ inner products of derivatives of cardinal B-splines, it was proved in [56, Lemma 4] that

$$\begin{aligned} \begin{aligned} \int _{\mathbb R}\phi _{[q_1]}^{(r_1)}(t)\phi _{[q_2]}^{(r_2)}(t+\tau )\mathrm{d}t&= (-1)^{r_1}\phi _{[q_1+q_2+1]}^{(r_1+r_2)}(q_1+1+\tau )\\&= (-1)^{r_2}\phi _{[q_1+q_2+1]}^{(r_1+r_2)}(q_2+1-\tau ) \end{aligned} \end{aligned}$$

(10.134)

for every $\tau \in \mathbb R$ and every $q_1, q_2, r_1, r_2\ge 0$. Equation (10.134) is a property of the more general family of box splines [118] and generalizes the result appearing in [32, p. 89]. Cardinal B-splines are of interest herein, because the so-called central basis functions $N_{i,[p]}$, $i=p+1,\ldots , n$, are uniformly shifted and scaled versions of the cardinal B-spline $\phi _{[p]}$. This is illustrated in Figs. 10.8 and 10.9 for $p=3$. In formulas, we have

$$\begin{aligned} N_{i,[p]}(t)=\phi _{[p]}(nt-i+p+1),\qquad t\in [0, 1],\qquad i=p+1,\ldots , n, \end{aligned}$$

(10.135)

and, consequently,

$$\begin{aligned} N'_{i,[p]}(t)&=n\,\phi _{[p]}'(nt-i+p+1),\qquad t\in [0, 1],\qquad i=p+1,\ldots , n,\end{aligned}$$

(10.136)

$$\begin{aligned} \quad N_{i,[p]}''(t)&=n^2\phi _{[p]}''(nt-i+p+1),\qquad t\in [0, 1],\qquad i=p+1,\ldots , n. \end{aligned}$$

(10.137)

Fig. 10.8

Graph of the B-splines $N_{i,[p]}$, $i=1,\ldots , n+p$, for $p=3$ and $n=10$

Fig. 10.9

Graph of the cubic cardinal B-spline $\phi _{[3]}$

Remark 10.10

For degree $p=1$, the central B-spline basis functions $N_{2,[1]},\ldots , N_{n,[1]}$ are the hat-functions $\varphi _1,\ldots ,\varphi _{n-1}$ corresponding to the grid points

$$\begin{aligned} x_i=ih,\qquad i=0,\ldots , n,\qquad h=\frac{1}{n}. \end{aligned}$$

To see this, simply write (10.122) for $p=1$ and compare it with (10.85). The graph of $N_{2,[1]},\ldots , N_{n,[1]}$ for $n=10$ is depicted in Fig. 10.4.

In view of (10.123) and (10.124), the Greville abscissa $\xi _{i,[p]}$ lies in the support of $N_{i,[p]}$,

$$\begin{aligned} \xi _{i,[p]}\in supp (N_{i,[p]})=[t_i, t_{i+p+1}],\qquad i=1,\ldots , n+p. \end{aligned}$$

(10.138)

The central Greville abscissae $\xi _{i,[p]}$, $i=p+1,\ldots , n$, which are the Greville abscissae associated with the central basis functions (10.135), simplify to

$$\begin{aligned} \xi _{i,[p]}=\frac{i}{n}-\frac{p+1}{2n},\qquad i=p+1,\ldots , n. \end{aligned}$$

(10.139)

The Greville abscissae are somehow equivalent, in an asymptotic sense, to the uniform knots in [0, 1]. More precisely,

$$\begin{aligned} \Bigl |\xi _{i,[p]}-\frac{i}{n+p}\Bigr |\le \frac{C_p}{n},\qquad i=1,\ldots , n+p, \end{aligned}$$

(10.140)

where $C_p$ depends only on p. The proof of (10.140) is a matter of straightforward computations; we leave the details to the reader.

B-spline IgA collocation matrices. In the IgA collocation approach based on (uniform) B-splines, the basis functions $\hat{\varphi }_1,\ldots ,\hat{\varphi }_N$ in (10.108) are chosen as the B-splines $N_{2,[p]},\ldots , N_{n+p-1,[p]}$ in (10.120), i.e.,

$$\begin{aligned} \hat{\varphi }_i=N_{i+1,[p]},\qquad i=1,\ldots , n+p-2. \end{aligned}$$

(10.141)

In this setting, $N=n+p-2$. Note that the boundary functions $N_{1,[p]}$ and $N_{n+p,[p]}$ are excluded because they do not vanish on the boundary $\partial ([0, 1])$; see also Fig. 10.8. As for the collocation points $\hat{\tau }_1\ldots ,\hat{\tau }_N$ in (10.109), they are chosen as the Greville abscissae $\xi _{2,[p]},\ldots ,\xi _{n+p-1,[p]}$ in (10.123), i.e.,

$$\begin{aligned} \hat{\tau }_i=\xi _{i+1,[p]},\qquad i=1,\ldots , n+p-2. \end{aligned}$$

(10.142)

In what follows we assume $p\ge 2$, so as to ensure that $N_{j+1,[p]}''(\xi _{i+1,[p]})$ is defined for all $i, j=1,\ldots , n+p-2$. The collocation matrix (10.117) resulting from the choices of $\hat{\varphi }_i,\hat{\tau }_i$ as in (10.141) and (10.142) will be denoted by $A_{G, n}^{[p]}$, in order to emphasize its dependence on the geometry map G and the parameters n, p:

$$\begin{aligned} A_{G, n}^{[p]}&=\Bigl [-a_G(\xi _{i+1,[p]})N_{j+1,[p]}''(\xi _{i+1,[p]})+s_G(\xi _{i+1,[p]})N_{j+1,[p]}'(\xi _{i+1,[p]})\\&\qquad +c_G(\xi _{i+1,[p]})N_{j+1,[p]}(\xi _{i+1,[p]})\Bigr ]_{i, j=1}^{n+p-2}\\&=D_n^{[p]}(a_G)\, K_n^{[p]}+D_n^{[p]}(s_G)\, H_n^{[p]}+D_n^{[p]}(c_G)\, M_n^{[p]}, \end{aligned}$$

where

$$\begin{aligned} D_n^{[p]}(v)=\mathop {\mathrm{diag}}_{i=1,\ldots , n+p-2}v(\xi _{i+1,[p]}) \end{aligned}$$

is the diagonal sampling matrix containing the samples of the function $v:[0, 1]\rightarrow \mathbb R$ at the Greville abscissae, and

$$\begin{aligned} K_n^{[p]}&=\bigl [-N_{j+1,[p]}''(\xi _{i+1,[p]})\bigr ]_{i, j=1}^{n+p-2},\\ H_n^{[p]}&=\bigl [N_{j+1,[p]}'(\xi _{i+1,[p]})\bigr ]_{i, j=1}^{n+p-2},\\ M_n^{[p]}&=\bigl [N_{j+1,[p]}(\xi _{i+1,[p]})\bigr ]_{i, j=1}^{n+p-2}. \end{aligned}$$

Note that $A_{G, n}^{[p]}$ can be decomposed as follows:

$$\begin{aligned} A_{G, n}^{[p]}=K_{G, n}^{[p]}+Z_{G, n}^{[p]}, \end{aligned}$$

where

$$\begin{aligned} K_{G, n}^{[p]}=\left[ -a_G(\xi _{i+1,[p]})N_{j+1,[p]}''(\xi _{i+1,[p]})\right] _{i, j=1}^{n+p-2}=D_n^{[p]}(a_G)\, K_n^{[p]} \end{aligned}$$

is the diffusion matrix, i.e., the matrix resulting from the discretization of the higher-order (diffusion) term in (10.106), and

$$\begin{aligned} Z_{G, n}^{[p]}&=\left[ s_G(\xi _{i+1,[p]})N_{j+1,[p]}'(\xi _{i+1,[p]})+c_G(\xi _{i+1,[p]})N_{j+1,[p]}(\xi _{i+1,[p]})\right] _{i, j=1}^{n+p-2}\\&=D_n^{[p]}(s_G)\, H_n^{[p]}+D_n^{[p]}(c_G)\, M_n^{[p]} \end{aligned}$$

is the matrix resulting from the discretization of the terms in (10.106) with lower-order derivatives (i.e., the convection and reaction terms). As already noticed in the previous sections about FD and FE discretizations, the matrix $Z_{G, n}^{[p]}$ can be regarded as a ‘residual term’, since it comes from the discretization of the lower-order differential operators. Indeed, we shall see that the norm of $Z_{G, n}^{[p]}$ is negligible with respect to the norm of the diffusion matrix $K_{G, n}^{[p]}$ when the discretization parameter n is large, because, after normalization by $n^2$, it will turn out that $\Vert n^{-2}Z_{G, n}^{[p]}\Vert$ tends to 0 as $n\rightarrow \infty$ (contrary to $\Vert n^{-2}K_{G, n}^{[p]}\Vert$, which remains bounded away from 0 and $\infty$).

Let us now provide an approximate construction of $K_n^{[p]}$, $M_n^{[p]}$, $H_n^{[p]}$. This is necessary for the GLT analysis of this section. We only construct the submatrices

$$\begin{aligned} \bigl [(K_n^{[p]})_{ij}\bigr ]_{i, j=p}^{n-1},\qquad \bigl [(H_n^{[p]})_{ij}\bigr ]_{i, j=p}^{n-1},\qquad \bigl [(M_n^{[p]})_{ij}\bigr ]_{i, j=p}^{n-1}, \end{aligned}$$

(10.143)

which are determined by the central basis functions (10.135) and by the central Greville abscissae (10.139). Note that the submatrix $[(K_n^{[p]})_{ij}]_{i, j=p}^{n-1}$, when embedded in any matrix of size $n+p-2$ at the right place (identified by the row and column indices $p,\ldots , n-1$), provides an approximation of $K_n^{[p]}$ up to a low-rank correction. A similar consideration also applies to the submatrices $[(H_n^{[p]})_{ij}]_{i, j=p}^{n-1}$ and $[(M_n^{[p]})_{ij}]_{i, j=p}^{n-1}$. A direct computation based on (10.133), (10.135)–(10.137) and (10.139) shows that, for $i, j=p,\ldots , n-1$,

$$\begin{aligned} (K_n^{[p]})_{ij}&=-n^2\phi _{[p]}''\Bigl (\frac{p+1}{2}+i-j\Bigr )=-n^2\phi _{[p]}''\Bigl (\frac{p+1}{2}-i+j\Bigr ),\\ (H_n^{[p]})_{ij}&=n\,\phi _{[p]}'\Bigl (\frac{p+1}{2}+i-j\Bigr )=-n\,\phi _{[p]}'\Bigl (\frac{p+1}{2}-i+j\Bigr ),\\ (M_n^{[p]})_{ij}&=\phi _{[p]}\Bigl (\frac{p+1}{2}+i-j\Bigr )=\phi _{[p]}\Bigl (\frac{p+1}{2}-i+j\Bigr ). \end{aligned}$$

Since their entries depend only on the difference $i-j$, the submatrices (10.143) are Toeplitz matrices, and precisely

$$\begin{aligned} \bigl [(K_n^{[p]})_{ij}\bigr ]_{i, j=p}^{n-1}&=n^2\left[ -\phi _{[p]}''\Bigl (\frac{p+1}{2}-i+j\Bigr )\right] _{i, j=p}^{n-1}=n^2\, T_{n-p}(f_p),\end{aligned}$$

(10.144)

$$\begin{aligned} \bigl [(H_n^{[p]})_{ij}\bigr ]_{i, j=p}^{n-1}&=n\left[ -\phi _{[p]}'\Bigl (\frac{p+1}{2}-i+j\Bigr )\right] _{i, j=p}^{n-1}=n\, I\, T_{n-p}(g_p),\end{aligned}$$

(10.145)

$$\begin{aligned} \bigl [(M_n^{[p]})_{ij}\bigr ]_{i, j=p}^{n-1}&=\left[ \phi _{[p]}\Bigl (\frac{p+1}{2}-i+j\Bigr )\right] _{i, j=p}^{n-1}=T_{n-p}(h_p), \end{aligned}$$

(10.146)

where

$$\begin{aligned} f_p(\theta )&=\sum _{k\in \mathbb Z}-\phi _{[p]}''\Bigl (\frac{p+1}{2}-k\Bigr )\,\mathrm{e}^{Ik\theta }\nonumber \\&=-\phi _{[p]}''\Bigl (\frac{p+1}{2}\Bigr )-2\sum _{k=1}^{\lfloor p/2\rfloor }\phi _{[p]}''\Bigl (\frac{p+1}{2}-k\Bigr )\cos (k\theta ),\end{aligned}$$

(10.147)

$$\begin{aligned} g_p(\theta )&=-I\sum _{k\in \mathbb Z}-\phi _{[p]}'\Bigl (\frac{p+1}{2}-k\Bigr )\,\mathrm{e}^{Ik\theta }\nonumber \\&=-2\sum _{k=1}^{\lfloor p/2\rfloor }\phi _{[p]}'\Bigl (\frac{p+1}{2}-k\Bigr )\sin (k\theta ),\end{aligned}$$

(10.148)

$$\begin{aligned} h_p(\theta )&=\sum _{k\in \mathbb Z}\phi _{[p]}\Bigl (\frac{p+1}{2}-k\Bigr )\,\mathrm{e}^{Ik\theta }\nonumber \\&=\phi _{[p]}\Bigl (\frac{p+1}{2}\Bigr )+2\sum _{k=1}^{\lfloor p/2\rfloor }\phi _{[p]}\Bigl (\frac{p+1}{2}-k\Bigr )\cos (k\theta ); \end{aligned}$$

(10.149)

note that we used (10.132) and (10.133) to simplify the expressions of $f_p(\theta )$, $g_p(\theta )$, $h_p(\theta )$. It follows from (10.144) that $T_{n-p}(f_p)$ is the principal submatrix of both $n^{-2}K_n^{[p]}$ and $T_{n+p-2}(f_p)$ corresponding to the set of indices $p,\ldots , n-1$. Similar results follow from (10.145) and (10.146), and so we obtain

$$\begin{aligned} n^{-2}K_n^{[p]}&=T_{n+p-2}(f_p)+R_n^{[p]},&\mathrm{rank}(R_n^{[p]})&\le 4(p-1),\end{aligned}$$

(10.150)

$$\begin{aligned} -\mathrm{i}\, n^{-1}H_n^{[p]}&=T_{n+p-2}(g_p)+S_n^{[p]},&\mathrm{rank}(S_n^{[p]})&\le 4(p-1),\end{aligned}$$

(10.151)

$$\begin{aligned} M_n^{[p]}&=T_{n+p-2}(h_p)+V_n^{[p]},&\mathrm{rank}(V_n^{[p]})&\le 4(p-1). \end{aligned}$$

(10.152)

To better appreciate the above construction, let us see two examples. We only consider the case of the matrix $K_n^{[p]}$ because for $H_n^{[p]}$ and $M_n^{[p]}$ the situation is the same. In the first example, we fix $p=3$. The matrix $K_n^{[3]}$ is given by

The submatrix $T_{n-2}(f_3)$ appears in correspondence of the highlighted box and we have

$$\begin{aligned} f_3(\theta )=\frac{1}{6}(-6\mathrm{e}^{I\theta }+12-6\mathrm{e}^{-I\theta })=2-2\cos \theta , \end{aligned}$$

as given by (10.147) for $p=3$. In the second example, we fix $p=4$. The matrix $K_n^{[4]}$ is given by

The submatrix $T_{n-3}(f_4)$ appears in correspondence of the highlighted box and we have

$$\begin{aligned} f_4(\theta )&=\frac{1}{96}(-12\mathrm{e}^{2I\theta }-48\mathrm{e}^{I\theta }+120-48\mathrm{e}^{-I\theta }-12\mathrm{e}^{-2I\theta }) \\&=\frac{5}{4}-\cos \theta -\frac{1}{4}\cos (2\theta ), \end{aligned}$$

as given by (10.147) for $p=4$.

Before passing to the GLT analysis of the collocation matrices $A_{G, n}^{[p]}$, we prove the existence of an n-independent bound for the spectral norms of $n^{-2}K_n^{[p]}$, $n^{-1}H_n^{[p]}$, $M_n^{[p]}$. Actually, one could also prove that the components of $n^{-2}K_n^{[p]}$, $n^{-1}H_n^{[p]}$, $M_n^{[p]}$ do not depend on n as illustrated above for the matrix $n^{-2}K_n^{[p]}$ in the cases $p=3, 4$. However, for our purposes it suffices to show that, for every $p\ge 2$, there exists a constant $C^{[p]}$ such that, for all n,

$$\begin{aligned} \Vert n^{-2}K_n^{[p]}\Vert \le C^{[p]},\qquad \Vert n^{-1}H_n^{[p]}\Vert \le C^{[p]},\qquad \Vert M_n^{[p]}\Vert \le C^{[p]}. \end{aligned}$$

(10.153)

To prove (10.153), we note that $K_n^{[p]}$, $H_n^{[p]}$, $M_n^{[p]}$ are banded, with bandwidth bounded by $2p+1$. Indeed, if $|i-j|>p$, one can show that $(K_n^{[p]})_{ij}=(H_n^{[p]})_{ij}=(M_n^{[p]})_{ij}{=0}$ by using (10.138), which implies that $\xi _{i+1,[p]}$ lies outside or on the border of $\mathrm{supp}(N_{j+1,[p]})$, whose intersection with $\mathrm{supp}(N_{i+1,[p]})$ consists of at most one of the knots $t_k$. Moreover, by (10.126)–(10.129), for all $i, j=1,\ldots , n+p-2$ we have

$$\begin{aligned} |(K_n^{[p]})_{ij}|&=|N_{j+1,[p]}''(\xi _{i+1,[p]})|\le 4p(p-1)n^2,\\ |(H_n^{[p]})_{ij}|&=|N_{j+1,[p]}'(\xi _{i+1,[p]})|\le 2pn,\\ |(M_n^{[p]})_{ij}|&=|N_{j+1,[p]}(\xi _{i+1,[p]})|\le 1. \end{aligned}$$

Hence, (10.153) follows from (2.31).

GLT analysis of the B-spline IgA collocation matrices. Assuming that the geometry map G possesses some regularity properties, we show that, for any $p\ge 2$, the sequence of normalized IgA collocation matrices $\{n^{-2}A_{G, n}^{[p]}\}_n$ is a GLT sequence whose symbol describes both its singular value and spectral distribution.

Theorem 10.14

Let $\varOmega$ be a bounded open interval of $\mathbb R$, let $a\in C^1(\overline{\varOmega })$ and $b, c\in C(\overline{\varOmega })$. Let $p\ge 2$ and let $G:[0, 1]\rightarrow \overline{\varOmega }$ be such that $G\in C^2([0, 1])$ and $G'(\hat{x})\ne 0$ for all $\hat{x}\in [0, 1]$. Then

$$\begin{aligned} \{n^{-2}A_{G, n}^{[p]}\}_n\sim _\mathrm{GLT}f_{G, p} \end{aligned}$$

(10.154)

and

$$\begin{aligned} \{n^{-2}A_{G, n}^{[p]}\}_n\sim _{\sigma ,\,\lambda }f_{G, p}, \end{aligned}$$

(10.155)

where

$$\begin{aligned} f_{G, p}(\hat{x},\theta )=a_G(\hat{x})f_p(\theta )=\frac{a(G(\hat{x}))}{(G'(\hat{x}))^2}f_p(\theta ) \end{aligned}$$

(10.156)

and $f_p(\theta )$ is defined in (10.147).

Proof

The proof consists of the following steps. Throughout the proof, the letter C will denote a generic constant independent of n.

Step 1. We show that

$$\begin{aligned} \Vert n^{-2}K_{G, n}^{[p]}\Vert \le C \end{aligned}$$

(10.157)

and

$$\begin{aligned} \Vert n^{-2}Z_{G, n}^{[p]}\Vert \le C/n. \end{aligned}$$

(10.158)

To prove (10.157), it suffices to use the regularity of G and (10.153):

$$\begin{aligned} \Vert n^{-2}K_{G, n}^{[p]}\Vert =\Vert n^{-2}D_n^{[p]}(a_G)K_n^{[p]}\Vert \le \Vert a_G\Vert _\infty C^{[p]}\le \frac{C^{[p]}\Vert a\Vert _\infty }{\min _{\hat{x}\in [0, 1]}|G'(\hat{x})|^2}. \end{aligned}$$

The proof of (10.158) is similar. It suffices to use the fact that $G\in C^2([0, 1])$ and (10.153):

$$\begin{aligned} \Vert n^{-2}Z_{G, n}^{[p]}\Vert&=\Vert n^{-2}D_n^{[p]}(s_G)H_n^{[p]}+n^{-2}D_n^{[p]}(c_G)M_n^{[p]}\Vert \\&\le n^{-1}C^{[p]}\biggl (\frac{\Vert a\Vert _\infty \Vert G''\Vert _\infty }{\min _{\hat{x}\in [0, 1]}|G'(\hat{x})|^3}+\frac{\Vert a'\Vert _\infty +\Vert b\Vert _\infty }{\min _{\hat{x}\in [0, 1]}|G'(\hat{x})|}\biggr )+n^{-2}C^{[p]}\Vert c\Vert _\infty . \end{aligned}$$

Step 2. Define the symmetric matrix

$$\begin{aligned} \tilde{K}_{G, n}^{[p]}=S_{n+p-2}(a_G)\circ n^2\, T_{n+p-2}(f_p), \end{aligned}$$

(10.159)

where we recall that $S_m(v)$ is the mth arrow-shaped sampling matrix generated by v (see (10.26)), and consider the following decomposition of $n^{-2}A_{G, n}^{[p]}$:

$$\begin{aligned} n^{-2}A_{G, n}^{[p]}=n^{-2}\tilde{K}_{G, n}^{[p]}+\bigl (n^{-2}K_{G, n}^{[p]}-n^{-2}\tilde{K}_{G, n}^{[p]}\bigr )+n^{-2}Z_{G, n}^{[p]}. \end{aligned}$$

(10.160)

We know from Theorem 10.4 that $\Vert n^{-2}\tilde{K}_{G, n}^{[p]}\Vert \le C$ and $\{n^{-2}\tilde{K}_{G, n}^{[p]}\}_n\sim _\mathrm{GLT}f_{G, p}(\hat{x},\theta )$.

Step 3. We show that

$$\begin{aligned} \Vert n^{-2}K_{G, n}^{[p]}-n^{-2}\tilde{K}_{G, n}^{[p]}\Vert _1=o(n). \end{aligned}$$

(10.161)

Once this is done, the thesis is proved. Indeed, from (10.161) and (10.158) we obtain

$$\begin{aligned}&\Vert (n^{-2}K_{G, n}^{[p]}-n^{-2}\tilde{K}_{G, n}^{[p]})+n^{-2}Z_{G, n}^{[p]}\Vert _1\\&\le \Vert n^{-2}K_{G, n}^{[p]}-n^{-2}\tilde{K}_{G, n}^{[p]}\Vert _1+\Vert n^{-2}Z_{G, n}^{[p]}\Vert n=o(n), \end{aligned}$$

hence $\{(n^{-2}K_{G, n}^{[p]}-n^{-2}\tilde{K}_{G, n}^{[p]})+n^{-2}Z_{G, n}^{[p]}\}_n$ is zero-distributed by Z 2. Thus, the GLT relation (10.154) follows from the decomposition (10.160) and $\mathbf{GLT\, 3}\!-\!\mathbf{GLT\, 4}$, the singular value distribution in (10.155) follows from GLT 1, and the eigenvalue distribution in (10.155) follows from GLT 2.

To prove (10.161), we decompose the difference $n^{-2}K_{G, n}^{[p]}-n^{-2}\tilde{K}_{G, n}^{[p]}$ as follows:

$$\begin{aligned}&n^{-2}K_{G, n}^{[p]}-n^{-2}\tilde{K}_{G, n}^{[p]}=n^{-2}D_n^{[p]}(a_G)\, K_n^{[p]}-S_{n+p-2}(a_G)\circ T_{n+p-2}(f_p)\nonumber \\&=n^{-2}D_n^{[p]}(a_G)\, K_n^{[p]}-D_n^{[p]}(a_G)\, T_{n+p-2}(f_p)\end{aligned}$$

(10.162)

$$\begin{aligned}&\quad \,\,+D_n^{[p]}(a_G)\, T_{n+p-2}(f_p)-D_{n+p-2}(a_G)\, T_{n+p-2}(f_p)\end{aligned}$$

(10.163)

$$\begin{aligned}&\quad \,\,+D_{n+p-2}(a_G)\, T_{n+p-2}(f_p)-S_{n+p-2}(a_G)\circ T_{n+p-2}(f_p). \end{aligned}$$

(10.164)

We consider separately the three matrices in (10.162)–(10.164) and we show that their trace-norms are o(n).

• By (10.150), the rank of the matrix (10.162) is bounded by $4(p-1)$. By the regularity of G, the inequality (10.153) and T 3, the spectral norm of (10.162) is bounded by C. Thus, the trace-norm of (10.162) is o(n) (actually, O(1)) by (2.42).

• By (10.140), the continuity of $a_G$ and T 3, the spectral norm of the matrix (10.163) is bounded by $C\omega _{a_G}(n^{-1})$, so it tends to 0. Hence, the trace-norm of (10.163) is o(n) by (2.42).

• By Theorem 10.4, the spectral norm of the matrix (10.164) is bounded by $C\omega _{a_G}(n^{-1})$, so it tends to 0. Hence, the trace-norm of (10.164) is o(n) by (2.42).

In conclusion, $\Vert n^{-2}K_{G, n}^{[p]}-n^{-2}\tilde{K}_{G, n}^{[p]}\Vert _1=o(n)$. $\square$

Fig. 10.10

Graph of $f_p/M_{f_p}$ for $p=2,\ldots , 10$

Remark 10.11

(formal structure of the symbol) We invite  the reader to compare the symbol (10.156) with the transformed problem (10.113). It is clear that the higher-order  operator $-a_G(\hat{x})\hat{u}''(\hat{x})$ has a discrete spectral counterpart $a_G(\hat{x})f_p(\theta )$ which looks formally the same (as in the FD and FE cases; see Remarks 10.1 and 10.9). To better appreciate the formal analogy, note that $f_p(\theta )$ is the trigonometric polynomial in the Fourier variable coming from the B-spline IgA collocation discretization of the second derivative $-\hat{u}''(\hat{x})$ on the parametric domain [0, 1]. Indeed, $f_p(\theta )$ is the symbol of the sequence of B-spline IgA collocation diffusion matrices $\{n^{-2}K_n^{[p]}\}_n$, which arises from the B-spline IgA collocation approximation of (10.106) in the case where $a(x)=1$, $b(x)=c(x)=0$ identically, $\varOmega =(0, 1)$ and G is the identity map over [0, 1]; note that in this case (10.106) is the same as (10.113), $x=G(\hat{x})=\hat{x}$ and $u=\hat{u}$.

Remark 10.12

(nonnegativity and order of the zero at $\theta =0$) Fig. 10.10 shows  the graph of $f_p(\theta )$ normalized by its maximum $M_{f_p}=\max _{\theta \in [-\pi ,\pi ]}f_p(\theta )$ for $p=2,\ldots , 10$. Note that $f_2(\theta )=f_3(\theta )=2-2\cos \theta$. We see from the figure (and it was proved in [41]) that $f_p(\theta )$ is nonnegative over $[-\pi ,\pi ]$ and has a unique zero of order 2 at $\theta =0$ because

$$\lim _{\theta \rightarrow 0}\frac{f_p(\theta )}{\theta ^2}=1.$$

This reflects the fact that, as observed in Remark 10.11, $f_p(\theta )$ arises from the B-spline IgA collocation discretization of the second derivative $-\hat{u}''(\hat{x})$ on the parametric domain [0, 1], which is a differential operator of order 2 (and it is nonnegative on $\{v\in C^2([0, 1]):\ v(0)=v(1)=0\}$); see also Remarks 10.2, 10.5 and 10.6.

Further properties of the functions $f_p(\theta )$, $g_p(\theta )$, $h_p(\theta )$ can be found in [41, Sect. 3]. In particular, it was proved therein that $f_p(\pi )/M_{f_p}\rightarrow 0$ exponentially as $p\rightarrow \infty$. Moreover, observing that $h_p(\theta )$ is defined by (10.149) for all $p\ge 0$ (and we have $h_0(\theta )=h_1(\theta )=1$ identically) provided that we use the standard convention that an empty sum like $\sum _{k=1}^0\phi _{[1]}(1-k)\cos (k\theta )$ equals 0,³ it was proved in [41] that, for all $p\ge 2$ and $\theta \in [-\pi ,\pi ]$,

$$\begin{aligned} f_p(\theta )=(2-2\cos \theta )h_{p-2}(\theta ),\quad \ \left( \frac{2}{\pi }\right) ^{p-1}\le h_{p-2}(\theta )\le h_{p-2}(0)=1. \end{aligned}$$

(10.165)

Fig. 10.11

Comparison between the spectrum of $n^{-2}A_{G, n}^{[p]}$ and the symbol $f_{G, p}(\hat{x},\theta )=f_p(\theta )$ for $p=8$, $n=75$ and $\varOmega =(0, 1)$, $G=\text {identity\ map}$, $a(x)=1$, $b(x)=c(x)=0$

Example 10.7

Consider the case where $\varOmega =(0, 1)$ and G is the identity map on $[0, 1]=\overline{\varOmega }$. Fix $a(x)=1$ and $b(x)=c(x)=0$ identically. In this situation, the symbol

$$\begin{aligned} f_{G, p}(\hat{x},\theta )=f_p(\theta ) \end{aligned}$$

only depends on the Fourier variable $\theta$ and is symmetric with respect to $\theta =0$. This means that a rearranged version of $f_{G, p}(\hat{x},\theta )$ is simply $f_p(\theta )$ regarded as a function from $[0,\pi ]$ to $\mathbb R$. In Fig. 10.11 we plotted the graph of $f_p(\theta )$ over $[0,\pi ]$ and the smallest $n-2$ eigenvalues of $n^{-2}A_{G, n}^{[p]}$ for $p=8$ and $n=75$. The eigenvalues, which turn out to be real despite the nonsymmetry of the matrix, have been sorted so as to match as much as possible the graph of $f_p(\theta )$ and have been placed at the points $(\frac{k\pi }{n-2^{}},\lambda _k(n^{-2}A_{G, n}^{[p]})),\ k=1,\ldots , n-2$. The figure shows an excellent agreement between the spectrum and the symbol, in accordance with the informal meaning behind the eigenvalue distribution $\{n^{-2}A_{G, n}^{[p]}\}_n\sim _\lambda f_p$; see Remark 3.2. The largest 8 eigenvalues of $n^{-2}A_{G, n}^{[p]}$ have not been plotted in Fig. 10.11 because they are outliers: they do not match the graph of $f_p(\theta )$. The 8 outliers are pairwise equal and their values are approximately given by 1.3277, 1.3277, 2.5144, 2.5144, 6.6862, 6.6862, 31.5746, 31.5746. Further numerical experiments reveal that the outliers substantially do not change with increasing n as their number remains equal to 8 and their values remain approximately equal to 1.3277, 1.3277, 2.5144, 2.5144, 6.6862, 6.6862, 31.5746, 31.5746.

10.7.2 Galerkin B-Spline IgA Discretization of Convection-Diffusion-Reaction Equations

Consider the convection-diffusion-reaction problem

$$\begin{aligned} \left\{ \begin{array}{ll} -(a(x)u'(x))'+b(x)u'(x)+c(x)u(x) = f(x), &{}\quad x\in \varOmega ,\\ u(x)=0, &{}\quad x\in \partial \varOmega , \end{array}\right. \end{aligned}$$

(10.166)

where $\varOmega$ is a bounded open interval of $\mathbb R$, $f\in L^2(\varOmega )$ and $a, b, c\in L^\infty (\varOmega )$. Problem (10.166) is the same as (10.105), except for the assumptions on a, b, c, f. We consider the isogeometric Galerkin approximation of (10.166) based on uniform B-splines of degree $p\ge 1$. This approximation technique is described below in some detail. For more on IgA Galerkin methods, see [34, 74].

Isogeometric Galerkin approximation. The  weak form of (10.166) reads as follows: find $u\in H^1_0(\varOmega )$ such that

$$\begin{aligned} a (u, v)=f (v),\qquad \forall \, v\in H_0^1(\varOmega ), \end{aligned}$$

where

$$\begin{aligned} a (u, v)&=\int _\varOmega \bigl (a(x)u'(x)v'(x)+b(x)u'(x)v(x)+c(x)u(x)v(x)\bigr )\mathrm{d}x,\\ f (v)&=\int _\varOmega f(x)v(x)\mathrm{d}x. \end{aligned}$$

In the standard Galerkin method, we look for an approximation $u_{\mathscr {W}}$ of u by choosing a finite dimensional vector space $\mathscr {W}\subset H^1_0(\varOmega )$, the so-called approximation space, and by solving the following (Galerkin) problem: find $u_{\mathscr {W}}\in {\mathscr {W}}$ such that

$$\begin{aligned} a (u_{\mathscr {W}}, v)=f (v),\qquad \forall \, v\in \mathscr {W}. \end{aligned}$$

If $\{\varphi _1,\ldots ,\varphi _N\}$ is a basis of ${\mathscr {W}}$, then we can write $u_{\mathscr {W}}=\sum _{j=1}^N u_j\varphi _j$ for a unique vector $\mathbf {u}=(u_1,\ldots , u_N)^T$, and, by linearity, the computation of $u_{\mathscr {W}}$ (i.e., of $\mathbf {u}$) reduces to solving the linear system

$$\begin{aligned} A\mathbf {u}=\mathbf {f}, \end{aligned}$$

where $\mathbf {f}=\bigl [f (\varphi _i)\bigr ]_{i=1}^N$ and

$$\begin{aligned} A&=\left[ a (\varphi _j,\varphi _i)\right] _{i, j=1}^N\nonumber \\&=\left[ \int _\varOmega \bigl (a(x)\varphi _j'(x)\varphi _i'(x)+b(x)\varphi _j'(x)\varphi _i(x)+c(x)\varphi _j(x)\varphi _i(x)\bigr )\mathrm{d}x\right] _{i, j=1}^N \end{aligned}$$

(10.167)

is the stiffness matrix.

Now, suppose that the physical domain $\varOmega$ can be described by a global  geometry function $G:[0, 1]\rightarrow \overline{\varOmega }$, which is invertible and satisfies $G(\partial ([0, 1]))=\partial \overline{\varOmega }$. Let $\{\hat{\varphi }_1,\ldots ,\hat{\varphi }_N\}$ be a set of basis functions defined on the parametric (or reference) domain [0, 1] and vanishing on the boundary $\partial ([0, 1])$. In the isogeometric Galerkin approach, we find an approximation $u_{\mathscr {W}}$ of u by using the standard Galerkin method, in which the approximation space is chosen as ${\mathscr {W}}=span (\varphi _1,\ldots ,\varphi _N)$, where

$$\begin{aligned} \varphi _i(x)=\hat{\varphi }_i(G^{-1}(x))=\hat{\varphi }_i(\hat{x}), \qquad x=G(\hat{x}). \end{aligned}$$

(10.168)

The resulting stiffness matrix A is given by (10.167), with the basis functions $\varphi _i$ defined as in (10.168). Assuming that G and $\hat{\varphi }_i$, $i=1,\ldots , N$, are sufficiently regular, we can apply  standard differential calculus to obtain the following expression for A in terms of G and $\hat{\varphi }_i$, $i=1,\ldots , N$:

$$\begin{aligned} A=\biggl [\int _{[0, 1]}&\Bigl (a_G(\hat{x})\hat{\varphi }_j'(\hat{x})\hat{\varphi }_i'(\hat{x})+\frac{b(G(\hat{x}))}{G'(\hat{x})}\hat{\varphi }_j'(\hat{x})\hat{\varphi }_i(\hat{x})\nonumber \\&\ \;+c(G(\hat{x}))\hat{\varphi }_j(\hat{x})\hat{\varphi }_i(\hat{x})\Bigr )|G'(\hat{x})|\mathrm{d}\hat{x}\biggr ]_{i, j=1}^N, \end{aligned}$$

(10.169)

where $a_G(\hat{x})$ is the same as in (10.114),

$$\begin{aligned} a_G(\hat{x})=\frac{a(G(\hat{x}))}{(G'(\hat{x}))^2}. \end{aligned}$$

(10.170)

In the IgA framework, the functions $\hat{\varphi }_i$ are usually B-splines or NURBS. Here, the role of the $\hat{\varphi }_i$ will be played by B-splines over uniform knot sequences.

Galerkin B-spline IgA discretization matrices. As in the IgA collocation framework considered in Sect. 10.7.1, in the Galerkin B-spline IgA based on (uniform) B-splines, the functions $\hat{\varphi }_1,\ldots ,\hat{\varphi }_N$ are chosen as the B-splines $N_{2,[p]},\ldots , N_{n+p-1,[p]}$ defined in (10.120)–(10.122), i.e.,

$$\begin{aligned} \hat{\varphi }_i=N_{i+1,[p]},\qquad i=1,\ldots , n+p-2. \end{aligned}$$

The boundary functions $N_{1,[p]}$ and $N_{n+p,[p]}$ are excluded because they do not vanish on $\partial ([0, 1])$; see also Fig. 10.8. The stiffness matrix (10.169) resulting from this choice of the $\hat{\varphi }_i$ will be denoted by $A_{G, n}^{[p]}$:

$$\begin{aligned} A_{G, n}^{[p]}=\biggl [\int _{[0, 1]}&\Bigl (a_G(\hat{x})N_{j+1,[p]}'(\hat{x})N_{i+1,[p]}'(\hat{x})+\frac{b(G(\hat{x}))}{G'(\hat{x})}N_{j+1,[p]}'(\hat{x})N_{i+1,[p]}(\hat{x})\nonumber \\&\ \;+c(G(\hat{x}))N_{j+1,[p]}(\hat{x})N_{i+1,[p]}(\hat{x})\Bigr )|G'(\hat{x})|\mathrm{d}\hat{x}\biggr ]_{i, j=1}^{n+p-2}. \end{aligned}$$

(10.171)

Note that $A_{G, n}^{[p]}$ can be decomposed as follows:

$$\begin{aligned} A_{G, n}=K_{G, n}^{[p]}+Z_{G, n}^{[p]}, \end{aligned}$$

(10.172)

where

$$\begin{aligned} K_{G, n}^{[p]}=\left[ \int _{[0, 1]}a_G(\hat{x})|G'(\hat{x})|\, N_{j+1,[p]}'(\hat{x})N_{i+1,[p]}'(\hat{x})\mathrm{d}\hat{x}\right] _{i, j=1}^{n+p-2} \end{aligned}$$

(10.173)

is the diffusion matrix, resulting from the discretization of the higher-order (diffusion) term in (10.166), and

$$\begin{aligned} Z_{G, n}^{[p]}=\biggl [\int _{[0, 1]}&\Bigl (\frac{b(G(\hat{x}))}{G'(\hat{x})}N_{j+1,[p]}'(\hat{x})N_{i+1,[p]}(\hat{x})\nonumber \\&\ \;+c(G(\hat{x}))N_{j+1,[p]}(\hat{x})N_{i+1,[p]}(\hat{x})\Bigr )|G'(\hat{x})|\mathrm{d}\hat{x}\biggr ]_{i, j=1}^{n+p-2} \end{aligned}$$

(10.174)

is the matrix resulting from the discretization of the lower-order (convection and reaction) terms. We will see that, as usual, the GLT analysis of a properly scaled version of the sequence $\{A_{G, n}^{[p]}\}_n$ reduces to the GLT analysis of its ‘diffusion part’ $\{K_{G, n}^{[p]}\}_n$, because $\Vert Z_{G, n}^{[p]}\Vert$ is negligible with respect to $\Vert K_{G, n}^{[p]}\Vert$ as $n\rightarrow \infty$.

Let

$$\begin{aligned} K_n^{[p]}&=\left[ \int _{[0, 1]}N_{j+1,[p]}'(\hat{x})N_{i+1,[p]}'(\hat{x})\mathrm{d}\hat{x}\right] _{i, j=1}^{n+p-2},\end{aligned}$$

(10.175)

$$\begin{aligned} H_n^{[p]}&=\left[ \int _{[0, 1]}N_{j+1,[p]}'(\hat{x})N_{i+1,[p]}(\hat{x})\mathrm{d}\hat{x}\right] _{i, j=1}^{n+p-2},\end{aligned}$$

(10.176)

$$\begin{aligned} M_n^{[p]}&=\left[ \int _{[0, 1]}N_{j+1,[p]}(\hat{x})N_{i+1,[p]}(\hat{x})\mathrm{d}\hat{x}\right] _{i, j=1}^{n+p-2}. \end{aligned}$$

(10.177)

These matrices will play an important role in the GLT analysis of this section. In particular, it is necessary to understand their approximate structure. This is achieved by (approximately) construct them. We only construct their central submatrices

$$\begin{aligned} \bigl [(K_n^{[p]})_{ij}\bigr ]_{i, j=p}^{n-1},\qquad \bigl [(H_n^{[p]})_{ij}\bigr ]_{i, j=p}^{n-1},\qquad \bigl [(M_n^{[p]})_{ij}\bigr ]_{i, j=p}^{n-1}, \end{aligned}$$

(10.178)

which are determined by the central basis functions in (10.135). For all $i, j=p,\ldots , n-1$, noting that $[-i+p,\, n-i+p]\supseteq \text {supp}(\phi _{[p]})=[0, p+1]$ and using (10.133), (10.134) and (10.136), we obtain

$$\begin{aligned} (K_n^{[p]})_{ij}&=\int _{[0, 1]}N_{j+1,[p]}'(\hat{x})N_{i+1,[p]}'(\hat{x})\mathrm{d}\hat{x}\\&=n^2\int _{[0, 1]}\phi _{[p]}'(n\hat{x}-j+p)\phi _{[p]}'(n\hat{x}-i+p)\mathrm{d}\hat{x}\\&=n\int _{[-i+p,\, n-i+p]}\phi _{[p]}'(t+i-j)\phi _{[p]}'(t)\mathrm{d}t=n\int _{\mathbb R}\phi _{[p]}'(t+i-j)\phi _{[p]}'(t)\mathrm{d}t\\&=-n\,\phi _{[2p+1]}''(p+1+i-j)=-n\,\phi _{[2p+1]}''(p+1-i+j), \end{aligned}$$

and similarly

$$\begin{aligned} (H_n^{[p]})_{ij}&=\phi _{[2p+1]}'(p+1+i-j)=-\phi _{[2p+1]}'(p+1-i+j),\\ (M_n^{[p]})_{ij}&=\frac{1}{n}\phi _{[2p+1]}(p+1+i-j)=\frac{1}{n}\phi _{[2p+1]}(p+1-i+j). \end{aligned}$$

Since their entries depend only on the difference $i-j$, the submatrices (10.178) are Toeplitz matrices. More precisely,

$$\begin{aligned} \bigl [(K_n^{[p]})_{ij}\bigr ]_{i, j=p}^{n-1}&=n\bigl [-\phi _{[2p+1]}''(p+1-i+j)\bigr ]_{i, j=p}^{n-1}=n\, T_{n-p}(f_p),\end{aligned}$$

(10.179)

$$\begin{aligned} \bigl [(H_n^{[p]})_{ij}\bigr ]_{i, j=p}^{n-1}&=\bigl [-\phi _{[2p+1]}'(p+1-i+j)\bigr ]_{i, j=p}^{n-1}=I\, T_{n-p}(g_p),\end{aligned}$$

(10.180)

$$\begin{aligned} \bigl [(M_n^{[p]})_{ij}\bigr ]_{i, j=p}^{n-1}&=\frac{1}{n}\bigl [\phi _{[2p+1]}(p+1-i+j)\bigr ]_{i, j=p}^{n-1}=\frac{1}{n}T_{n-p}(h_p), \end{aligned}$$

(10.181)

where

$$\begin{aligned} f_p(\theta )&=\sum _{k\in \mathbb Z}-\phi _{[2p+1]}''(p+1-k)\,\mathrm{e}^{Ik\theta }\nonumber \\&=-\phi _{[2p+1]}''(p+1)-2\sum _{k=1}^p\phi _{[2p+1]}''(p+1-k)\cos (k\theta ),\end{aligned}$$

(10.182)

$$\begin{aligned} g_p(\theta )&=-I\sum _{k\in \mathbb Z}-\phi _{[2p+1]}'(p+1-k)\,\mathrm{e}^{Ik\theta }\nonumber \\&=-2\sum _{k=1}^p\phi _{[2p+1]}'(p+1-k)\sin (k\theta ),\end{aligned}$$

(10.183)

$$\begin{aligned} h_p(\theta )&=\sum _{k\in \mathbb Z}\phi _{[2p+1]}(p+1-k)\,\mathrm{e}^{Ik\theta }\nonumber \\&=\phi _{[2p+1]}(p+1)+2\sum _{k=1}^p\phi _{[2p+1]}(p+1-k)\cos (k\theta ); \end{aligned}$$

(10.184)

note that we used (10.132) and (10.133) to simplify the expressions of $f_p(\theta )$, $g_p(\theta )$, $h_p(\theta )$. It follows from (10.179) that $T_{n-p}(f_p)$ is the principal submatrix of both $n^{-1}K_n^{[p]}$ and $T_{n+p-2}(f_p)$ corresponding to the set of indices $p,\ldots , n-1$. Similar results follow from (10.180) and (10.181), and so

$$\begin{aligned} n^{-1}K_n^{[p]}&=T_{n+p-2}(f_p)+R_n^{[p]},\qquad \mathrm{rank}(R_n^{[p]})\le 4(p-1),\end{aligned}$$

(10.185)

$$\begin{aligned} -\mathrm{i} H_n^{[p]}&=T_{n+p-2}(g_p)+S_n^{[p]},\qquad \mathrm{rank}(S_n^{[p]})\le 4(p-1),\end{aligned}$$

(10.186)

$$\begin{aligned} n M_n^{[p]}&=T_{n+p-2}(h_p)+V_n^{[p]},\qquad \mathrm{rank}(V_n^{[p]})\le 4(p-1). \end{aligned}$$

(10.187)

Let us see two examples. In the case $p=2$, the matrix $K_n^{[2]}$ is given by

The submatrix $T_{n-2}(f_2)$ appears in correspondence of the highlighted box and we have

$$\begin{aligned} f_2(\theta )=\frac{1}{6}(-\mathrm{e}^{2I\theta }-2\mathrm{e}^{I\theta }+6-2\mathrm{e}^{-I\theta }-\mathrm{e}^{-2I\theta })=1-\frac{2}{3}\cos \theta -\frac{1}{3}\cos (2\theta ), \end{aligned}$$

as given by (10.182) for $p=2$. In the case $p=3$, the matrix $K_n^{[3]}$ is given by

The submatrix $T_{n-3}(f_3)$ appears in correspondence of the highlighted box and we have

$$\begin{aligned} f_3(\theta )&=\frac{1}{240}(-2\mathrm{e}^{3I\theta }-48\mathrm{e}^{2I\theta }-30\mathrm{e}^{I\theta }+160-30\mathrm{e}^{-I\theta }-48\mathrm{e}^{-2I\theta }-2\mathrm{e}^{-3I\theta })\\&=\frac{2}{3}-\frac{1}{4}\cos \theta -\frac{2}{5}\cos (2\theta )-\frac{1}{60}\cos (3\theta ), \end{aligned}$$

as given by (10.182) for $p=3$.

Remark 10.13

For every degree $q\ge 1$, the functions $f_q(\theta )$, $g_q(\theta )$, $h_q(\theta )$ defined by (10.182)–(10.184) for $p=q$ coincide with the functions $f_{2q+1}(\theta )$, $g_{2q+1}(\theta )$, $h_{2q+1}(\theta )$ defined by (10.147)–(10.149) for odd degree $p=2q+1$.

GLT analysis of the Galerkin B-spline IgA discretization matrices.  Assuming that the geometry map G is regular, i.e., $G\in C^1([0, 1])$ and $G'(\hat{x})\ne 0$ for every $\hat{x}\in [0, 1]$, we show that, for any $p\ge 1$, $\{n^{-1}A_{G, n}^{[p]}\}_n$ is a GLT sequence whose symbol describes both its singular value and spectral distribution.

Theorem 10.15

Let $\varOmega$ be a bounded open interval of $\mathbb R$ and let $a, b, c\in L^\infty (\varOmega )$. Let $p\ge 1$ and let $G:[0, 1]\rightarrow \overline{\varOmega }$ be such that $G\in C^1([0, 1])$ and $G'(\hat{x})\ne 0$ for all $\hat{x}\in [0, 1]$. Then

$$\begin{aligned} \{n^{-1}A_{G, n}^{[p]}\}_n\sim _\mathrm{GLT}f_{G, p} \end{aligned}$$

(10.188)

and

$$\begin{aligned} \{n^{-1}A_{G, n}^{[p]}\}_n\sim _{\sigma ,\,\lambda }f_{G, p}, \end{aligned}$$

(10.189)

where

$$\begin{aligned} f_{G, p}(\hat{x},\theta )=a_G(\hat{x})|G'(\hat{x})|f_p(\theta )=\frac{a(G(\hat{x}))}{|G'(\hat{x})|}f_p(\theta ) \end{aligned}$$

(10.190)

and $f_p(\theta )$ is defined in (10.182).

Proof

We follow the same argument as in the proof of Theorem 10.12. Throughout the proof, the letter C will denote a generic constant independent of n.

Step 1. We show that

$$\begin{aligned} \Vert n^{-1}K_{G, n}^{[p]}\Vert \le C \end{aligned}$$

(10.191)

and

$$\begin{aligned} \Vert n^{-1}Z_{G, n}^{[p]}\Vert \le C/n. \end{aligned}$$

(10.192)

To prove (10.191), we note that $K_{G, n}^{[p]}$ is a banded matrix, with bandwidth at most equal to $2p+1$. Indeed, due to the local support property (10.124), if $|i-j|>p$ then the supports of $N_{i+1,[p]}$ and $N_{j+1,[p]}$ intersect in at most one point, hence $(K_{G, n}^{[p]})_{ij}=0$. Moreover, by (10.124) and (10.128), for all $i, j=1,\ldots , n+p-2$ we have

$$\begin{aligned} |(K_{G, n}^{[p]})_{ij}|&=\left| \int _{[0, 1]}a_G(\hat{x})|G'(\hat{x})|N_{j+1,[p]}'(\hat{x})\, N_{i+1,[p]}'(\hat{x})\mathrm{d}\hat{x}\right| \\&=\left| \int _{[t_{i+1},\, t_{i+p+2}]}\frac{a(G(\hat{x}))}{|G'(\hat{x})|}N_{j+1,[p]}'(\hat{x})\, N_{i+1,[p]}'(\hat{x})\mathrm{d}\hat{x}\right| \\&\le \frac{4p^2n^2\Vert a\Vert _{L^\infty }}{\min _{\hat{x}\in [0, 1]}|G'(\hat{x})|}\int _{[t_{i+1},\, t_{i+p+2}]}\mathrm{d}\hat{x}\le \frac{4p^2(p+1)n\Vert a\Vert _{L^\infty }}{\min _{\hat{x}\in [0, 1]}|G'(\hat{x})|}, \end{aligned}$$

where in the last inequality we used the fact that $t_{k+p+1}-t_k\le (p+1)/n$ for all $k=1,\ldots , n+p$; see (10.118) and (10.119). In conclusion, the components of the banded matrix $n^{-1}K_{G, n}^{[p]}$ are bounded (in modulus) by a constant independent of n, and (10.191) follows from (2.31).

To prove (10.192), we follow the same argument as for the proof of (10.191). Due to the local support property (10.124), $Z_{G, n}^{[p]}$ is banded and, precisely, $(Z_{G, n}^{[p]})_{ij}=0$ whenever $|i-j|>p$. Moreover, by (10.124) and (10.126)–(10.128), for all $i, j=1,\ldots , n+p-2$ we have

$$\begin{aligned} |(Z_{G, n}^{[p]})_{ij}|&=\biggl |\int _{[t_{i+1},\, t_{i+p+2}]}\Bigl (\frac{b(G(\hat{x}))}{G'(\hat{x})}N_{j+1,[p]}'(\hat{x})\, N_{i+1,[p]}(\hat{x})\\&\qquad \qquad \qquad \ +c(G(\hat{x}))N_{j+1,[p]}(\hat{x})\, N_{i+1,[p]}(\hat{x})\Bigr )|G'(\hat{x})|\mathrm{d}\hat{x}\biggr |\\&\le 2p(p+1)\Vert b\Vert _{L^\infty }+\frac{(p+1)\Vert c\Vert _{L^\infty }\Vert G'\Vert _\infty }{n}, \end{aligned}$$

and (10.192) follows from (2.31).

Step 2. Consider the linear  operator $K_n^{[p]}(\cdot ):L^1([0, 1])\rightarrow \mathbb R^{(n+p-2)\times (n+p-2)}$,

$$\begin{aligned} K_n^{[p]}(g)=\left[ \int _{[0, 1]}g(\hat{x})N_{j+1,[p]}'(\hat{x})N_{i+1,[p]}'(\hat{x})\mathrm{d}\hat{x}\right] _{i, j=1}^{n+p-2}. \end{aligned}$$

By (10.173), we have $K_{G, n}^{[p]}=K_n^{[p]}(a_G|G'|)$. The next three steps are devoted to show that

$$\begin{aligned} \{n^{-1}K_n^{[p]}(g)\}_n\sim _\mathrm{GLT}g(\hat{x})f_p(\theta ),\qquad \forall \, g\in L^1([0, 1]). \end{aligned}$$

(10.193)

Once this is done, the theorem is proved. Indeed, by applying (10.193) with $g=a_G|G'|$ we immediately get $\{n^{-1}K_{G, n}^{[p]}\}_n\sim _\mathrm{GLT}f_{G, p}(\hat{x},\theta )$. Since $\{n^{-1}Z_{G, n}^{[p]}\}_n$ is zero-distributed by Step 1, (10.188) follows from the decomposition

$$\begin{aligned} n^{-1}A_{G, n}^{[p]}=n^{-1}K_{G, n}^{[p]}+n^{-1}Z_{G, n}^{[p]} \end{aligned}$$

(10.194)

and from $\mathbf{GLT\, 3}\!-\!\mathbf{GLT\, 4}$; and the singular value distribution in (10.189) follows from GLT 1. If $b(x)=0$ identically, then $n^{-1}A_{G, n}^{[p]}$ is symmetric and also the spectral distribution in (10.189) follows from GLT 1. If b(x) is not identically 0, the spectral distribution in (10.189) follows from GLT 2 applied to the decomposition (10.194), taking into account what we have seen in Step 1.

Step 3. We first prove (10.193) in the constant-coefficient case $g(\hat{x})=1$. In this case, we note that $K_n^{[p]}(1)=K_n^{[p]}$. Hence, the desired GLT relation $\{n^{-1}K_n^{[p]}(1)\}_n\sim _\mathrm{GLT}f_p(\theta )$ follows from (10.185) and $\mathbf{GLT\, 3}\!-\!\mathbf{GLT\, 4}$, taking into account that $\{R_n^{[p]}\}_n$ is zero-distributed by Z 1.

Step 4. Now we prove (10.193) in the case where $g\in C([0, 1])$. As in Step 4 of Sect. 10.6.1, the proof is based on the fact that the B-spline basis functions $N_{2,[p]},\ldots , N_{n+p-1,[p]}$ are ‘locally supported’. Indeed, the width of the support $[t_{i+1}, t_{i+p+2}]$ of the ith basis function $N_{i+1,[p]}$ is bounded by $(p+1)/n$ and goes to 0 as $n\rightarrow \infty$. Moreover, the support itself is located near the point $\frac{i}{n+p-2^{}}$, because

$$\begin{aligned} \max _{\hat{x}\in [t_{i+1},\, t_{i+p+2}]}\biggl |\hat{x}-\frac{i}{n+p-2}\biggr |\le \frac{C_p}{n} \end{aligned}$$

(10.195)

for all $i=2,\ldots , n+p-1$ and for some constant $C_p$ depending only on p. By (10.128) and (10.195), for all $i, j=1,\ldots , n+p-2$ we have

$$\begin{aligned}&\left| (K_n^{[p]}(g))_{ij}-(D_{n+p-2}(g)K_n^{[p]}(1))_{ij}\right| \\&=\left| \int _{[0, 1]}\Bigl [g(\hat{x})-g\Bigl (\frac{i}{n+p-2}\Bigr )\Bigr ]N_{j+1,[p]}'(\hat{x})N_{i+1,[p]}'(\hat{x})\mathrm{d}\hat{x}\right| \\&\le 4p^2n^2\int _{[t_{i+1},\, t_{i+p+2}]}\Bigl |g(\hat{x})-g\Bigl (\frac{i}{n+p-2}\Bigr )\Bigr |\mathrm{d}\hat{x}\le 4p^2(p+1)n\,\omega _g\Bigl (\frac{C_p}{n}\Bigr ). \end{aligned}$$

It follows that each entry of $Z_n=n^{-1}K_n^{[p]}(g)-n^{-1}D_{n+p-2}(g)K_n^{[p]}(1)$ is bounded in modulus by $C\omega _g(1/n)$. Moreover, $Z_n$ is banded with bandwidth at most $2p+1$, due to the local support property of the B-spline basis functions $N_{i,[p]}$. By (2.31) we conclude that $\Vert Z_n\Vert \le C\omega _g(1/n)\rightarrow 0$ as $n\rightarrow \infty$. Thus,  $\{Z_n\}_n\sim _\sigma 0$, which implies (10.193) by Step 3 and $\mathbf{GLT\, 3}\!-\!\mathbf{GLT\, 4}$.

Step 5. Finally, we prove (10.193) in the general case where $g\in L^1([0, 1])$. By the density of C([0, 1]) in $L^1([0, 1])$, there exist continuous functions $g_m\in C([0, 1])$ such that $g_m\rightarrow g$ in $L^1([0, 1])$. By Step 4,

$$\begin{aligned} \{n^{-1}K_n^{[p]}(g_m)\}_n\sim _\mathrm{GLT}g_m(\hat{x})f_p(\theta ). \end{aligned}$$

Moreover,

$$\begin{aligned} g_m(\hat{x})f_p(\theta )\rightarrow g(\hat{x})f_p(\theta ) \ \text {in measure}. \end{aligned}$$

We show that

$$\begin{aligned} \{n^{-1}K_n^{[p]}(g_m)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{n^{-1}K_n^{[p]}(g)\}_n. \end{aligned}$$

Using (10.128) and (2.43), we obtain

$$\begin{aligned} \Vert K_n^{[p]}(g)-K_n^{[p]}(g_m)\Vert _1&\le \sum _{i, j=1}^{n+p-2}\left| (K_n^{[p]}(g))_{ij}-(K_n^{[p]}(g_m))_{ij}\right| \\&=\sum _{i, j=1}^{n+p-2}\left| \int _{[0, 1]}\bigl [g(\hat{x})-g_m(\hat{x})\bigr ]N_{j+1,[p]}'(\hat{x})N_{i+1,[p]}'(\hat{x})\mathrm{d}\hat{x}\right| \\&\le \int _{[0, 1]}|g(\hat{x})-g_m(\hat{x})|\sum _{i, j=1}^{n+p-2}|N_{j+1,[p]}'(\hat{x})|\,|N_{i+1,[p]}'(\hat{x})|\mathrm{d}\hat{x}\\&\le 4p^2n^2\int _{[0, 1]}|g(\hat{x})-g_m(\hat{x})|\mathrm{d}\hat{x} \end{aligned}$$

and

$$\begin{aligned} \Vert n^{-1}K_n^{[p]}(g)-n^{-1}K_n^{[p]}(g_m)\Vert _1\le 4p^2n\Vert g-g_m\Vert _{L^1}. \end{aligned}$$

Thus, $\{n^{-1}K_n^{[p]}(g_m)\}_n\mathop {\longrightarrow }\limits ^\mathrm{a.c.s.}\{n^{-1}K_n^{[p]}(g)\}_n$ by ACS 6. The relation (10.193) now follows from GLT 7. $\square$

Remark 10.14

(formal structure of the symbol) Problem (10.166) can be formally rewritten as in (10.106). If, for any $u:\overline{\varOmega }\rightarrow \mathbb R$, we define $\hat{u}:[0, 1]\rightarrow \mathbb R$ as in (10.112), then u satisfies (10.106) if and only if $\hat{u}$ satisfies the corresponding transformed problem (10.113), in which the higher-order operator takes the form $-a_G(\hat{x})\hat{u}''(\hat{x})$. It is then clear that, similarly to the collocation case (see Remark 10.11), even in the Galerkin case the symbol $f_{G, p}(\hat{x},\theta )=a_G(\hat{x})|G'(\hat{x})|f_p(\theta )$ preserves the formal structure of the higher-order operator associated with the transformed problem (10.113). However, in this Galerkin context we notice the appearance of the factor $|G'(\hat{x})|$, which is not present in the collocation setting; cf. (10.190) with (10.156).

Exercise 10.5

The matrix $A_{G, n}^{[p]}$ in (10.171), which we decomposed as in (10.172), can also be decomposed as follows, according to the diffusion, convection and reaction terms:

$$\begin{aligned} A_{G, n}^{[p]}=K_{G, n}^{[p]}+H_{G, n}^{[p]}+M_{G, n}^{[p]}, \end{aligned}$$

where the diffusion, convection and reaction matrices are given by

$$\begin{aligned} K_{G, n}^{[p]}&=\left[ \int _{[0, 1]}\frac{a(G(\hat{x}))}{|G'(\hat{x})|}N_{j+1,[p]}'(\hat{x})N_{i+1,[p]}'(\hat{x})\mathrm{d}\hat{x}\right] _{i, j=1}^{n+p-2},\end{aligned}$$

(10.196)

$$\begin{aligned} H_{G, n}^{[p]}&=\left[ \int _{[0, 1]}\frac{b(G(\hat{x}))|G'(\hat{x})|}{G'(\hat{x})}N_{j+1,[p]}'(\hat{x})N_{i+1,[p]}(\hat{x})\mathrm{d}\hat{x}\right] _{i, j=1}^{n+p-2},\end{aligned}$$

(10.197)

$$\begin{aligned} M_{G, n}^{[p]}&=\left[ \int _{[0, 1]}c(G(\hat{x}))|G'(\hat{x})|N_{j+1,[p]}(\hat{x})N_{i+1,[p]}(\hat{x})\mathrm{d}\hat{x}\right] _{i, j=1}^{n+p-2}; \end{aligned}$$

(10.198)

note that the diffusion matrix is the same as in (10.173). Let $\varOmega$ be a bounded open interval of $\mathbb R$ and let $p\ge 1$. Prove the following results.

• (a) Suppose

  $$\begin{aligned} \frac{a(G(\hat{x}))}{|G'(\hat{x})|}\in L^1([0, 1]); \end{aligned}$$

  then $\{n^{-1}K_{G, n}^{[p]}\}_n\sim _\mathrm{GLT}f_{G, p}$ and $\{n^{-1}K_{G, n}^{[p]}\}_n\sim _{\sigma ,\,\lambda }f_{G, p}$, where

  $$\begin{aligned} f_{G, p}(\hat{x},\theta )=\frac{a(G(\hat{x}))}{|G'(\hat{x})|}f_p(\theta ) \end{aligned}$$

  (10.199)

  and $f_p(\theta )$ is defined in (10.182); note that $f_{G, p}(\hat{x},\theta )$ is the same as in (10.190).

• (b) Suppose

  $$\begin{aligned} \frac{b(G(\hat{x}))|G'(\hat{x})|}{G'(\hat{x})}\in C([0, 1]); \end{aligned}$$

  then $\{-IH_{G, n}^{[p]}\}_n\sim _\mathrm{GLT}g_{G, p}$ and $\{-IH_{G, n}^{[p]}\}_n\sim _{\sigma ,\,\lambda }g_{G, p}$, where

  $$\begin{aligned} g_{G, p}(\hat{x},\theta )=\frac{b(G(\hat{x}))|G'(\hat{x})|}{G'(\hat{x})}g_p(\theta ) \end{aligned}$$

  (10.200)

  and $g_p(\theta )$ is defined in (10.183).

• (c) Suppose

  $$\begin{aligned} c(G(\hat{x}))|G'(\hat{x})|\in L^1([0, 1]); \end{aligned}$$

  then $\{nM_{G, n}^{[p]}\}_n\sim _\mathrm{GLT}h_{G, p}$ and $\{nM_{G, n}^{[p]}\}_n\sim _{\sigma ,\,\lambda }h_{G, p}$, where

  $$\begin{aligned} h_{G, p}(\hat{x},\theta )=c(G(\hat{x}))|G'(\hat{x})|h_p(\theta ) \end{aligned}$$

  (10.201)

  and $h_p(\theta )$ is defined in (10.184).

Exercise 10.6

Let $\varOmega$ be a bounded open interval of $\mathbb R$ and let $p\ge 1$. Suppose the convection term is not present, i.e., $b(x)=0$ identically. Using the result of Exercise 10.5, show that (10.188) and (10.189)  continue to hold under the only assumption that

$$\begin{aligned} a_G(\hat{x})|G'(\hat{x})|=\frac{a(G(\hat{x}))}{|G'(\hat{x})|}\in L^1([0, 1]),\qquad c(G(\hat{x}))|G'(\hat{x})|\in L^1([0, 1]). \end{aligned}$$

(10.202)

This assumption is satisfied if, for example, one of the following conditions is met:

• $a, c\in L^1(\varOmega )$ and G is regular, i.e., $G\in C^1([0, 1])$ and $G'(\hat{x})\ne 0$ for all $\hat{x}\in [0, 1]$;

• $c\in L^1(\varOmega )$, $a\in L^\infty (\varOmega )$, $G\in C^1([0, 1])$ and $|G'|^{-1}\in L^1([0, 1])$.

Note that the condition on G in the second item is met even in some cases in which G is not regular. For instance, it is met if $\varOmega =(0, 1)$ and $G(\hat{x})=\hat{x}^q$, $1<q<2$; in this case, $G'(0)=0$ (so G is singular), and the mapping of the uniform mesh $\frac{i}{n}$, $i=0,\ldots , n$, through the function G is the non-uniform grid $(\frac{i}{n})^q$, $i=0,\ldots , n$, whose points rapidly accumulate at $x=0$. This induces a local  refinement around the site $x=0$, and the choice of G is then a way to better approximate the solution in a neighborhood of $x=0$; see also the discussion in Remark 10.8.

10.7.3 Galerkin B-Spline IgA Discretization of Second-Order Eigenvalue Problems

Let  $\mathbb R^+$ be the set of positive real numbers. Consider the following second-order eigenvalue problem: find eigenvalues $\lambda _j\in \mathbb R^+$ and eigenfunctions $u_j$, for $j=1, 2,\ldots ,\infty$, such that

$$\begin{aligned} \left\{ \begin{array}{ll} -(a(x)u_j'(x))'=\lambda _jc(x)u_j(x), &{}\quad x\in \varOmega ,\\ u_j(x)=0, &{}\quad x\in \partial \varOmega , \end{array}\right. \end{aligned}$$

(10.203)

where $\varOmega$ is a bounded open interval of $\mathbb R$ and we assume that $a, c\in L^1(\varOmega )$ and $a, c>0$ a.e. in $\varOmega$. It can be shown that the eigenvalues $\lambda _j$ must necessarily be real and positive. This can be formally seen by multiplying (10.203) by $u_j(x)$ and integrating over $\varOmega$:

$$\begin{aligned} \lambda _j=\frac{-\int _\varOmega (a(x)u'_j(x))'u_j(x)\mathrm{d}x}{\int _\varOmega c(x)(u_j(x))^2\mathrm{d}x}=\frac{\int _\varOmega a(x)(u'_j(x))^2\mathrm{d}x}{\int _\varOmega c(x)(u_j(x))^2\mathrm{d}x}>0. \end{aligned}$$

Isogeometric Galerkin approximation. The weak form of (10.203) reads as follows: find eigenvalues $\lambda _j\in \mathbb R^+$ and eigenfunctions $u_j\in H^1_0(\varOmega )$, for $j=1, 2,\ldots ,\infty$, such that

$$\begin{aligned} \mathrm{a}(u_j, w)=\lambda _j(c\, u_j, w),\qquad \forall \, w\in H^1_0(\varOmega ), \end{aligned}$$

where

$$\begin{aligned} \mathrm{a}(u_j, w)&=\int _\varOmega a(x)u_j'(x)w'(x)\mathrm{d}x,\\ (c\, u_j, w)&=\int _\varOmega c(x)u_j(x)w(x)\mathrm{d}x. \end{aligned}$$

In the standard Galerkin method, we choose a finite dimensional vector space $\mathscr {W}\subset H^1_0(\varOmega )$, the so-called approximation space, we let $N=\dim \mathscr {W}$ and we look for approximations of the eigenpairs $(\lambda _j, u_j)$, $j=1, 2,\ldots ,\infty$, by solving the following discrete (Galerkin) problem: find $\lambda _{j,\mathscr {W}}\in \mathbb R^+$ and $u_{j,\mathscr {W}}\in \mathscr {W}$, for $j=1,\ldots , N$, such that

$$\begin{aligned} \mathrm{a}(u_{j,\mathscr {W}}, w)=\lambda _{j,\mathscr {W}}(c\, u_{j,\mathscr {W}}, w),\qquad \forall \, w\in \mathscr {W}. \end{aligned}$$

(10.204)

Assuming that both the exact and  numerical eigenvalues are arranged in non-decreasing order, the pair $(\lambda _{j,\mathscr {W}}, u_{j,\mathscr {W}})$ is taken as an approximation to the pair $(\lambda _j, u_j)$ for all $j=1, 2,\ldots , N$. The numbers $\lambda _{j,\mathscr {W}}/\lambda _j-1,\ j=1,\ldots , N$, are referred to as the (relative)  eigenvalue errors. If $\{\varphi _1,\ldots ,\varphi _N\}$ is a basis of $\mathscr {W}$, we can identify each $w\in \mathscr {W}$ with its coefficient vector relative to this basis. With this identification in mind, solving the discrete problem (10.204) is equivalent to solving the  generalized eigenvalue problem

$$\begin{aligned} K\mathbf {u}_{j,\mathscr {W}}=\lambda _{j,\mathscr {W}}M\mathbf {u}_{j,\mathscr {W}}, \end{aligned}$$

(10.205)

where $\mathbf {u}_{j,\mathscr {W}}$ is the coefficient vector of $u_{j,\mathscr {W}}$ with respect to $\{\varphi _1,\ldots ,\varphi _N\}$ and

$$\begin{aligned} K&=\left[ \int _\varOmega a(x)\varphi _j'(x)\varphi _i'(x)\mathrm{d}x\right] _{i, j=1}^N,\end{aligned}$$

(10.206)

$$\begin{aligned} M&=\left[ \int _\varOmega c(x)\varphi _j(x)\varphi _i(x)\mathrm{d}x\right] _{i, j=1}^N. \end{aligned}$$

(10.207)

The matrices K and M are referred to as the stiffness  and mass matrix, respectively. Due to our assumption that $a, c>0$ a.e., both K and M are SPD, regardless of the chosen basis functions $\varphi _1,\ldots ,\varphi _N$. Moreover, it is clear from (10.205) that the numerical eigenvalues $\lambda _{j,\mathscr {W}}$, $j=1,\ldots , N$, are just the eigenvalues of the matrix

$$\begin{aligned} L=M^{-1}K. \end{aligned}$$

(10.208)

In the isogeometric Galerkin method, we assume that the physical domain $\varOmega$ is described by a global geometry  function $G:[0, 1]\rightarrow \overline{\varOmega }$, which is invertible and satisfies $G(\partial ([0, 1]))=\partial \overline{\varOmega }$. We fix a set of basis functions $\{\hat{\varphi }_1,\ldots ,\hat{\varphi }_N\}$ defined on the reference (parametric)  domain [0, 1] and vanishing on the boundary $\partial ([0, 1])$, and we find approximations to the exact eigenpairs $(\lambda _j, u_j)$, $j=1, 2,\ldots ,\infty$, by using the standard Galerkin method described above, in which the approximation space is chosen as ${\mathscr {W}}=span (\varphi _1,\ldots ,\varphi _N)$, where

$$\begin{aligned} \varphi _i(x)=\hat{\varphi }_i(G^{-1}(x))=\hat{\varphi }_i(\hat{x}), \qquad x=G(\hat{x}). \end{aligned}$$

(10.209)

The resulting stiffness and mass matrices K and M are given by (10.206) and (10.207), with the basis functions $\varphi _i$ defined as in (10.209). If we assume that G and $\hat{\varphi }_i$, $i=1,\ldots , N$, are sufficiently regular, we can apply  standard differential calculus to obtain for K and M the following expressions:

$$\begin{aligned} K&=\left[ \int _{[0, 1]}\frac{a(G(\hat{x}))}{|G'(\hat{x})|}\hat{\varphi }_j'(\hat{x})\hat{\varphi }_i'(\hat{x})\mathrm{d}\hat{x}\right] _{i, j=1}^N,\end{aligned}$$

(10.210)

$$\begin{aligned} M&=\left[ \int _{[0, 1]}c(G(\hat{x}))|G'(\hat{x})|\hat{\varphi }_j(\hat{x})\hat{\varphi }_i(\hat{x})\mathrm{d}\hat{x}\right] _{i, j=1}^N. \end{aligned}$$

(10.211)

GLT analysis of the Galerkin B-spline IgA discretization matrices.  Following the approach of Sects. 10.7.1–10.7.2, we choose the basis functions $\hat{\varphi }_i$, $i=1,\ldots , N$, as the B-splines $N_{i+1,[p]}$, $i=1,\ldots , n+p-2$. The resulting stiffness and mass matrices (10.210) and (10.211) are given by

$$\begin{aligned} K_{G, n}^{[p]}&=\left[ \int _{[0, 1]}\frac{a(G(\hat{x}))}{|G'(\hat{x})|}N_{j+1,[p]}'(\hat{x})N_{i+1,[p]}'(\hat{x})\mathrm{d}\hat{x}\right] _{i, j=1}^{n+p-2},\\ M_{G, n}^{[p]}&=\left[ \int _{[0, 1]}c(G(\hat{x}))|G'(\hat{x})|N_{j+1,[p]}(\hat{x})N_{i+1,[p]}(\hat{x})\mathrm{d}\hat{x}\right] _{i, j=1}^{n+p-2}, \end{aligned}$$

and it is immediately seen that they are the same as the diffusion and reaction matrices in (10.196) and (10.198). The numerical eigenvalues will be henceforth denoted by $\lambda _{j, n}$, $j=1,\ldots , n+p-2$; as noted above, they are simply the eigenvalues of the matrix

$$\begin{aligned} L_{G, n}^{[p]}=(M_{G, n}^{[p]})^{-1}K_{G, n}^{[p]}. \end{aligned}$$

Theorem 10.16

Let $\varOmega$ be a bounded open interval of $\mathbb R$ and let $a, c\in L^1(\varOmega )$ with $a, c>0$ a.e. Let $p\ge 1$ and let $G:[0, 1]\rightarrow \overline{\varOmega }$ be such that

$$\begin{aligned} \frac{a(G(\hat{x}))}{|G'(\hat{x})|}\in L^1([0, 1]). \end{aligned}$$

Then

$$\begin{aligned} \{n^{-2}L_{G, n}^{[p]}\}_n\sim _\mathrm{GLT}e_{G, p}(\hat{x},\theta ) \end{aligned}$$

(10.212)

and

$$\begin{aligned} \{n^{-2}L_{G, n}^{[p]}\}_n\sim _{\sigma ,\,\lambda }e_{G, p}(\hat{x},\theta ), \end{aligned}$$

(10.213)

where

$$\begin{aligned} e_{G, p}(\hat{x},\theta )&=(h_{G, p}(\theta ))^{-1}f_{G, p}(\theta )=\frac{a(G(\hat{x}))}{c(G(\hat{x}))(G'(\hat{x}))^2}\, e_p(\theta ),\end{aligned}$$

(10.214)

$$\begin{aligned} e_p(\theta )&=(h_p(\theta ))^{-1}f_p(\theta ), \end{aligned}$$

(10.215)

and $f_p(\theta )$, $h_p(\theta )$, $f_{G, p}(\hat{x},\theta )$, $h_{G, p}(\hat{x},\theta )$ are given by (10.182), (10.184), (10.199), (10.201), respectively.

Proof

We have $a(G(\hat{x}))/|G'(\hat{x})|\in L^1([0, 1])$ by assumption and $c(G(\hat{x}))|G'(\hat{x})|\in L^1([0, 1])$ because $c\in L^1(\varOmega )$ by assumption and

$$\begin{aligned} \int _{[0, 1]}c(G(\hat{x}))|G'(\hat{x})|\mathrm{d}\hat{x}=\int _\varOmega c(x)\mathrm{d}x. \end{aligned}$$

Hence, by Exercise 10.5,

$$\begin{aligned} \{n^{-1}K_{G, n}^{[p]}\}_n\sim _\mathrm{GLT}f_{G, p},\qquad \{nM_{G, n}^{[p]}\}_n\sim _\mathrm{GLT}h_{G, p}, \end{aligned}$$

and the relations (10.212) and (10.213) follow from Exercise 8.4, taking into account that $h_{G, p}(\hat{x},\theta )\ne 0$ a.e. by our assumption that $c(x)>0$ a.e. and by the positivity of $h_p(\theta )$; see (10.165) and Remark 10.13. $\square$

For $p=1, 2, 3, 4$, Eq. (10.215) gives

$$\begin{aligned} e_1(\theta )&=\frac{6(1-\cos \theta )}{2+\cos \theta },\\ e_2(\theta )&=\frac{20(3-2\cos \theta -\cos (2\theta ))}{33+26\cos \theta +\cos (2\theta )},\\ e_3(\theta )&=\frac{42(40-15\cos \theta -24\cos (2\theta )-\cos (3\theta ))}{1208+1191\cos \theta +120\cos (2\theta )+\cos (3\theta )},\\ e_4(\theta )&=\frac{72(1225-154\cos \theta -952\cos (2\theta )-118\cos (3\theta )-\cos (4\theta ))}{78095+88234\cos \theta +14608\cos (2\theta )+502\cos (3\theta )+\cos (4\theta )}. \end{aligned}$$

These equations are the analogs of formulas (117), (130), (135), (140) obtained by  engineers in [75]; see also formulas (32), (33) in [35], formulas (23), (56) in [76], and formulas (23), (24) in [92]. We may therefore conclude that (10.214) is a generalization of these formulas to any degree $p\ge 1$.

Remark 10.15

Contrary to the B-spline IgA discretizations investigated herein and in [75], the authors of [35, 76, 92] considered NURBS IgA discretizations. However, the same formulas are obtained in both cases. This can be easily explained in view of the results of [59], where it is shown that the symbols $f_p$, $g_p$, $h_p$ in (10.182)–(10.184) are exactly the same in the B-spline and NURBS IgA frameworks.

Example 10.8

Consider the case where $\varOmega =(0, 1)$, G is the identity map on $[0, 1]=\overline{\varOmega }$, and $a(x)=c(x)=1$. In this case, (10.203) reduces to the classical eigenvalue problem for the (negative) Laplacian operator,

$$ \left\{ \begin{array}{ll} -u_j''(x)=\lambda _ju_j(x), &{}\quad x\in (0, 1),\\ u_j(0)=u_j(1)=0. \end{array}\right. $$

The exact eigenpairs are explicitly known and are given by

$$\begin{aligned} \lambda _j=j^2\pi ^2,\qquad u_j(x)=\sin (j\pi x),\qquad j=1, 2,\ldots ,\infty . \end{aligned}$$

The symbol of the matrix-sequence $\{n^{-2}L_{G, n}^{[p]}\}_n$, namely

$$\begin{aligned} e_{G, p}(\hat{x},\theta )=e_p(\theta ), \end{aligned}$$

only depends on the Fourier variable $\theta$ and is symmetric with respect to $\theta =0$. One of its rearranged versions is then $e_p:[0,\pi ]\rightarrow \mathbb R$.

Fig. 10.12

Comparison between the spectrum of $n^{-2}L_{G, n}^{[p]}$ and the symbol $e_{G, p}(\hat{x},\theta )=e_p(\theta )$ for $p=1, 2, 3$, $n=40$ and $\varOmega =(0, 1)$, $G=\text {identity map}$, $a(x)=c(x)=1$

For $p=1, 2, 3$, in Fig. 10.12 we plotted the graph of $e_p(\theta )$ over $[0,\pi ]$ and the eigenvalues of $n^{-2}L_{G, n}^{[p]}$ for $n=40$. The eigenvalues have been sorted in non-decreasing order (so as to match the graph of $e_p(\theta )$) and have been placed at the points $(\frac{j\pi }{n},\lambda _j(n^{-2}L_{G, n}^{[p]}))$, $j=1,\ldots , n+p-2$. The figure shows an excellent agreement between the spectrum and the symbol, although we also observe the presence of 2 outliers at the right end of the spectrum for $p=3$.

In view of Fig. 10.12, almost all the eigenvalues

$$\begin{aligned} \lambda _j(n^{-2}L_{G, n}^{[p]}),\qquad j=1,\ldots , n+p-2, \end{aligned}$$

are approximated by the uniform samples

$$\begin{aligned} e_p\Bigl (\frac{j\pi }{n}\Bigr ),\qquad j=1,\ldots , n. \end{aligned}$$

Hence, almost all the numerical eigenvalues

$$\begin{aligned} \lambda _{j, n}=\lambda _j(L_{G, n}^{[p]}),\qquad j=1,\ldots , n+p-2, \end{aligned}$$

are approximated by the values

$$\begin{aligned} n^2e_p\Bigl (\frac{j\pi }{n}\Bigr ),\qquad j=1,\ldots , n. \end{aligned}$$

Consequently, we have

$$\begin{aligned} \frac{\lambda _{j, n}}{\lambda _j}-1\approx \frac{e_p(\frac{j\pi }{n})}{(\frac{j\pi }{n})^2}-1 \end{aligned}$$

for almost all $j=1,\ldots , n+p-2$. For $p=1, 2, 3$, in Fig. 10.13 we plotted the eigenvalue errors $\lambda _{j, n}/\lambda _j-1$ and their analytical predictions $e_p(\frac{j\pi }{n})/(\frac{j\pi }{n})^2-1$ versus $j/(n+p-2)$, for $j=1,\ldots , n+p-2$ and $n=500$. Clearly, the analytical prediction is excellent, except at the right end of the spectrum for $p=3$, where two outliers (already observed in Fig. 10.12 for $n=40$) produce a mismatch.

Fig. 10.13

Eigenvalue errors $\lambda _{j, n}/\lambda _{j-1}$ and analytical predictions $e_p(\frac{j \pi }{n})/(\frac{j \pi }{n})^{2}-1$ versus $j/(n+p-2)$, $j=1,\ldots , n+p-2$, for $p=1, 2, 3$ and $n=500$

For an extension of the results obtained in this section, we refer the reader to the engineering paper [55].