Abstract
In this chapter we present several applications of the theory of multilevel GLT sequences to the computation of the singular value and eigenvalue distribution of matrix-sequences arising from the numerical discretization of PDEs. In order to understand the content of this chapter, it is enough that the reader knows the summary of Chap. 6 and possesses the necessary prerequisites, most of which have been addressed in Chap. 2 and [22]. Indeed, our arguments/derivations in this chapter will never refer to Chaps. 1–5, i.e., they will only rely on the summary of Chap. 6. For more applications than the ones presented herein, we refer the reader to [22, Sect. 1.1], where specific pointers to the available literature are provided.
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Notes
- 1.
Note that the boundary conditions (Dirichlet, Neumann, etc.) have not been specified precisely because they only produce a small-rank perturbation \(R_{\varvec{n}}\) in the resulting discretization matrix \(A_{\varvec{n}}\); see also the discussion in [22, p. 116] and the 2nd part of [22, Sect. 10.5.2].
- 2.
We say that a differential operator is separable if it is obtained by multiplying a given function with a product of partial derivatives. The general separable differential operator can be written as
$$ a\frac{\partial ^{r_1+\cdots +r_d}u}{\partial x_1^{r_1}\cdots \partial {x_d^{r_d}}^{ {1}}}. $$An example of a non-separable differential operator is the Laplacian, which, however, can be written (just like any other linear differential operator) as a sum of separable differential operators:
$$ \varDelta u=\sum _{k=1}^d\frac{\partial ^2u}{\partial x_k^2}. $$As evidenced by the forthcoming discussion, the discretization of a separable differential operator gives rise to a GLT (actually, a sLT) sequence. For instance, after a suitable normalization that we here ignore, the matrix-sequences \(\{K_{{\varvec{n}},\ell k}(a_{\ell k})\}_n\), \(\{H_{{\varvec{n}}, k}(b_k)\}_n\), \(\{I_{\varvec{n}}(c)\}_n\) are GLT (actually, sLT) sequences. As a consequence, the discretization of an arbitrary linear differential operator (a sum of separable differential operators) gives rise to a sum of GLT (actually, sLT) sequences, i.e., again a GLT sequence.
- 3.
Recall that operations involving d-indices that have no meaning in \(\mathbb Z^d\) must be interpreted in the componentwise sense. In the present case, given \({\varvec{n}}=(n_1,\ldots , n_d)\) and \({\varvec{j}}=(j_1,\ldots , j_d)\), the vector of discretization steps \({\varvec{h}}=\frac{\mathbf {1}}{{\varvec{n}}+\mathbf {1}}\) and the grid point \(\mathbf {x}_{\varvec{j}}={\varvec{j}}{\varvec{h}}\) are given by \({\varvec{h}}=(\frac{1}{n_1+1},\ldots ,\frac{1}{n_d+1})=(h_1,\ldots , h_d)\) and \(\mathbf {x}_{\varvec{j}}=(j_1h_1,\ldots , j_dh_d)\).
- 4.
Actually, the argument used in Step 1 of the proof of Theorem 7.7 to prove the inequality \(\Vert n^{d-2}K_{{\mathbf {G}},{\varvec{n}}}^{[{\varvec{p}}]}\Vert \le C\) can be used to prove that \(\Vert n^{d-2}K_{\varvec{n}}^{[{\varvec{p}}]}(L)\Vert \le C\) for all functions \(L:[0,1]^d\rightarrow \mathbb R^{d\times d}\) such that \(L_{ij}\in L^\infty ([0,1]^d)\) for all \(i, j=1,\ldots , d\). Recall also that \(K_{{\mathbf {G}},{\varvec{n}}}^{[{\varvec{p}}]}=K_{\varvec{n}}^{[{\varvec{p}}]}(|\mathrm{det}(J_{\mathbf {G}})|A_{\mathbf {G}})\) and \((|\mathrm{det}(J_{\mathbf {G}})|A_{\mathbf {G}})_{ij}\in L^\infty ([0,1]^d)\) for all \(i, j=1,\ldots , d\) under the assumptions of both Theorem 7.7 and the present theorem.
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Garoni, C., Serra-Capizzano, S. (2018). Applications. In: Generalized Locally Toeplitz Sequences: Theory and Applications. Springer, Cham. https://doi.org/10.1007/978-3-030-02233-4_7
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DOI: https://doi.org/10.1007/978-3-030-02233-4_7
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