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Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

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Abstract

We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics, a nonlocal formulation of continuum mechanics. We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation. A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero. We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible. We find that, for polynomial degrees greater than or equal to two, there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal diffusion are asymptotically compatible. We verify these findings through extensive numerical experiments.

Introduction

We study the convergence and asymptotic compatibility of higher order collocation methods used to discretize nonlocal operators that are used in the nonlocal wave propagation [1, 2] and nonlocal diffusion [4, 9, 10] with the local Dirichlet or Neumann boundary conditions (BC) denoted by \(\mathtt{BC}\in \{ \mathtt{D}, \mathtt{N}\}\), respectively,

$$\begin{aligned} \begin{aligned} &u_{tt}^\mathtt{BC}(x,t) + \mathcal {M}_\mathtt{BC}u^\mathtt{BC}(x,t) =\, b^\mathtt{BC}(x,t), \quad (x,t) \in \Omega \times (0,T), \\ &u^\mathtt{D}(\pm 1, t) =\, \alpha _{\pm }^\mathtt{D}(t) \quad \text {or} \quad u_x^\mathtt{N}(\pm 1, t) = \alpha _{\pm }^\mathtt{N}(t), \quad t \in (0,T),\\ &u^\mathtt{BC}(x,0) =\, \phi _\mathtt{BC}(x), \quad x \in \Omega, \\ &u_t^\mathtt{BC}(x,0) = \, \psi _\mathtt{BC}(x), \quad x \in \Omega . \end{aligned} \end{aligned}$$
(1)

The operators are inspired by the theory of peridynamics, a nonlocal formulation of continuum mechanics developed by Silling [20]. Furthermore, the operators \(\mathcal {M}_\mathtt{D}\) and \(\mathcal {M}_\mathtt{N}\) agree with the original bond-based peridynamic operator in the bulk of the domain \(\Omega :=(-1,1)\) and simultaneously enforce the local Dirichlet or Neumann BC, respectively.

We studied various aspects of local BC in nonlocal problems [1,2,3,4,5, 7,8,9,10, 12]. Building on [12], the results in \(\mathbb {R}\) were generalized to a bounded domain [7, 8], a critical feature for all practical applications. In [8], the theoretical foundations were formulated and in [7], the foundations were applied to numerically solve wave propagation problems using local BC. In [4], we constructed the first 1D operators \(\mathcal {M}_\mathtt{BC}\). We carried out numerical experiments by utilizing \(\mathcal {M}_\mathtt{BC}\) as governing operators in [7]. The operators were extended to higher dimensions in [10]. In [9], we methodically applied functional calculus to general nonlocal problems. In [5], the conditioning of nonlocal operators was studied together with the error analysis. The collocation method with piecewise linear basis functions was applied to the operator \(\mathcal {M}_\mathtt{BC}\) for the first time in [1, 2]. In this paper, we extend the method to basis functions with arbitrary order polynomials.

The peridynamic operator is connected to the classical operator via the scaling and a calibration process. It is desired to maintain this connection at the discrete level. One way to identify the presence of such a connection is by the asymptotic compatibility of discretization schemes [23, 24]. Asymptotically compatible schemes provide a freedom of choice in the way one couples modeling parameters with discretization parameters. This freedom ensures that nonlocal solutions converge to the correct local solution as the horizon parameter \(\delta \rightarrow 0\). At the analytical level, there is a direct connection between our nonlocal operators and the local one. The governing operator \(\mathcal {M}_\mathtt{BC}\) is constructed by the functional calculus [12] and, hence, is a function of the classical operator \(-\Delta _\mathtt{BC}:= -\frac{\mathrm {d}^2{}}{\mathrm {d}x^2}\) equipped with boundary conditions [7, 8]. The directness of the connection enables us to determine the calibration constant independent of \(\delta\) and h, which is critical to obtaining asymptotically compatible schemes.

The asymptotic compatibility of discretization schemes was introduced for nonlocal problems with homogeneous nonlocal BC in the pioneering work of Du and Tian [23, 24]. Our study is novel in the way that we apply the asymptotic compatibility to nonlocal problems with inhomogeneous local BC. Hence, our study extends the applicability of asymptotic compatibility to nonlocal problems with local BC. We emphasize that the BC in our construction are local. The local nature of the BC and the fact that the BC are satisfied for any \(\delta >0\), not just as \(\delta \rightarrow 0\), are two distinguishing features from various other approaches. Inhomogeneous BC can be easily treated in our framework, which were also considered in [13, 22], and recently [14, 28]. We also note that another distinguishing feature of the present work is the higher orders used in collocation methods. Different discretization schemes were the subject of asymptotic compatibility in the study of nonlocal problems with homogeneous nonlocal BC; see [23] for finite element and finite difference methods with \(p=0,1\), [24] for the discontinuous finite element method with \(p=1\) and [18] for the arbitrary p, and [29] for the collocation method with \(p=1\), where p denotes the order of discretization. In this paper, we study higher order collocation discretizations and show that higher order (\(p \geqslant 2\)) collocation schemes enjoy the asymptotic compatibility, whereas the lower order (\(p=0,1\)) schemes do not.

Several aspects of the nonlocal diffusion and asymptotic compatibility of discretizations have been studied [15, 16, 18, 25,26,27]. The asymptotic compatibility relies on the symmetric positive definiteness and M-matrix property of stiffness matrices. Similar properties were reported in [6] as well. One difference in our study is that the stiffness matrices are centrosymmetric (symmetric with respect to the center). Similar to the rigorous treatment in [24], a relevant future research avenue would be to develop an abstract mathematical framework for the asymptotic compatibility that can accommodate the centrosymmetry. In [17], local BC have been explored in the study of coupling nonlocal and local models. Numerical results free of surface effects have been reported. Since our operators enforce local BC, they altogether avoid surface effects and can also be used to couple nonlocal and local models.

The rest of the paper is structured as follows. In Sect. 2, we explain the construction of the governing operators using the functional calculus. In Sect. 3, we introduce the kernel functions employed and explain the reason for the zero row sum property. In Sect. 4, we explain the direct connection between the operators \(\mathcal {M}_\mathtt{BC}\) and \(-\Delta _\mathtt{BC}\) at the analytical level. In Sect. 5, we introduce the collocation methods with arbitrary order polynomial degree p and carry out their error analysis in Sect. 6. In Sect. 7, we show how to scale and calibrate the discretized operator to obtain the asymptotic compatibility. In Sect. 8, we provide examples of obtaining the appropriate calibration constants using Taylor series expansions. In Sect. 9, we present numerical results to verify our theoretical results on the convergence of the collocation methods. In Sect. 10, we carry out numerical experiments to ascertain the asymptotic compatibility of the collocation methods. We conclude in Sect. 11.

The Construction of the Governing Operators

In this section, we explain the key steps in construction of the governing operator \(\mathcal {M}_\mathtt{BC}\). We observe that the peridynamic governing operator contains a convolution operator. First, we construct the convolution operators \(\mathcal {C}_\mathtt{a}\) and \(\mathcal {C}_\mathtt{p}\) with antiperiodic and periodic BC, respectively, using the eigenfunctions

$$\begin{aligned} e_k^\mathtt{a}(x) := \frac{1}{\sqrt{2}} {\text {e}}^{{\text{i}} \pi (k+\frac{1}{2})x}, \quad k \in \mathbb {N} \quad \text {and} \quad e_k^\mathtt{p}(x) := \frac{1}{\sqrt{2}} {\text {e}}^{{\text{i}} \pi k x}, \quad k \in \mathbb {N}\end{aligned}$$
(2)

of the classical operator \(-\Delta _\mathtt{a}\) and \(-\Delta _\mathtt{p}\) in which the BC information is already encoded. For a given kernel function \(C \in L^2(\Omega )\), the convolution operator, for \(u \in L^2(\Omega )\), is defined as

$$\begin{aligned} \mathcal {C}_\mathtt{BC}u(x) := \sqrt{2} \sum _{k \in \mathbb {N}} \langle e_k^\mathtt{BC}|C\rangle \langle e_k^\mathtt{BC}|u\rangle e_k^\mathtt{BC}(x), \quad \mathtt{BC}\in \{\mathtt{a}, \mathtt{p}\}, \end{aligned}$$

where \(\langle \cdot |\cdot \rangle\) denotes the \(L^2(\Omega )\) inner product. We define \(\mathbb {N}_\mathtt{D}:= \mathbb {N}{\setminus } \{0\}\) and \(\mathbb {N}_\mathtt{N}:= \mathbb {N}\). The operators \(\mathcal {C}_\mathtt{BC}\) turn out to be bounded functions of the classical operator \(-\Delta _\mathtt{BC}\), thereby maintaining the connection to \(-\Delta _\mathtt{BC}\).

In this study, we consider only the operators \(\mathcal {M}_\mathtt{D}\) and \(\mathcal {M}_\mathtt{N}\) where \(\mathtt{D}\) and \(\mathtt{N}\) denote the Dirichlet and Neumann BC. Hence, in the rest of the discussion, we set \(\mathtt{BC}\in \{\mathtt{D}, \mathtt{N}\}\). The operator \(\mathcal {M}_\mathtt{BC}\) is constructed using the functional calculus on the classical self-adjoint operator \(-\Delta _\mathtt{BC}\). We are in search of a suitable regulating function \(f_\mathtt{BC}:\sigma (-\Delta _\mathtt{BC}) \rightarrow \mathbb {R}\) that would connect the nonlocal operator \(\mathcal {M}_\mathtt{BC}\) to \(-\Delta _\mathtt{BC}\), i.e., \(\mathcal {M}_\mathtt{BC}= f_\mathtt{BC}(-\Delta _\mathtt{BC})\). This regulating function should be bounded so that the end product \(\mathcal {M}_\mathtt{BC}\) is a bounded operator. Eventually, we end up with the nonlocal governing operator \(\mathcal {M}_\mathtt{BC}\) that is densely defined in \(L^2(\Omega )\) with a domain that encodes the prescribed BC, linear, bounded, and self-adjoint. Therefore, the operator \(\mathcal {M}_\mathtt{BC}\) has a unique bounded extension to \(L^2(\Omega )\). We find that a construction involving densely defined operators provides a suitable framework for treating local BC in the nonlocal wave equation. Throughout the paper, we assume that the solution \(u^\mathtt{BC}\) and the kernel functions C satisfy

$$\begin{aligned} u^\mathtt{BC}, C \in L^2(\Omega ). \end{aligned}$$

In this work, the choice of \(f_\mathtt{BC}\) is inspired by the theory of peridynamics. In prior work, we discovered that the peridynamic governing operator for the case \(\Omega = \mathbb {R}\) is a function of the classical operator [12]. We reuse that the regulating function for the case of \(\Omega = (-1,1)\). Our choice of regulating functions \(f_\mathtt{BC}: \sigma (-\Delta _\mathtt{BC}) \rightarrow \mathbb {R}\) is

$$\begin{aligned} f_\mathtt{BC}(\lambda _k^\mathtt{BC}) = \langle 1|C\rangle - \sqrt{2} {\left\{ \begin{array}{ll} \langle e_{k/2}^{\mathtt{p}}|C\rangle &{} \text {if }\; k \in \mathbb {N}_\mathtt{BC}\text { is even,} \\ \langle e_{(k-1)/2}^\mathtt{a}|C\rangle &{} \text {if }\; k \in \mathbb {N}_\mathtt{BC}\text { is odd,} \end{array}\right. } \end{aligned}$$
(3)

where \(\lambda _k^\mathtt{BC}\in \sigma (-\Delta _\mathtt{BC})\) denotes the kth eigenvalue of the operator \(-\Delta _\mathtt{BC}\). Utilizing the convolution operators \(\mathcal {C}_\mathtt{a}\) and \(\mathcal {C}_\mathtt{p}\) obtained by functional calculus on \(-\Delta _\mathtt{a}\) and \(-\Delta _\mathtt{p}\), respectively, defining

$$\begin{aligned} c := \langle 1|C\rangle = \int _\Omega C(x)\,\mathop {}\!\mathrm {d}x, \end{aligned}$$
(4)

we proved in [4, 7] that

$$\begin{aligned} f_\mathtt{D}(-\Delta _\mathtt{D}) u^\mathtt{D}& = \big ( c - \mathcal {C}_\mathtt{a}P_{\text{e}} - \mathcal {C}_\mathtt{p}P_{\text{o}} \big ) u^\mathtt{D}= {} \, \mathcal {M}_\mathtt{D}u^\mathtt{D}, \\ f_\mathtt{N}(-\Delta _\mathtt{N}) u^\mathtt{N} & = \big ( c - \mathcal {C}_\mathtt{p}P_{\text{e}} - \mathcal {C}_\mathtt{a}P_{\text{o}} \big ) u^\mathtt{N}= {} \mathcal {M}_\mathtt{N}u^\mathtt{N}, \end{aligned}$$

where we denote the orthogonal projections that give the even and odd parts, respectively, by \(P_{\text{e}},P_{\text{o}}:~L^2(\Omega )~\rightarrow ~L^2(\Omega ),\) whose definitions are

$$\begin{aligned} P_{\text{e}} u(x) := \frac{u(x) + u(-x)}{2}, \quad P_{\text{o}} u(x) := \frac{u(x) - u(-x)}{2}. \end{aligned}$$

The crucial step in the construction of \(\mathcal {M}_\mathtt{BC}\) is the application of the spectral theorem for bounded operators. Namely, for \(u^\mathtt{BC}= \sum _k \langle e_k^\mathtt{BC}|u^\mathtt{BC}\rangle e_k^\mathtt{BC}\), we have

$$\begin{aligned} \mathcal {M}_\mathtt{BC}u^\mathtt{BC}= f_\mathtt{BC}(-\Delta _\mathtt{BC}) u^\mathtt{BC}= \sum _{k \in \mathbb {N}_\mathtt{BC}} f_\mathtt{BC}(\lambda _k^\mathtt{BC}) \langle e_k^\mathtt{BC}|u^\mathtt{BC}\rangle e_k^\mathtt{BC}. \end{aligned}$$
(5)

For an extended discussion on the treatment of general nonlocal problems using functional calculus, see [9].

An integral representation of the series (5) is more convenient for implementation. We gave such representations in [7] and the governing operators take the form

$$\begin{aligned}\big({\mathcal{M}}_{\mathtt{BC}}- c \big ) u^{\mathtt{BC}}(x,t)= - \int _\Omega K_{\mathtt{BC}}(x,x')u^{\mathtt{BC}}(x',t) \mathop {}\!\mathrm {d}x', \end{aligned}$$
(6)
$$\begin{aligned} K_\mathtt{D}(x,x'):= {} \, \frac{1}{2} \big \{ \big [ \widehat{C}_\mathtt{a}(x'-x) + \widehat{C}_\mathtt{a}(x'+x) \big ] + \big [ \widehat{C}_\mathtt{p}(x'-x) - \widehat{C}_\mathtt{p}(x'+x) \big ] \big \}, \nonumber \\ K_\mathtt{N}(x,x'):= {} \frac{1}{2} \big \{ \big [ \widehat{C}_\mathtt{p}(x'-x) + \widehat{C}_\mathtt{p}(x'+x) \big ] + \big [ \widehat{C}_\mathtt{a}(x'-x) - \widehat{C}_\mathtt{a}(x'+x) \big ] \big \}, \end{aligned}$$

where we denote the periodic and antiperiodic extensions of C(x) from \((-1,1)\) to \((-2,2)\), respectively, as follows:

$$\begin{aligned} \widehat{C}_\mathtt{p}(x):= &\left\{ {\begin{array}{ll} C(x+2), & x \in (-2,-1), \\ C(x), & x \in (-1,1), \\ C(x-2), & x \in (1,2), \end{array}} \right. \\ \widehat{C}_\mathtt{a}(x):= &\left\{ {\begin{array} {ll} {-C(x+2)}, & {x \in (-2,-1)}, \\ {C(x),} & {x \in (-1,1),} \\ {-C(x-2),} & {x \in (1,2)}. \end{array} }\right. \end{aligned}$$

In addition, \(u_x^\mathtt{N}\) represents the strain in (1).

The Kernel Functions and Zero Row Sum Property

Fig. 1
figure1

Plot of the univariate kernel function C(x) with \(\delta =2^{-1}\)

Fig. 2
figure2

Plot of the bivariate kernel functions \(K_\mathtt{D}(x,x')\) (left) and \(K_\mathtt{N}(x,x')\) (right) with \(\delta =2^{-1}\)

A family of kernel functions with horizon size \(\delta\) is chosen as

$$C_1(x) := \left\{ {\begin{array}{ll} 1 \\ 0 \\ \end{array} } \right.\begin{array}{ll} {{\text{if}}\;x \in ( - \delta ,\delta ),} \\ {{\text{otherwise}}}, \\ \end{array}$$
(7)
$$C_{2,s}(x) := \left\{ {\begin{array}{ll} 1 - |\frac{x}{\delta } |^s \\ 0 \\ \end{array} } \right.\begin{array}{ll} {{\text{if}}\;x \in ( - \delta ,\delta ),} \\ {{\text{otherwise}}} \\ \end{array}$$
(8)

with \(s>0\). Next, we want to elaborate on the choice of kernel functions. Univariate and bivariate kernel functions are illustrated in Figs. 1 and 2, respectively. First note that

$$\begin{aligned} \lim _{s \rightarrow \infty } C_{2,s}(x) = C_1(x). \end{aligned}$$

Namely, \(C_{2,s}\) is chosen a family that approaches \(C_1\). The parameter s associated with the \(C_{2,s}\) family provides a way to monitor the matrix properties as \(s \rightarrow \infty\).

Next, we elaborate on a crucial property of the discrete version of the operator \(\mathcal {M}_\mathtt{BC}\). First, note that the constant function \(u(x,t) \equiv k\) is in the kernel of the \(\mathcal {M}_\mathtt{N}\) operator. Since

$$\begin{aligned} \mathcal {M}_\mathtt{N}\,k \equiv 0, \end{aligned}$$
(9)

any discrete version of the operator \(\mathcal {M}_\mathtt{N}\) should satisfy

$$\begin{aligned} \mathcal {M}_\mathtt{N}\, 1_h = 0_h. \end{aligned}$$

In other words, it has the zero row sum property for all of its rows. Since \(\mathcal {M}_\mathtt{N}\) and \(\mathcal {M}_\mathtt{D}\) agree in the bulk, the zero row sum property holds for the discrete version of \(\mathcal {M}_\mathtt{D}\) for all rows corresponding to the bulk. We pay special attention to maintain the zero row sum property at machine precision. Otherwise, for small \(\delta\), round-off errors spoil the zero row sum property.

Let us discuss why a fitted grid is essential for the asymptotic compatibility in our numerical method; the utilized fitted grids are illustrated in Fig. 3. The boundary data reside in the first and last degrees of freedom (DOF). Hence, the first and last columns of the stiffness matrix are the most important columns for propagating data into the domain [1, 2]. For instance, consider the Dirichlet problem with the kernel \(C_1\). With a uniform grid, the first/last column entries corresponding to the DOF between the boundary and the bulk vanish simply due to the structure of the kernel function, see Fig. 2. As a result, boundary data cannot propagate into the domain. When one chooses \(C_{2,s}\) as the kernel function, the entries in the first/last column become nonzero. But, this is a partial fix because now the rows corresponding to the DOF do not satisfy the zero row sum property, which is a requirement of the kernel of the discrete version of \(\mathcal {M}_\mathtt{BC}\). In summary, the zero row sum property rules out the presence of a DOF outside of the bulk except the local boundary and, hence, a fitted grid becomes a requirement for our discretization.

Fig. 3
figure3

Fitted grids with 6 nodes used to construct an approximate solution with polynomial order p. The boundary nodes are always degrees of freedom due to enforcing local boundary conditions. The shape of kernel functions and the requirement of zero row sum property for the stiffness matrix rule out the presence of a degree of freedom in the \(\delta\)-neighborhood (horizon) of boundary nodes, which forces us to utilize fitted grids. Here \(\delta = 3 \,h\) for each polynomial order

Eigenvalues of the Governing Operators

We mentioned that there exists a direct connection between the operators \(\mathcal {M}_\mathtt{BC}\) and \(-\Delta _\mathtt{BC}\) at the analytical level. Now, we explain what we mean by a direct connection. Since \(\mathcal {M}_\mathtt{BC}\) is constructed as a function of \(-\Delta _\mathtt{BC}\), one can explicitly express the eigenvalues of \(\mathcal {M}_\mathtt{BC}\) in terms of those of \(-\Delta _\mathtt{BC}\) [1, 2, 9] using the regulating function \(f_\mathtt{BC}\) in (3) employed in functional calculus:

$$\begin{aligned} \lambda _k(\mathcal {M}_\mathtt{BC}) = f_\mathtt{BC}(\lambda _k(-\Delta _\mathtt{BC})). \end{aligned}$$

The eigenvalues of the classical operator are given by

$$\begin{aligned} \lambda _k(-\Delta _\mathtt{BC}) = \frac{\pi ^2}{4} k^2, \quad k \in \mathbb {N}_\mathtt{BC}. \end{aligned}$$

Next, we explain how the calibration constant \(\gamma ^{C}\) is defined at the analytical level for a generic kernel function C. To fix ideas, let us present the case of \(C=C_{2,s}\). One employs a calibration constant \(\gamma ^{C_{2,s}}\) defined in such a way that the dominant term of \(\lambda _k(\mathcal {M}_\mathtt{BC})\) becomes equal to \(\lambda _k(-\Delta _\mathtt{BC})\). More precisely,

$$\begin{aligned} \lambda _k(\gamma ^{C_{2,s}} \delta ^{-3} \mathcal {M}_\mathtt{BC}^{C_{2,s}}) = \frac{\pi ^2}{4} k^2 + {\mathcal {O}}(\delta k). \end{aligned}$$

The calibration process takes place at the discrete level in practice in the study of asymptotic compatibility. When the operator \(\mathcal {M}_\mathtt{BC}\) is discretized, the calibration constant depends on the polynomial degree p and is determined by a Taylor expansion. A detailed discussion of the effect of discretization on the calibration constant will be provided in Sect. 8.

In this section, when we write \(\mathcal {M}_\mathtt{BC}\) without attaching a superscript, we mean that the scaling and calibration have already been incorporated. When we want to indicate, the kernel function and its calibration constant explicitly, for instance, \(C_{2,s}\) and \(\gamma\), we write the governing operator with a notation reflecting this choice:

$$\begin{aligned} \gamma ^{C_{2,s}} \; \delta ^{-3} \mathcal {M}_{\mathtt{BC}}^{C_{2,s}}. \end{aligned}$$

The explicit expressions of the eigenvalues for the kernel functions \(C_1\) and \(C_{2,s}\) with \(s=1,2,5\) are

$$\begin{aligned} &\lambda _k(3\delta ^{-3} \mathcal {M}_\mathtt{BC}^{C_1}) =\, 3\delta ^{-3} \left[ 4\delta - \frac{4}{k \pi } \sin \left( \displaystyle \frac{k \pi \delta }{2}\right) \right], \\ &\lambda _k\left( 12 \delta ^{-3} \mathcal {M}_{\mathtt{BC}}^{C_{2,1}}\right) =\, 12 \delta ^{-3} \left[ \delta - \frac{8}{k^2 \pi ^2 \delta } \left( 1 - \cos \left( \displaystyle \frac{k \pi \delta }{2}\right) \right) \right], \\ &\lambda _k\left( \frac{15}{2}\delta ^{-3} \mathcal {M}_{\mathtt{BC}}^{C_{2,2}}\right) =\, \frac{15}{2} \delta ^{-3} \left[ \displaystyle \frac{4 \delta }{3} + \frac{16}{k^3 \pi ^3 \delta ^2} \left( k \pi \delta \cos \left( \displaystyle \frac{k \pi \delta }{2}\right) - 2 \sin \left( \displaystyle \frac{k \pi \delta }{2}\right) \right) \right], \\ &\lambda _k\left( \frac{24}{5} \delta ^{-3} \mathcal {M}_{\mathtt{BC}}^{C_{2,5}}\right) =\, \frac{24}{5} \delta ^{-3}\ \Bigg [\displaystyle \frac{5 \delta }{3} - \frac{40}{k^6 \pi ^6 \delta ^5} \Bigg (384 - (384- 48 k^2 \pi ^2 \delta ^2 +k^4 \pi ^4 \delta ^4) \cos \left( \displaystyle \frac{k \pi \delta }{2}\right) \\ &\qquad \qquad \qquad \qquad + 8 k \pi \delta (-24 + k^2 \pi ^2 \delta ^2) \sin \left( \displaystyle \frac{k \pi \delta }{2}\right) \Bigg ) \Bigg ]. \end{aligned}$$
(10)

It is instructive to examine the Taylor series expansions of the eigenvalues in (10) to clearly see the connection to the eigenvalues of the classical operator. Assuming \(\delta k < 1\), the Taylor series expansions are

$$\begin{aligned} \begin{aligned} &\lambda _k(3\delta ^{-3} \mathcal {M}_\mathtt{BC}^{C_1})= \frac{\pi ^2}{4} k^2 \Big (1 - \frac{\pi ^2}{80} (\delta k)^2 + \frac{\pi ^2}{13\,440} (\delta k)^4 + {\mathcal {O}}((\delta k)^6) \Big ), \\ &\lambda _k\left( 12 \delta ^{-3} \mathcal {M}_{\mathtt{BC}}^{C_{2,1}}\right)= \frac{\pi ^2}{4} k^2 \Big (1 - \frac{\pi ^2}{120} (\delta k)^2 + \frac{\pi ^2}{26\,880} (\delta k)^4 + {\mathcal {O}}((\delta k)^6) \Big ), \\ &\lambda _k\left( \frac{15}{2}\delta ^{-3} \mathcal {M}_{\mathtt{BC}}^{C_{2,2}}\right)= \frac{\pi ^2}{4} k^2 \Big (1 - \frac{\pi ^2}{112} (\delta k)^2 + \frac{\pi ^2}{24\,192} (\delta k)^4 + {\mathcal {O}}((\delta k)^6) \Big ), \\ &\lambda _k\left( \frac{24}{5} \delta ^{-3} \mathcal {M}_{\mathtt{BC}}^{C_{2,5}}\right)= \frac{\pi ^2}{4} k^2 \Big (1 - \frac{\pi ^2}{100} (\delta k)^2 + \frac{\pi ^2}{20\,160} (\delta k)^4 + {\mathcal {O}}((\delta k)^6) \Big ). \end{aligned} \end{aligned}$$
(11)

The preceding equations suggest that as s increases, the eigenvalues of \(\gamma ^{C_{2,s}} \delta ^{-3} \mathcal {M}_{\mathtt{BC}}^{C_{2,s}}\) approach to those of \(\gamma ^{C_1} \delta ^{-3} \mathcal {M}_{\mathtt{BC}}^{C_1}\). A rigorous and detailed discussion is the subject of ongoing work. Note also that the eigenvalues of the nonlocal operators all approach the classical eigenvalue from below, see (11). As a result, the dispersion relations of the nonlocal operator approach the classical (linear) dispersion relation from below. This indicates that the nonlocal wave speeds are less than the classical wave speed, a direct consequence of dispersion. Similar behavior has been observed for other families of kernel functions [19, Sec. 4.1.1]. It is worth noting that the expressions in (11) are all \({\mathcal {O}}((\delta k)^2)\) approximations to \(\lambda _k(-\Delta _\mathtt{BC})\). We will allude to the effects of this \({\mathcal {O}}(\delta ^2)\) perturbation on the order of asymptotic convergence of the collocation methods in Sect. 10.

The Collocation Methods

In this section, we describe the collocation methods we are utilizing. We begin with discretizing the domain \(\Omega = (-1,1)\) by a set of \(n+1\)nodes

$$\begin{aligned} \{z_1 = -1< z_2< z_3< \cdots< z_n < z_{n+1}=1 \} \end{aligned}$$

and choose a polynomial order \(p \geqslant 0\). We also define the stencil size

$$\begin{aligned} h_i = {\left\{ \begin{array}{ll} z_{i+1}-z_i \quad &{}{\text {if }}\; p = 0, \\ \frac{1}{p}(z_{i+1}-z_i) &{}\text{ if } \; p \geqslant 1, \end{array}\right. } \end{aligned}$$

the grid size

$$\begin{aligned} \mathtt{h}_i = z_{i+1}-z_i = {\left\{ \begin{array}{ll} h_i \quad &{}\text{ if } \; p = 0, \\ p\,h_i &{}\text{ if } \; p \geqslant 1, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} I_i := (z_i,z_{i+1}),\quad i=1,\cdots ,n. \end{aligned}$$

We set

$$\begin{aligned} h = \max _{1 \leqslant i \leqslant n}h_i \quad {\text {and}}\quad \mathtt{h}= \max _{1 \leqslant i \leqslant n}\mathtt{h}_i. \end{aligned}$$

Next, we introduce the internal DOF depending on the polynomial degree. For \(p=0\), we take \(n-2\) internal and two boundary DOF, denoted by \(\{\xi _1, \xi _2,\cdots ,\xi _n\}\), where

$$\begin{aligned} \xi _i = {\left\{ \begin{array}{ll} -1 \quad &{}\text{ if } \; i = 1, \\ \frac{1}{2}(z_i+z_{i+1}) &{}\text{ if } \; i = 2,\cdots ,n-1, \\ 1 &{}\text{ if } \; i = n. \end{array}\right. } \end{aligned}$$

For \(p=1\), no internal DOF is needed, and for \(p \geqslant 2\), we introduce \(p-1\) internal DOF between any two consecutive nodes as follows:

$$\begin{aligned} \xi _i^j = z_i + jh_i, \quad j = 1,2,\cdots ,p-1. \end{aligned}$$

The set of all DOF involved in the discretization is the union of the set of nodes and the set of the internal DOF. More explicitly,

$$\begin{aligned} \mathcal {E}_h:= {\left\{ \begin{array}{ll} \{\xi _1, \cdots ,\xi _n\} \quad &{}\text{ if } p = 0, \\ \{z_1, \cdots , z_{n+1}\} &{}\text{ if } p = 1, \\ \{z_1, \cdots , z_{n+1}\}\cup \{\xi _i^j : 1 \leqslant i \leqslant n,\, 1\leqslant j\leqslant p-1\} &{}\text{ if } p \geqslant 2. \end{array}\right. } \end{aligned}$$

Observe that the local boundary points \(\{-1,1\}\) are always contained in the set of the DOF. This is a requirement for the nonlocal operators to enforce local BC. The total number of the DOF is denoted by

$$\begin{aligned} N := {\left\{ \begin{array}{ll} n \quad &{}\text{ if } \; p = 0, \\ np+1 &{}\text{ if } \; p \geqslant 1. \end{array}\right. } \end{aligned}$$
(12)

For convenience of notation, we relabel the DOF as

$$\begin{aligned} \mathcal {E}_h= \{x_1,x_2,\cdots ,x_N\}. \end{aligned}$$

Next, we introduce the Lagrange basis function \(\phi _j(x)\) associated with the jth DOF. More explicitly, \(\phi _j\) is the piecewise polynomial of degree p on \(\Omega\) such that

$$\begin{aligned} \text {supp}(\phi _j) = {\left\{ \begin{array}{ll} \overline{I}_{j-1} \cup \overline{I}_j \quad &{}\text{ if } \; x_j \text{ is } \text{ not } \text{ an } \text{ internal } \text{ DOF}, \\ \overline{I}_i \quad &{}\text{ if } \; x_j \text{ is } \text{ an } \text{ internal } \text{ DOF } \text{ such } \text{ that } x_j \in I_i \end{array}\right. } \end{aligned}$$

with obvious modifications for \(j = 1\) and \(j = n\). For \(x_i \in \mathcal {E}_h\), the basis function \(\phi _j(x)\) satisfies

$$\begin{aligned} \phi _j(x_i) = {\left\{ \begin{array}{ll} 1 \quad &{}\text{ if } \; i = j, \\ 0 &{}\text{ if } \; i \ne j. \end{array}\right. } \end{aligned}$$

For a plot of basis functions for \(p=0,1,2,3\), see Fig. 4.

Fig. 4
figure4

Basis functions. Square and circle represent a node and an internal DOF, respectively. If a square is empty, it means it is not a DOF, which occurs only for \(p=0\)

The collocation methods seek an approximation

$$\begin{aligned} u_h^\mathtt{BC}(x) = \sum _{j=1}^{N} u_j^\mathtt{BC}\phi _j(x) \end{aligned}$$

to the solution \(u^\mathtt{BC}(x)\) of

$$\begin{aligned} \begin{aligned} &\mathcal {M}_\mathtt{BC}u^\mathtt{BC}(x) =\, b^\mathtt{BC}(x), \quad x \in \Omega , \\ &u^\mathtt{D}(\pm 1) =\, \alpha _{\pm }^\mathtt{D}\quad \text {or} \quad \frac{\mathrm {d}{u^\mathtt{N}}}{\mathrm {d}x}(\pm 1) = \alpha _{\pm }^\mathtt{N}\end{aligned} \end{aligned}$$
(13)

such that

$$\begin{aligned} \mathcal {M}_\mathtt{BC}u_h^\mathtt{BC}(x_i) = b^\mathtt{BC}(x_i) \quad {\text {for }}\; i = 1,2,\cdots ,N. \end{aligned}$$

Note that (13) is the stationary version of (1). We restrict ourselves to this version because we are primarily concerned with nonlocality in space and, hence, the role of time dependence is ignored. For the remainder of the paper, the governing equation becomes (13). Inserting the expression of \(u_\mathtt{h}^\mathtt{BC}(x)\) and using the linearity of the operator \(\mathcal {M}_\mathtt{BC}\), we get

$$\begin{aligned} \sum _{j=1}^{N} u_j^\mathtt{BC}\mathcal {M}_\mathtt{BC}\phi _j(x_i) = b^\mathtt{BC}(x_i) \quad {\text {for} }\; i=1,2,\cdots ,N. \end{aligned}$$
(14)

The operator \(\mathcal {M}_\mathtt{D}\) is designed in a way that

$$\begin{aligned} c\frac{\gamma ^C}{\delta ^3}u^\mathtt{D}(\pm 1) = b^\mathtt{D}(\pm 1), \end{aligned}$$
(15)

and \(\mathcal {M}_\mathtt{N}\) is such that

$$\begin{aligned} c\frac{\gamma ^C}{\delta ^3}\frac{\mathrm {d}{u^\mathtt{N}}}{\mathrm {d}x}(\pm 1) = \frac{\mathrm {d}{b^\mathtt{N}}}{\mathrm {d}x}(\pm 1). \end{aligned}$$
(16)

For details, see [4, Sec. 5] and [10, Sec. 3.1]. The fact that the boundary nodes are always part of the DOF, the collocation method ensures that the numerical solution \(u_h^\mathtt{BC}\) inherits this property.

The operator \(\mathcal {M}_\mathtt{N}\) has a non-trivial kernel as pointed out in (9). To obtain a unisolvent linear system, we modify the stiffness matrix with a rank-one perturbation. More explicitly,

$$\begin{aligned} \big (\mathcal {M}_\mathtt{N}+ 1_h 1_h^t\big )u_h^\mathtt{N}= b_h^\mathtt{N}. \end{aligned}$$

This ensures that the solution satisfies

$$\begin{aligned} \int _{\Omega } u_h^\mathtt{N}(x)\,\mathop {}\!\mathrm {d}x = 0. \end{aligned}$$
(17)

Error Analysis

To utilize the Fredholm theory, we define the compact operator \(\mathcal {K}_\mathtt{BC}\) such that

$$\begin{aligned} \mathcal {K}_\mathtt{BC}u^\mathtt{BC}(x) := \int _\Omega K_\mathtt{BC}(x,x')u^\mathtt{BC}(x')\mathop {}\!\mathrm {d}x', \end{aligned}$$
(18)

so that \(\mathcal {M}_\mathtt{BC}= c - \mathcal {K}_\mathtt{BC}\) falls into the class of Fredholm operators of the second kind. A key tool for our analysis is the projection operator \(\mathcal {P}_h : C(\overline{\Omega }) \rightarrow V_h\), where

$$\begin{aligned} V_h := \text {span}\{\phi _1,\cdots ,\phi _N\}, \end{aligned}$$

where N is the total number of the DOF as defined in (12). Clearly, \(V_h\) is a subspace of \(C(\overline{\Omega })\) for \(p \geqslant 1\). Given \(u \in C(\overline{\Omega })\), define \(\mathcal {P}_h u\) to be that element of \(V_h\) which interpolates u at the DOF \(\mathcal {E}_n\). Since our choice of the basis functions \(\{\phi _j\}\) is nodal Lagrange, the projection \(\mathcal {P}_h\) is written as

$$\begin{aligned} \mathcal {P}_h u(x) = \sum _{j=1}^{N} u(x_j) \phi _j(x) \end{aligned}$$
(19)

with the operator norm [11, (12.1.9)]

$$\begin{aligned} \Vert \mathcal {P}_h\Vert _{\infty } = \max _{x \in \Omega } \sum _{j=1}^N |\phi _j(x)|. \end{aligned}$$
(20)

Note that the collocation method (14) can be expressed as

$$\begin{aligned} \mathcal {P}_h \mathcal {M}_\mathtt{BC}u_h^\mathtt{BC}= \mathcal {P}_h b^\mathtt{BC}\end{aligned}$$

or equivalently as

$$\begin{aligned} (c-\mathcal {P}_h\mathcal {K}_\mathtt{BC}) u_h^\mathtt{BC}= \mathcal {P}_h b^\mathtt{BC}. \end{aligned}$$
(21)

The following theorem states the approximation properties of the projection operator \(\mathcal {P}_h\). Its proof is a straightforward application of the Taylor series theorem with the remainder term.

Theorem 6.1

Let\(u \in C^{p+1}(\overline{\Omega })\)and let\(\mathcal {P}_h\)be the projection operator defined by (19). Then, there exists a constant\(\text{ C }\)independent ofhsuch that

$$\begin{aligned} \Vert u-\mathcal {P}_h u\Vert _{\infty } \leqslant \text{ C }h^{p+1}\Vert u^{(p+1)}\Vert _{\infty }. \end{aligned}$$

It turns out that the suitable spaces for the error analysis of the collocation methods for Dirichlet and Neumann BC are different. That is why we split the rest of the discussion into two subsections.

Existence, Uniqueness, and the Error Estimates for the Case of Dirichlet BC

We first prove that the operator \(\mathcal {M}_\mathtt{D}\) is invertible on \(L^2(\Omega )\). Then, we show that it is also invertible on \(C(\overline{\Omega })\), the suitable working space for Dirichlet BC. After that, we prove an error estimate in terms of the projection error. Optimal error estimates for the approximate solution follows from the approximation properties of the projection operator. We note that the case of \(p=0\) requires special attention which we detail below.

Theorem 6.2

The operator \(\mathcal {M}_\mathtt{D}: L^2(\Omega ) \rightarrow L^2(\Omega )\) is a bijection.

Proof

We start with the proof of injectivity of the \(\mathcal {M}_\mathtt{D}\) operator. We have proved in [9] that the eigenvalues of \(\mathcal {M}_\mathtt{D}\) are as follows:

$$\begin{aligned} \lambda _k(\mathcal {M}_\mathtt{D})&= \langle 1|C\rangle - \sqrt{2} {\left\{ \begin{array}{ll} \langle e_{k/2}^{\mathtt{p}}|C\rangle &{} \text {if }\; k \in \mathbb {N}^*\text { is even,} \\ \langle e_{(k-1)/2}^\mathtt{a}|C\rangle &{} \text {if }\; k \in \mathbb {N}^*\text { is odd,} \end{array}\right. } \end{aligned}$$

where \(\mathbb {N}^*:= \mathbb {N}{\setminus } \{0\}\), and the eigenfunctions \(e_k^\mathtt{a}(x)\) and \(e_k^\mathtt{a}(x)\) are given in (2).

Since \(C(x) \geqslant 0\), it follows that

$$\begin{aligned} \lambda _k(\mathcal {M}_\mathtt{D}) > 0, \quad k \in \mathbb {N}^*. \end{aligned}$$
(22)

To see this, let \(k = 2k' \in \mathbb {N}^*\) be even. Then,

$$\begin{aligned} \lambda _k(\mathcal {M}_\mathtt{D})&= \langle 1|C\rangle - \sqrt{2}\langle e_{k/2}^{\mathtt{p}}|C\rangle \\&= \langle 1|C\rangle - \int _{\Omega } {\text {e}}^{{\text{i}}\pi k'x}C(x){\text {d}}x \\&= \int _{\Omega } (1-\cos (\pi k'x))C(x){\text {d}}x \quad {\text {since } } C \text{ is } \text{ even}, \\&> 0 \end{aligned}$$

since \((1-\cos (\pi k'x))C(x)\) is nonnegative and not identically zero. The case of \(k \in \mathbb {N}^*\) odd follows similarly.

Now, we will prove that

$$\begin{aligned} {\underline{\lambda }} := \inf _{k \in \mathbb {N}^*} \lambda _k(\mathcal {M}_\mathtt{D}) > 0. \end{aligned}$$
(23)

By the Riemann–Lebesgue lemma,

$$\begin{aligned} \lim _{k \rightarrow \infty } \lambda _k(\mathcal {M}_\mathtt{D}) = \langle 1|C\rangle = c. \end{aligned}$$

Since \(c > 0\), we can choose \(\epsilon >0\) such that \(c - \epsilon > 0\). Let \(m \in \mathbb {N}^*\) such that \(|\lambda _k(\mathcal {M}_\mathtt{D}) - c| < \epsilon\) for all \(k \geqslant m\). Then, \(\lambda _k(\mathcal {M}_\mathtt{D}) > c - \epsilon\) for all \(k \geqslant m\). Hence,

$$\begin{aligned} \lambda _*:= \inf _{k \geqslant m} \lambda _k(\mathcal {M}_\mathtt{D}) \geqslant c -\epsilon > 0. \end{aligned}$$

Consequently,

$$\begin{aligned} {\underline{\lambda }} := \min \{\lambda _1(\mathcal {M}_\mathtt{D}),\lambda _2(\mathcal {M}_\mathtt{D}), \cdots , \lambda _{m-1}(\mathcal {M}_\mathtt{D}), \lambda _*\} > 0, \end{aligned}$$
(24)

which proves (23).

Next we prove the surjectivity of \(\mathcal {M}_\mathtt{D}\). Let \(\big (e_k^\mathtt{D}\big )_{k \in \mathbb {N}^*}\) be the Hilbert (complete and orthonormal) basis of \(L^2(\Omega )\) associated with \(\mathcal {M}_\mathtt{D}\) [7], where the eigenfunctions are given by

$$\begin{aligned} e_k^\mathtt{D}(x) := \sin \left( \frac{k \pi }{2}(x+1) \right) . \end{aligned}$$
(25)

Given \(b \in L^2(\Omega )\), on the one hand, we can express b(x) in the basis \(\big (e_k^\mathtt{D}\big )_{k \in \mathbb {N}^*}\),

$$\begin{aligned} b(x) = \sum _{k \in \mathbb {N}^*} \langle e_k^\mathtt{D}|b\rangle e_k^\mathtt{D}(x). \end{aligned}$$
(26)

On the other hand, for any \(v \in L^2(\Omega )\), by the spectral theorem for bounded operators, the action of \(\mathcal {M}_\mathtt{D}\) on v can be expressed as

$$\begin{aligned} \mathcal {M}_\mathtt{D}v(x) = \sum _{k \in \mathbb {N}^*} \lambda _k(\mathcal {M}_\mathtt{D}) \langle e_k^\mathtt{D}|v\rangle e_k^\mathtt{D}(x). \end{aligned}$$
(27)

Note that the coefficients determine v uniquely by the orthonormality of \(\big ( e_k^\mathtt{D}\big )_{k \in \mathbb {N}^*}\). Choosing \(v=u\) with coefficients

$$\begin{aligned} \langle e_k^\mathtt{D}|u\rangle = \frac{\langle e_k^\mathtt{D}|b\rangle }{\lambda _k(\mathcal {M}_\mathtt{D})}, \end{aligned}$$
(28)

and inserting them in (27) show that

$$\begin{aligned} \mathcal {M}_\mathtt{D}u (x) = b(x) \end{aligned}$$

by the virtue of (26). The choice of (28) is possible because \(\lambda _k(\mathcal {M}_\mathtt{D}) > 0\) for all \(k \in \mathbb {N}^*\), see (22). It remains to show that

$$\begin{aligned} u(x) = \sum _{k \in \mathbb {N}^*} \frac{\langle e_k^\mathtt{D}|b\rangle }{\lambda _k(\mathcal {M}_\mathtt{D})} e_k^\mathtt{D}(x) \end{aligned}$$

belongs to \(L^2(\Omega )\). By (23), we have

$$\begin{aligned} \left( \frac{\langle e_k^\mathtt{D}|b\rangle }{\lambda _k(\mathcal {M}_\mathtt{D})} \right) ^2 \leqslant \frac{\langle e_k^\mathtt{D}|b\rangle ^2}{{\underline{\lambda }}^2}. \end{aligned}$$

Since \(b \in L^2(\Omega )\), the sequence \(\big ( \langle e_k^\mathtt{N}|b\rangle \big )_{k \in \mathbb {N}^*}\) is square summable. Consequently, the sequence \(\big ( \langle e_k^\mathtt{D}|u\rangle \big )_{k \in \mathbb {N}^*}\) is also square summable.

Corollary 6.3

The operator \(\mathcal {M}_\mathtt{D}: C(\overline{\Omega }) \rightarrow C(\overline{\Omega })\) is a bijection.

Proof

The injectivity of \(\mathcal {M}_\mathtt{D}: C(\overline{\Omega }) \rightarrow C(\overline{\Omega })\) follows from that of \(\mathcal {M}_\mathtt{D}: L^2(\Omega )~\rightarrow ~L^2(\Omega )\) and viewing the eigenfunctions in (25) as members of \(C(\overline{\Omega })\).

To prove the surjectivity of \(\mathcal {M}_\mathtt{D}\), let \(b \in C(\overline{\Omega })\). Since \(C(\overline{\Omega }) \subset L^2(\Omega )\), by the surjectivity of \(\mathcal {M}_\mathtt{D}\) in \(L^2(\Omega )\), there exists \(u \in L^2(\Omega )\) such that \(\mathcal {M}_\mathtt{D}u(x) = b(x)\). Then,

$$\begin{aligned} u(x) = \frac{1}{c} \mathcal {K}_\mathtt{D}u(x) + \frac{1}{c} b(x). \end{aligned}$$
(29)

We know that for any \(u(x) \in L^2(\Omega )\), \(\mathcal {K}_\mathtt{D}u (x)\) has an extension to a function \(\widehat{\mathcal {K}_\mathtt{D}u}(x)\) that is continuous on \(\overline{\Omega }\). Consequently, the function

$$\begin{aligned} \widehat{u}(x):= \frac{1}{c} \widehat{\mathcal {K}_\mathtt{D}u}(x) + \frac{1}{c} b(x) \end{aligned}$$
(30)

belongs to \(C(\overline{\Omega })\) and it satisfies

$$\begin{aligned} \mathcal {M}_\mathtt{D}\widehat{u}(x) = b(x), \quad x \in \overline{\Omega }. \end{aligned}$$
(31)

Note that by (30), (31) holds if and only if

$$\begin{aligned} \widehat{\mathcal {K}_\mathtt{D}u}(x) = \mathcal {K}_\mathtt{D}\widehat{u}(x), \quad x \in \overline{\Omega }. \end{aligned}$$
(32)

Since the restriction of \(\widehat{\mathcal {K}_\mathtt{D}u}(x)\) to \(\Omega\) is \(\mathcal {K}_\mathtt{D}u(x)\), the restriction of \(\widehat{u}(x)\) to \(\Omega\) is u(x) in comparison to (29). It remains to show that (32) holds on \(\partial \Omega\) as well. Since \(\widehat{\mathcal {K}_\mathtt{D}u}(x)\) is a continuous extension of \(\mathcal {K}_\mathtt{D}u(x)\),

$$\begin{aligned} \lim _{x \rightarrow \pm 1} \widehat{\mathcal {K}_\mathtt{D}u}(x) = \lim _{x \rightarrow \pm 1} \mathcal {K}_\mathtt{D}u(x) = \big ( \mathcal {K}_\mathtt{D}u\big )(\pm 1), \end{aligned}$$

where we used the Hilbert–Schmidt [7, 8] property of \(\mathcal {K}_\mathtt{D}\) in the second equality. On the other hand, again using the Hilbert–Schmidt property of \(\mathcal {K}_\mathtt{D}\), we have

$$\begin{aligned} \lim _{x \rightarrow \pm 1} \mathcal {K}_\mathtt{D}\widehat{u}(x) = \big ( \mathcal {K}_\mathtt{D}\widehat{u} \big ) (\pm 1) = \big ( \mathcal {K}_\mathtt{D}u\big )(\pm 1) \end{aligned}$$

since the integral operator is indifferent to the values of \(\widehat{u}\) and on \(\partial \Omega\). This concludes the proof of (32).

The following result is an instance of [11, Lemma 12.1.4].

Lemma 6.4

The operator\(\mathcal {K}_\mathtt{D}: C(\overline{\Omega }) \rightarrow C(\overline{\Omega })\)defined in (18) satisfies

$$\begin{aligned} \Vert \mathcal {K}_\mathtt{D}-\mathcal {P}_h\mathcal {K}_\mathtt{D}\Vert _{\infty } \rightarrow 0 \quad as \, h \rightarrow 0. \end{aligned}$$

Proof

By (20), \(\mathcal {P}_h\) is a bounded projection, and by Theorem 6.1, \(\mathcal {P}_h u \rightarrow u\) as \(h \rightarrow 0\) for any \(u~\in ~C(\overline{\Omega })\). Since \(\mathcal {K}_\mathtt{D}\) is a Hilbert–Schmidt operator [7, 8], it is compact. The result readily follows.

The following theorem is an application of [11, Thm. 12.1.2] and it proves the existence and uniqueness of the approximate solution and provides its error estimate.

Theorem 6.5

For sufficiently smallhand\(p \geqslant 1\), the solution to (14) or equivalently to (21) with\(\mathtt{BC}= \mathtt{D}\)exists and is unique. It satisfies the error estimate

$$\begin{aligned} \Vert u^\mathtt{D}- u_h^\mathtt{D}\Vert _{\infty } \leqslant \text{ C }\Vert u^\mathtt{D}-\mathcal {P}_h u^\mathtt{D}\Vert _{\infty } \end{aligned}$$

for some constant\(\text{ C }\)independent ofh.

Proof

The hypotheses of [11, Thm. 12.1.2] require the boundedness of \(\mathcal {K}_\mathtt{D}\), the bijectivity of \(\mathcal {M}_\mathtt{D}: C(\overline{\Omega }) \rightarrow C(\overline{\Omega })\), and the following convergence property:

$$\begin{aligned} \Vert \mathcal {K}_\mathtt{D}-\mathcal {P}_h\mathcal {K}_\mathtt{D}\Vert _{\infty } \rightarrow 0 \quad {\text {as }} h \rightarrow 0. \end{aligned}$$

The first hypothesis is satisfied since \(\mathcal {K}_\mathtt{D}= \mathcal {M}_\mathtt{D}-c\) is a Hilbert–Schmidt operator. The second and the last hypotheses were proved in Corollary 6.3 and Lemma 6.4, respectively. The result follows from [11, Thm. 12.1.2].

The following is a corollary to Theorem 6.5 and it follows by the approximation properties of the projection operator \(\mathcal {P}_h\) given in Theorem 6.1.

Corollary 6.6

Suppose that the hypotheses of Theorem 6.5hold. In addition, suppose that\(u^\mathtt{D}\in C^{p+1}(\overline{\Omega })\). Then,

$$\begin{aligned} \Vert u^\mathtt{D}-u_h^\mathtt{D}\Vert _{\infty } \leqslant \text{ C }h^{p+1}\left\| {\frac{\mathop {}\!\mathrm {d}^{p+1}u^\mathtt{D}}{\mathop {}\!\mathrm {d}x^{p+1}}}\right\| _{\infty } \end{aligned}$$

for some constant\(\text{ C }\)independent ofh.

For \(p=0\), the fact that \(\mathcal {P}_h\) does not map into a subspace of \(C(\overline{\Omega })\) prevents us from utilizing Corollary 6.3 and hence we cannot prove the existence and uniqueness of the approximation. Thus, Theorem 6.5 does not apply. To circumvent this difficulty, we introduce the following space:

$$\begin{aligned} \Omega _h:= \big\{I_1,\cdots ,I_N\big\}, \quad C(\Omega _h) := \big\{u \in L^2(\Omega ) \;|\,\; u|_{\overline{I}} \in C(\overline{I}),\, I \in \Omega _h\big\}. \end{aligned}$$

Observe that the space \(C(\Omega _h)\) allows jump discontinuities at grid nodes \(\{z_2,\cdots ,z_{N-1}\}\). This flexibility will allow us to treat the case of \(p=0\). We outline the details in the remainder of this subsection.

First, note that the projection operator \(\mathcal {P}_h : C(\Omega _h) \rightarrow V_h\) is well defined by (19) since the DOF \(x_i\) do not coincide with the potential jump discontinuities of a function in \(C(\Omega _h)\). Next, observe that Theorem 6.2 holds independently of the polynomial degree p. We state and prove the required bijectivity property of \(\mathcal {M}_\mathtt{D}\) on \(C(\Omega _h)\) which is analogous to Corollary 6.3.

Corollary 6.7

The operator \(\mathcal {M}_\mathtt{D}: C(\Omega _h) \rightarrow C(\Omega _h)\) is a bijection.

Proof

The injectivity of \(\mathcal {M}_\mathtt{D}: C(\Omega _h)~\rightarrow ~C(\Omega _h)\) follows from that of \(\mathcal {M}_\mathtt{D}: L^2(\Omega )~\rightarrow ~L^2(\Omega )\) and viewing the eigenfunctions in (25) as members of \(C(\Omega _h)\). To prove the surjectivity of \(\mathcal {M}_\mathtt{D}\), let \(b \in C(\Omega _h)\). Since \(C(\Omega _h) \subset L^2(\Omega )\), by the surjectivity of \(\mathcal {M}_\mathtt{D}\) in \(L^2(\Omega )\), there exists \(u \in L^2(\Omega )\) such that \(\mathcal {M}_\mathtt{D}u(x) = b(x)\). Then,

$$\begin{aligned} u(x) = \frac{1}{c} \mathcal {K}_\mathtt{D}u(x) + \frac{1}{c} b(x). \end{aligned}$$
(33)

By the Hilbert–Schmidt property of \(\mathcal {K}_\mathtt{D}\), \(\mathcal {K}_\mathtt{D}u \in C(\overline{\Omega })\) and since \(b \in C(\Omega _h)\), (33) implies that \(u \in C(\Omega _h)\).

Next, we state a slight variation of Theorem 6.1 for \(p=0\). To this end, we need the following notation:

$$\begin{aligned} C^1(\Omega _h) := \big\{u \in C(\overline{\Omega }) \; : \; u|_{\overline{I}} \in C^1(\overline{I}),\, I \in \Omega _h\big\}. \end{aligned}$$
(34)

Note that the functions in \(C^1(\Omega _h)\) are piecewise \(C^1\) and their piecewise derivatives are allowed to have jump discontinuities. The definition of \(\Vert u'\Vert _{\infty }\) for \(u \in C^1(\Omega _h)\) is generalized by considering the piecewise values of \(u'\). More explicitly,

$$\begin{aligned} \Vert u'\Vert _{\infty } := \max _{I \in \Omega _h} \big (\max _{x \in \overline{I}} |u'(x)|\big ). \end{aligned}$$

Theorem 6.8

Let\(u \in C^1(\Omega _h)\)and let\(\mathcal {P}_h\)be the projection operator defined by (19). Then, there exists a constant\(\text{ C }\)independent of h such that

$$\begin{aligned} \Vert u-\mathcal {P}_h u\Vert _{\infty } \leqslant \text{ C }h\Vert u'\Vert _{\infty }. \end{aligned}$$

In light of Theorem 6.8, Lemma 6.4 readily applies to \(\mathcal {K}_\mathtt{D}: C(\Omega _h) \rightarrow C(\Omega _h)\). Consequently, Theorem 6.5 also holds true for \(p=0\) which leads to the following error estimate:

Corollary 6.9

Let\(p = 0\)and suppose that the hypotheses of Theorem 6.8hold. In addition, suppose that \(u \in C^1(\Omega _h)\). Then,

$$\begin{aligned} \Vert u^\mathtt{D}-u_h^\mathtt{D}\Vert _{\infty } \leqslant \text{ C }h\left\| \frac{\mathop {}\!\mathrm {d}u^\mathtt{D}}{\mathop {}\!\mathrm {d}x}\right\| _{\infty } \end{aligned}$$

for some constant\(\text{ C }\)independent ofh.

Existence, Uniqueness, and the Error Estimates for the Case of Neumann BC

To prove a result analogous to Theorem 6.2 for the operator \(\mathcal {M}_\mathtt{N}\), we begin with defining the Hilbert basis of \(L^2(\Omega )\) associated with \(\mathcal {M}_\mathtt{N}\), denoted by \(\big (e_k^\mathtt{N}\big )_{k \in \mathbb {N}}\) [7] where the eigenfunctions are given by

$$\begin{aligned} e_k^\mathtt{N}(x) := {\left\{ \begin{array}{ll} \frac{1}{\sqrt{2}}, &{} k = 0, \\ \cos \big (\frac{k \pi }{2}(x+1)\big ), &{} k \in \mathbb {N}^*. \end{array}\right. } \end{aligned}$$
(35)

Since the operator \(\mathcal {M}_\mathtt{N}\) has the nontrivial kernel, i.e., \(\text {span}\{e_0^\mathtt{N}\}\), we will establish its bijectivity on the space

$$\begin{aligned} \mathring{L}^2(\Omega ) := \big (e_0^\mathtt{N}\big )^\perp = \text {span}\{e_k^\mathtt{N}\}_{k \in \mathbb {N}^*}. \end{aligned}$$
(36)

We denote the restriction of the operator \(\mathcal {M}_\mathtt{N}\) to \(\mathring{L}^2(\Omega )\) by \(\mathring{\mathcal {M}}_\mathtt{N}\).

Theorem 6.10

The operator\(\mathring{\mathcal {M}}_\mathtt{N}: \mathring{L}^2(\Omega ) \rightarrow \mathring{L}^2(\Omega )\)is a bijection.

Proof

We have proved in [9] that the eigenvalues of \(\mathcal {M}_\mathtt{N}\) are as follows:

$$\begin{aligned} \lambda _k(\mathcal {M}_\mathtt{N}) = \langle 1|C\rangle - \sqrt{2} {\left\{ \begin{array}{ll} 0 &{} \text {if }\; k= 0, \\ \langle e_{k/2}^{\mathtt{p}}|C\rangle &{} \text {if }\; k \in \mathbb {N}^*\text { is even,} \\ \langle e_{(k-1)/2}^\mathtt{a}|C\rangle &{} \text {if }\; k \in \mathbb {N}^*\text { is odd.} \end{array}\right. } \end{aligned}$$

Since \(\lambda _0(\mathcal {M}_\mathtt{N}) = 0\), we have that

$$\begin{aligned} \lambda _k(\mathring{\mathcal {M}}_\mathtt{N}) = \lambda _k(\mathcal {M}_\mathtt{D}), \quad k \in \mathbb {N}^*\end{aligned}$$
(37)

and the proof of injectivity of \(\mathring{\mathcal {M}}_\mathtt{N}\) reduces to that of \(\mathcal {M}_\mathtt{D}\).

To show the surjectivity of \(\mathring{\mathcal {M}}_\mathtt{N}\), first we show that, if \(b \in \text {Range}(\mathcal {M}_\mathtt{N}),\) then \(b \in \mathring{L}^2(\Omega )\). Since \(b \in \text {Range}(\mathcal {M}_\mathtt{N})\), there exists \(u \in L^2(\Omega )\) such that \(\mathcal {M}_\mathtt{N}u = b\). Then, using the self-adjointness of \(\mathcal {M}_\mathtt{N}\),

$$\begin{aligned} \langle e_0^\mathtt{N}|b\rangle = \langle e_0^\mathtt{N}|\mathcal {M}_\mathtt{N}u\rangle = \langle \mathcal {M}_\mathtt{N}e_0^\mathtt{N}|u\rangle = 0. \end{aligned}$$

Hence, \(b \in \mathring{L}^2(\Omega )\). Furthermore, for \(b \in \mathring{L}^2(\Omega )\), we have

$$\begin{aligned} \mathring{\mathcal {M}}_\mathtt{N}u(x) = \sum _{k \in \mathbb {N}^*} \lambda _k(\mathring{\mathcal {M}}_\mathtt{N}) \langle e_k^\mathtt{N}|u\rangle e_k^\mathtt{N}(x) = \sum _{k \in \mathbb {N}^*} \langle e_k^\mathtt{N}|b\rangle e_k^\mathtt{N}(x) = b(x). \end{aligned}$$

As in the proof of Theorem 6.2, the coefficients of u are

$$\begin{aligned} \langle e_k^\mathtt{N}|u\rangle = \frac{\langle e_k^\mathtt{N}|b\rangle }{\lambda _k(\mathring{\mathcal {M}}_\mathtt{N})}, \end{aligned}$$

and we obtain the following explicit expression for u:

$$\begin{aligned} u(x) = \sum _{k \in \mathbb {N}^*} \frac{\langle e_k^\mathtt{N}|b\rangle }{\lambda _k(\mathring{\mathcal {M}}_\mathtt{N})} e_k^\mathtt{N}(x). \end{aligned}$$

The sequence \(\big ( \langle e_k^\mathtt{N}|u\rangle \big )_{k \in \mathbb {N}^*}\) is square summable because

$$\begin{aligned} \left( \frac{\langle e_k^\mathtt{N}|b\rangle }{\lambda _k(\mathring{\mathcal {M}}_\mathtt{N})} \right) ^2 \leqslant \frac{\langle e_k^\mathtt{N}|b\rangle ^2}{{\underline{\lambda }}^2}, \end{aligned}$$

where \({\underline{\lambda }}\) is defined in (24) and we have used (37). Finally, \(\big ( \langle e_k^\mathtt{N}|b\rangle \big )_{k \in \mathbb {N}^*}\) is square summable since \(b \in \mathring{L}^2(\Omega )\).

Next, we prove an analogous version of Theorem 6.2 for \(\mathcal {M}_\mathtt{N}\) on the space

$$\begin{aligned} \mathring{C}^1(\Omega _h) := C^1(\Omega _h) \cap \mathring{L}^2, \end{aligned}$$

where \(C^1(\Omega _h)\) and \(\mathring{L}^2\) are defined in (34) and (36), respectively.

Corollary 6.11

The operator \(\mathring{\mathcal {M}}_\mathtt{N}: \mathring{C}^1(\Omega _h) \rightarrow \mathring{C}^1(\Omega _h)\) is a bijection.

Proof

The injectivity of \(\mathring{\mathcal {M}}_\mathtt{N}: \mathring{C}^1(\Omega _h)~\rightarrow ~\mathring{C}^1(\Omega _h)\) follows from that of \(\mathring{\mathcal {M}}_\mathtt{N}: \mathring{L}^2(\Omega )~\rightarrow ~\mathring{L}^2(\Omega )\) and viewing the eigenfunctions in (35), for \(k \in \mathbb {N}^*\), as members of \(\mathring{C}^1(\Omega _h)\). To prove the surjectivity of \(\mathring{\mathcal {M}}_\mathtt{N}\), let \(b \in \mathring{C}^1(\Omega _h)\). Since \(\mathring{C}^1(\Omega _h) \subset \mathring{L}^2(\Omega )\), by the surjectivity of \(\mathring{\mathcal {M}}_\mathtt{N}\) in \(\mathring{L}^2(\Omega )\), there exists \(u \in \mathring{L}^2(\Omega )\) such that \(\mathring{\mathcal {M}}_\mathtt{N}u(x) = b(x)\). Then,

$$\begin{aligned} u(x) = \frac{1}{c} \mathring{\mathcal {K}}_\mathtt{N}u(x) + \frac{1}{c} b(x), \end{aligned}$$
(38)

where \(\mathring{\mathcal {K}}_\mathtt{N}\) denotes the restriction of \(\mathcal {K}_\mathtt{N}\) to \(\mathring{L}^2(\Omega )\). By the Hilbert–Schmidt property of \(\mathring{\mathcal {K}}_\mathtt{N}\), \(\mathring{\mathcal {K}}_\mathtt{N}u \in \mathring{C}^1(\overline{\Omega }) \subset \mathring{C}^1(\Omega _h)\) and since \(b \in \mathring{C}^1(\Omega _h)\), (38) implies that \(u \in \mathring{C}^1(\Omega _h)\).

In what follows, we keep the same projection operator \(\mathcal {P}_h\) by replacing its domain with \(\mathring{C}^1(\Omega _h)\). Note that \(V_h\) is a subspace of \(\mathring{C}^1(\Omega _h)\) for \(p \geqslant 1\). The following result and its proof mimic those of Lemma 6.4.

Lemma 6.12

The operator \(\mathring{\mathcal {K}}_\mathtt{N}: \mathring{C}^1(\Omega _h) \rightarrow \mathring{C}^1(\Omega _h)\) satisfies

$$\begin{aligned} \Vert \mathring{\mathcal {K}}_\mathtt{N}-\mathcal {P}_h\mathring{\mathcal {K}}_\mathtt{N}\Vert _{\infty } \rightarrow 0 \quad as \; h \rightarrow 0. \end{aligned}$$

For \(p=0\), we encounter the same difficulties as we did in the case of Dirichlet BC because \(V_h\) is not a subspace of \(\mathring{C}^1(\Omega _h)\). However, all of these difficulties can be circumvented precisely as we did in the case of Dirichlet BC. Hence, the following theorem, its corollary, and their proofs are verbatim copies of Theorem 6.5 and Corollary 6.6.

Theorem 6.13

For sufficiently smallhand\(p \geqslant 0\), the solution to (14) or equivalently to (21) with\(\mathtt{BC}= \mathtt{N}\)exists and is unique. It satisfies the error estimate

$$\begin{aligned} \Vert u^\mathtt{N}- u_h^\mathtt{N}\Vert _{\infty } \leqslant \text{ C }\Vert u^\mathtt{N}-\mathcal {P}_h u^\mathtt{N}\Vert _{\infty } \end{aligned}$$

for some constant\(\text{ C }\)independent ofh.

Corollary 6.14

Suppose that the hypotheses of Theorem 6.13hold. In addition, suppose that\(u^\mathtt{N}\in C^{p+1}(\overline{\Omega })\). Then,

$$\begin{aligned} \Vert u^\mathtt{N}-u_h^\mathtt{N}\Vert _{\infty } \leqslant \text{ C }h^{p+1}\left\| {\frac{\mathop {}\!\mathrm {d}^{p+1}u^\mathtt{N}}{\mathop {}\!\mathrm {d}x^{p+1}}}\right\| _{\infty } \end{aligned}$$

for some constant\(\text{ C }\)independent ofh.

The Scaling and the Calibration

A guiding principle used to construct nonlocal governing operators involves a calibration process to match a local physical quantity. In peridynamics, the nonlocal value of the strain energy density is calibrated to match that from the classical theory. This calibration process gives rise to a default scaling of \(\displaystyle \frac{\gamma ^C}{\delta ^3}\) in 1D, for some calibration constant \(\gamma ^C\) [21, Eq. (2.9)].

As we mentioned in Sect. 4, since we study the asymptotic compatibility of collocation methods, our calibration process is performed at the discrete level. Hence, the calibration constant depends on the polynomial order p, which we denote by \(\gamma _p^C\). We proceed as follows. We choose the kernel function involved in \(\mathcal {M}_\mathtt{BC}\). Then we fix a polynomial degree \(p\geqslant 0\) and a grid with the stencil size h. The discretization leads to a stiffness matrix, and its entries before the calibration process are denoted by \(A_{i,j}\). Since the operator \(\mathcal {M}_\mathtt{BC}\) is designed as a function of \(-\Delta _\mathtt{BC}\), that function should include the factor of \(\frac{1}{\delta ^3}\). As a result, we determine the calibration constant \(\gamma _p^C\) that leads to an approximation of \(-\Delta\), the discrete Laplace operator

$$\begin{aligned} \frac{\gamma _p^C}{\delta ^3} \sum _{j \in {\mathcal {J}}_i} A_{i,j} v(x_j) \approx -\Delta v(x_i) \end{aligned}$$

for the sufficiently smooth v. For the definition \({\mathcal {J}}_i\), see (40). Note that the calibration constant should be independent of \(\delta\) and h so that the limit exists in the following definition of the asymptotic compatibility:

$$\begin{aligned} \lim _{(\delta ,\mathtt{h}) \rightarrow (0,0)} u_{\delta ,\mathtt{h}}^{\mathtt{BC}}(x) = u_0^\mathtt{BC}(x), \end{aligned}$$

where the limit is taken in the supremum norm and \(u_{\delta ,\mathtt{h}}^{\mathtt{BC}}(x)\) is the solution of

$$\begin{aligned} \begin{aligned} &\mathcal {M}_\mathtt{BC}u_{\delta ,\mathtt{h}}^{\mathtt{BC}}(x)= b_0(x), \quad x \in \Omega , \\ &u_{\delta ,\mathtt{h}}^{\mathtt{D}}(\pm 1)= \alpha _{\pm }^\mathtt{D}\quad \text {or} \quad \frac{\mathop {}\!\mathrm {d}u_{\delta ,\mathtt{h}}^{\mathtt{N}}}{\mathop {}\!\mathrm {d}x} (\pm 1) = \alpha _{\pm }^\mathtt{N}, \end{aligned} \end{aligned}$$
(39)

and \(u_0^\mathtt{BC}\) is the solution of

$$\begin{aligned} \begin{aligned} &- \Delta u_0^\mathtt{BC}(x)= b_0(x), \\ &u_0^\mathtt{D}(\pm 1)= \alpha _{\pm }^\mathtt{D}\quad \text {or} \quad \frac{\mathop {}\!\mathrm {d}u_0^{\mathtt{N}}}{\mathop {}\!\mathrm {d}x} (\pm 1) = \alpha _{\pm }^\mathtt{N}. \end{aligned} \end{aligned}$$

Remark 7.1

Due to the aforementioned independence of \(\gamma _p^C\) from \(\delta\) and h, the same calibration constant is valid for the DOF regardless of being associated with fitted or regular basis functions. This is a strength of functional calculus. For \(p \geqslant 2\), when \(x_i\) is associated with a regular basis function, we observe a second-order accurate approximation

$$\begin{aligned} \frac{\gamma _p^C}{\delta ^3} \sum _{j \in {\mathcal {J}}_i} A_{i,j} v(x_j) = -\Delta v(x_i) + {\mathcal {O}}(h^2). \end{aligned}$$

For \(p=2\), the order of accuracy drops to 1 for the DOF associated with fitted basis functions. On the other hand, for \(p=3\), the order accuracy goes up to 2 even for the DOF associated with fitted basis functions. Using higher order basis functions pays and one obtains a higher order approximation of \(-\Delta\) for DOF associated with both fitted and regular basis functions.

Taylor Series Expansions to Determine the Calibration Constant

In this section, we provide examples of how \(\gamma _p^C\) is obtained through Taylor series expansions performed in Mathematica. Several horizon and basis function scenarios are depicted in Figs. 56, and 7. We find the following calibration constant as a result of Taylor series expansions:

$$\begin{aligned} \gamma _p^{C_1}&= 3, \\ \gamma _p^{C_{2,s}}&= 3 + \frac{9}{s} \end{aligned}$$

for \(p \geqslant 2\). As we have shown in (11), it is possible to identify a calibration constant at the analytical level. This identification involves calibrating the dominant term of the Taylor series expansion of the eigenvalue. Calibration constants obtained at the analytical level agree with the ones obtained at the discrete level when the discretization space is rich enough, i.e., \(p \geqslant 2\). When the discretization space is poor, the calibration constant obtained at the analytical level does not lead to calibration constants independent of \(\delta\) and h at the discrete level. In fact, we were not able to find calibration constants independent of \(\delta\) and h for the cases of \(p=0,1\) that render the resulting collocation scheme asymptotically compatible.

Fig. 5
figure5

Horizon (indicated by shaded gray) and the basis functions intersecting the horizon

Fig. 6
figure6

Horizon (indicated by shaded gray) and the fitted basis functions intersecting the horizon

Expansions 1–4 and 5–7 are for regular and fitted basis functions, respectively, to show that the calibration constants are independent of \(\delta\) and h. We show that the same calibration constant \(\gamma _2 = \gamma _3 = 2\) is valid independent of \(\delta\) and h; see Expansions 1, 2, 5, and 6 for \(p=2\) and Expansions 3, 4, and 7 for \(p=3\). Fitted grids are depicted in Fig. 3.

Let us collect all the necessary ingredients for the calibration process. Let \(\text {supp}_j\) denote the support of the basis function \(\phi _j\) and \({\mathcal {H}}_i := (x_i-\delta ,x_i+\delta )\) denote the horizon of \(x_i \in \mathcal {E}_h\). A basis function is called fitted when the DOF in its support are not equispaced. Otherwise, it is called a regular basis function. For the purposes of calibration, a DOF \(x_i\) is said to be in the bulk if none of the basis functions intersecting its horizon \({\mathcal {H}}_i\) are fitted. For ease of notation, we assume that the ith DOF \(x_i\) is in the bulk. The index of basis functions intersecting the horizon of \(x_i\) is denoted by

$$\begin{aligned} {\mathcal {J}}_{i} := \{ j: {\mathcal {H}}_i \cap \text {supp}_j \ne \emptyset \}. \end{aligned}$$
(40)

The index set \({\mathcal {J}}_{i}\) contains the column indices of nonzero entries present in the ith row. The matrix entries of the unscaled version discretized operator \(\mathcal {M}_\mathtt{BC}\) are denoted by \(A_{i,j}\). The entries \({\mathbf {A}}_{{\mathcal {J}}_i}\) indicate the nonzero entries in the ith row. For the sake of simplicity, we assume that \(\delta\) is an integer multiple of h, so that \(R:= \displaystyle \frac{\delta }{h}\) is an integer. In fact, it is possible to treat cases, where R is not necessarily an integer. In such cases, we modify the definition of R simply as \(R:= \left\lceil \displaystyle \frac{\delta }{h}\right\rceil\), where \(\left\lceil \cdot \right\rceil\) denotes the ceiling function. Consequently, when \(j \in {\mathcal {J}}_i\) is such that \(x_j=\pm 1\), one needs to find \(\rho _\pm\) that satisfies \(\pm 1=x_i \pm \rho _\pm h\) and use them for \(v(\pm 1)\). Instances of such cases are exhibited in (54), (57), and (60). We show the related Taylor series expansions for various p and R values. For the sake of brevity, we present the \(C_1\) kernel. Similar expansions can be derived for the \(C_{2,s}\) kernel. To simplify the notation, we drop the superscript on the calibration constant \(\gamma _p^{C_1}\) and write \(\gamma _p\). In App. A, we give a simple example and provide the full details of determining a calibration constant. Hence, we omit the details in the examples below.

  • Expansion 1 The case of p = 2, R = 2, 5-point stencil.

Due to \(R=2\), one has \({\mathcal {J}}_i := \{ i-2, \cdots , i+2 \}\); see Fig. 5a. Obtain an explicit expression of \(A_{i,j}\) in h and define

$$\begin{aligned} {\mathbf {A}}_{{\mathcal {J}}_i}&:= \left[ A_{i,i-2}, \cdots , A_{i,i+2} \right] \end{aligned}$$
(41)
$$\begin{aligned}&= \left[ -\frac{h}{3}, -\frac{4h}{3}, \frac{10h}{3}, -\frac{4h}{3}, -\frac{h}{3} \right], \end{aligned}$$
(42)
$$\begin{aligned} {\mathbf {v}}_{{\mathcal {J}}_i}&:= \left[ v(x_{i-2}), \cdots , v(x_{i+2}) \right] = \left[ v(x_i-2h), \cdots , v(x_i+2h) \right] . \end{aligned}$$
(43)

Then,

$$\begin{aligned} \frac{\gamma _2}{\delta ^3} \sum _{j \in {\mathcal {J}}_i} A_{ij} v(x_j)&= \frac{\gamma _2}{\delta ^3} {\mathbf {A}}_{{\mathcal {J}}_i} \cdot {\mathbf {v}}_{{\mathcal {J}}_i} &&\text {by } (41) \text { and } (43), \\&= \frac{\gamma _2}{h^2} \left[ -\frac{1}{24}, -\frac{1}{6}, \frac{5}{12}, -\frac{1}{6}, -\frac{1}{24} \right] \cdot {\mathbf {v}}_{{\mathcal {J}}_i} &&\text {by } (42) \text { and since }\delta =2h, \\&=\frac{\gamma _2}{3h^2} \left[ -v''(x_i) h^2 + {\mathcal {O}}(h^4) \right] &&\text {by Taylor series expansion, }\\&= -v''(x_i) +{\mathcal {O}}(h^2) &&\text {by setting } \gamma _2=3. \end{aligned}$$
  • Expansion 2 The case of p = 2, R = 2, 7-point stencil.

Due to \(p=2, R=2\), one has \({\mathcal {J}}_i := \{ i-3, \cdots , i+3 \}\); see Fig. 5b. Obtain an explicit expression of \(A_{i,j}\) in h and define

$$\begin{aligned} {\mathbf {A}}_{{\mathcal {J}}_i}&:= \left[ A_{i,i-3}, \cdots , A_{i,i+3} \right] , \end{aligned}$$
(44)
$$\begin{aligned}&= \left[ \frac{h}{12}, -\frac{2h}{3}, -\frac{3h}{4}, \frac{8h}{3}, -\frac{3h}{4}, -\frac{2h}{3}, \frac{h}{12} \right] , \end{aligned}$$
(45)
$$\begin{aligned} {\mathbf {v}}_{{\mathcal {J}}_i}&:= \left[ v(x_{i-3}), \cdots , v(x_{i+3}) \right] = \left[ v(x_i-3h), \cdots , v(x_i+3h) \right] . \end{aligned}$$
(46)

Then,

$$\begin{aligned} \frac{\gamma _2}{\delta ^3} \sum _{j \in {\mathcal {J}}_i} A_{ij} v(x_j)&= \frac{\gamma _2}{\delta ^3} {\mathbf {A}}_{{\mathcal {J}}_i} \cdot {\mathbf {v}}_{{\mathcal {J}}_i} &&\text {by }(44) \text { and } (46), \\&= \frac{\gamma _2}{h^2} \left[ \frac{1}{96}, -\frac{1}{12}, -\frac{3}{32}, \frac{1}{3}, -\frac{3}{32}, -\frac{1}{12}, \frac{1}{96} \right] \cdot {\mathbf {v}}_{{\mathcal {J}}_i} &&\text {by } (45) \text { and since }\delta =2h, \\&=\frac{\gamma _2}{3h^2} \left[ -v''(x_i) h^2 + {\mathcal {O}}(h^4) \right] &&\text {by Taylor series expansion, }\\&= -v''(x_i) +{\mathcal {O}}(h^2) &&\text {by setting } \gamma _2=3. \end{aligned}$$
  • Expansion 3 The case of p = 3, R = 2, 7-point stencil.

Due to \(p=3, R=2\), one has \({\mathcal {J}}_i := \{ i-4, \cdots , i+2 \}\); see Fig. 5c. Obtain an explicit expression of \(A_{i,j}\) in h and define

$$\begin{aligned} {\mathbf {A}}_{{\mathcal {J}}_i}&:= \left[ A_{i,i-4}, \cdots , A_{i,i+2} \right] \end{aligned}$$
(47)
$$\begin{aligned}&= \left[ -\frac{h}{192}, \frac{5h}{192}, -\frac{19h}{192}, -\frac{3h}{32}, \frac{23h}{64}, -\frac{9h}{64}, -\frac{3h}{64} \right], \end{aligned}$$
(48)
$$\begin{aligned} {\mathbf {v}}_{{\mathcal {J}}_i}&:= \left[ v(x_{i-4}), \cdots , v(x_{i+2}) \right] = \left[ v(x_i-4h), \cdots , v(x_i+2h) \right] . \end{aligned}$$
(49)

Then,

$$\begin{aligned}\frac{\gamma _3}{\delta ^3} \sum _{j \in {\mathcal {J}}_i} A_{ij} v(x_j) &= \frac{\gamma _3}{\delta ^3} {\mathbf {A}}_{{\mathcal {J}}_i} \cdot {\mathbf {v}}_{{\mathcal {J}}_i} &&\text {by }(47) \text { and } (49),\\ &= \frac{\gamma _3}{h^2} \left[ -\frac{1}{1\,536}, \frac{5}{1\,536}, -\frac{19}{1\,536}, -\frac{3}{256}, \frac{23}{512}, -\frac{9}{512}, -\frac{3}{512} \right] \cdot {\mathbf {v}}_{{\mathcal {J}}_i} &&\text {by } (48) \text { and since } \delta =2h, \\& =\frac{\gamma _3}{3h^2} \left[ -v''(x_i) h^2 + {\mathcal {O}}(h^4) \right] && \text {by Taylor series expansion, }\\& = -v''(x_i) +{\mathcal {O}}(h^2) && \text {by setting } \gamma _3=3. \end{aligned}$$
  • Expansion 4 The case of p = 3, R = 3, 10-point stencil.

Due to \(p=3, R=3\), one has \({\mathcal {J}}_i := \{ i-5, \cdots , i+4 \}\); see Fig. 5d. Obtain an explicit expression of \(A_{i,j}\) in h and define

$$\begin{aligned} {\mathbf {A}}_{{\mathcal {J}}_i} :&= \left[ A_{i,i-5}, \cdots , A_{i,i+4} \right] \nonumber \\&= \left[ -\frac{h}{648}, \frac{5h}{648}, -\frac{19h}{648}, -\frac{h}{36}, -\frac{h}{24}, \frac{13h}{72}, -\frac{17h}{648}, -\frac{4h}{81}, -\frac{h}{81}, 0 \right], \end{aligned}$$
(50)
$$\begin{aligned} {\mathbf {v}}_{{\mathcal {J}}_i}&:= \left[ v(x_{i-5}), \cdots , v(x_{i+4}) \right] = \left[ v(x_i-5h), \cdots , v(x_i+4h) \right] . \end{aligned}$$
(51)

Then, using (50), (51), the fact that \(\delta =3h\), and setting \(\gamma _3=3\), we obtain

$$\begin{aligned}\frac{\gamma _3}{\delta ^3} \sum _{j \in {\mathcal {J}}_i} A_{ij} v(x_j)& = \frac{\gamma _3}{\delta ^3} {\mathbf {A}}_{{\mathcal {J}}_i} \cdot {\mathbf {v}}_{{\mathcal {J}}_i} \\ &= \frac{\gamma _3}{h^2} \left[ -\frac{1}{17\,496}, \frac{5}{17\,496}, -\frac{19}{17\,496}, -\frac{1}{972}, -\frac{1}{648}, \frac{13}{1\,944}, -\frac{17}{17\,496}, -\frac{4}{2\,187}, -\frac{1}{2\,187}, 0 \right] \cdot {\mathbf {v}}_{{\mathcal {J}}_i} \\& =\frac{\gamma _3}{3h^2} \left[ -v''(x_i) h^2 + {\mathcal {O}}(h^4) \right] \\& = -v''(x_i) +{\mathcal {O}}(h^2). \end{aligned}$$
  • Expansion 5 The case of p = 2, R = 2, 5-point stencil, associated with the DOF \(x_2\).

Due to \(p=2, R=2\), one has \({\mathcal {J}}_2 := \{ 1, \cdots , 5 \}\); see Fig. 6a. Obtain an explicit expression of \(A_{2,j}\) in h and define

$$\begin{aligned} {\mathbf {A}}_{{\mathcal {J}}_2}&:= \left[ A_{2,1}, \cdots , A_{2,5} \right] \end{aligned}$$
(52)
$$\begin{aligned}&= \left[ -\frac{3h}{4}, \frac{7h}{4}, -\frac{5h}{12}, -\frac{2h}{3}, \frac{h}{12} \right], \end{aligned}$$
(53)
$$\begin{aligned} {\mathbf {v}}_{{\mathcal {J}}_1}:=& \left[ v(x_{1}), \cdots , v(x_{5}) \right] \nonumber \\=& \left[ v(x_2 -2h), v(x_2), v(x_2+h), v(x_2+2h), v(x_2+3h) \right] . \end{aligned}$$
(54)

Then,

$$\begin{aligned} \frac{\gamma _2}{\delta ^3} \sum _{j \in {\mathcal {J}}_2} A_{2j} v(x_j)&= \frac{\gamma _2}{\delta ^3} {\mathbf {A}}_{{\mathcal {J}}_2} \cdot {\mathbf {v}}_{{\mathcal {J}}_2} &&\text {by } (52) \text { and } (54), \\&= \frac{\gamma _2}{h^2} \left[ -\frac{3}{32}, \frac{7}{32}, -\frac{5}{96}, -\frac{1}{12}, \frac{1}{96} \right] \cdot {\mathbf {v}}_{{\mathcal {J}}_2} &&\text {by } (53) \text { and since } \delta =2h, \\&=\frac{\gamma _2}{3h^2} \left[ -v''(x_2) h^2 + {\mathcal {O}}(h^3) \right] &&\text {by Taylor series expansion, }\\&= -v''(x_2) +{\mathcal {O}}(h) &&\text {by setting } \gamma _2=3. \end{aligned}$$
  • Expansion 6 The case of p = 2, R = 2, 5-point stencil, associated with the DOF \(x_3\).

Due to \(p=2, R=2\), one has \({\mathcal {J}}_3 := \{ 1, \cdots , 5 \}\); see Fig. 6b. Obtain an explicit expression of \(A_{3,j}\) in h and define

$$\begin{aligned} {\mathbf {A}}_{{\mathcal {J}}_3}&:= \left[ A_{3,1}, \cdots , A_{3,5} \right] \end{aligned}$$
(55)
$$\begin{aligned}&= \left[ -\frac{h}{9}, -\frac{5h}{3}, \frac{31h}{9}, -\frac{4h}{3}, -\frac{h}{3} \right], \end{aligned}$$
(56)
$$\begin{aligned} {\mathbf {v}}_{{\mathcal {J}}_3}&:= \left[ v(x_{1}), \cdots , v(x_{5}) \right] \nonumber \\&= \left[ v(x_3 -3h), v(x_3-h), v(x_3), v(x_3+h), v(x_3+2h) \right] . \end{aligned}$$
(57)

Then,

$$\begin{aligned} \frac{\gamma _2}{\delta ^3} \sum _{j \in {\mathcal {J}}_3} A_{3j} v(x_j)&= \frac{\gamma _2}{\delta ^3} {\mathbf {A}}_{{\mathcal {J}}_3} \cdot {\mathbf {v}}_{{\mathcal {J}}_3} &&\text {by } (55) \text { and } (57),\\&= \frac{\gamma _2}{h^2} \left[ -\frac{1}{72}, -\frac{5}{24}, \frac{31}{72}, -\frac{1}{6}, -\frac{1}{24} \right] \cdot {\mathbf {v}}_{{\mathcal {J}}_3} &&\text {by } (56) \text { and since } \delta =2h, \\&=\frac{\gamma _2}{3h^2} \left[ -v''(x_3) h^2 + {\mathcal {O}}(h^3) \right] &&\text {by Taylor series expansion, }\\&= -v''(x_3) +{\mathcal {O}}(h) &&\text {by setting } \gamma _2=3. \end{aligned}$$
  • Expansion 7 The case of p = 3, R = 2, 7-point stencil, associated with DOF \(x_5\).

Due to \(p=3, R=2\), one has \({\mathcal {J}}_5 := \{ 1, \cdots , 7 \}\); see Fig. 6c. Obtain an explicit expression of \(A_{5,j}\) in h and define

$$\begin{aligned} {\mathbf {A}}_{{\mathcal {J}}_5}&:= \left[ A_{5,1}, \cdots , A_{5,7} \right] \end{aligned}$$
(58)
$$\begin{aligned}&= \left[ -\frac{h}{96}, \frac{7h}{48}, -\frac{3h}{4}, -\frac{73h}{96}, \frac{23h}{8}, -\frac{9h}{8}, -\frac{3h}{8} \right], \end{aligned}$$
(59)
$$\begin{aligned} {\mathbf {v}}_{{\mathcal {J}}_5}&:= \left[ v(x_{1}), \cdots , v(x_{7}) \right] \nonumber \\&= \left[ v(x_5 -5h), v(x_5 -3h), v(x_5 -2h), v(x_5-h), v(x_5), v(x_5+h), v(x_5+2h) \right] . \end{aligned}$$
(60)

Then,

$$\begin{aligned}&\frac{\gamma _3}{\delta ^3} \sum _{j \in {\mathcal {J}}_5} A_{5j} v(x_j) = \frac{\gamma _3}{\delta ^3} {\mathbf {A}}_{{\mathcal {J}}_5} \cdot {\mathbf {v}}_{{\mathcal {J}}_5} &&\text {by } (58)\text { and }(60), \\&\quad = \frac{\gamma _3}{h^2} \left[ -\frac{1}{768}, \frac{7}{384}, -\frac{3}{32}, -\frac{73}{768}, \frac{23}{64}, -\frac{9}{64}, -\frac{3}{64} \right] \cdot {\mathbf {v}}_{{\mathcal {J}}_5} &&\text {by } (59) \text { and since } \delta =2h, \\&\quad =\frac{\gamma _3}{3h^2} \left[ -v''(x_5) h^2 + {\mathcal {O}}(h^4) \right] &&\text {by Taylor series expansion, }\\&\quad = -v''(x_5) +{\mathcal {O}}(h^2) &&\text {by setting } \gamma _3=3. \end{aligned}$$

History of Convergence Experiments

In this section, we present numerical results to verify the theoretical results of Sect. 6. To study the convergence of the method to the exact solution of the problem, we solve (39) with the method (14) with polynomial degrees \(p = 0,1,\cdots ,4.\) In each case, we choose an exact solution \(u^\mathtt{BC}(x)\) and insert it into (39) to compute the corresponding right-hand side function \(b^\mathtt{BC}(x)\). Then we compute an approximate solution \(u_\mathtt{h}^\mathtt{BC}\) and compute the supremum norm of the error \(u^\mathtt{BC}(x)-u_\mathtt{h}^\mathtt{BC}(x)\). We display numerical results corresponding to the two kernel functions \(C_1\) and \(C_{2,s}\) defined in (7) and (8), respectively.

In Table 1, we display numerical results for exact solutions with Dirichlet and Neumann BC. We use the kernel function \(C_1\) with fixed \(\delta = 2^{-1}\) and then with \(\delta = 2^{-3}\). We observe that the method converges with the optimal order of \(h^{p+1}\) for each \(p = 0,\cdots ,4\). The “Grid” column indicates the number of nodes in the grids used in the computations, more explicitly, Grid = i means that the corresponding grid has \(2^i+1\) nodes. The number of the DOF in the grid depends on the polynomial degree p of the approximation. For \(p=0\), there are \(2^i\) DOF and for \(p \geqslant 1\) there are \(2^i p+1\) DOF in the grid.

In Tables 2 and 3 , we carry out an analogous study with the kernel function \(C_{2,s}\). Since (8) defines a sequence of kernel functions, one for each nonnegative integer s, we display numerical results only for the extreme case \(s=1\) and with the moderate value of \(s=5\). Note that although the exact solutions are the same as those used in Table 1, the right-hand side functions used in (39) have to be recalculated for each problem since the exact solutions are inserted into different governing operators with each \(\delta\) as well as each s. We perform these calculations in Mathematica. Recall that the right-hand side of the governing operator \(\mathcal {M}_\mathtt{D}\) in (6) was denoted as \(\mathcal {K}_\mathtt{D}\) in (18) whose action on the exact solution \(u^\mathtt{D}(x)=1+\sin (\pi x)+\cos (\pi x)\) in Table 2 is given by

$$\begin{aligned} \mathcal {K}_\mathtt{D}u^\mathtt{D}(x) = {\left\{ \begin{array}{ll} \mathcal {K}_\mathtt{D}^{L}(x), \quad &{} x \in (-1,-1+\delta ], \\ \mathcal {K}_\mathtt{D}^{M}(x), &{} x \in (-1+\delta ,1-\delta ), \\ \mathcal {K}_\mathtt{D}^{R}(x), &{} x \in [1-\delta ,1), \end{array}\right. } \end{aligned}$$

where for \(s=1\) and \(\delta =2^{-3}\), we have

$$\begin{aligned} \mathcal {K}_\mathtt{D}^{L}(x) =&\, -\frac{2}{\pi ^2} \left ( 4 \pi ^2 x^2 +7 \pi ^2 x-8 \sin (\pi x)+4 \sin \left(\pi \left(x+{\frac{1}{8}}\right) \right)-4 \sin \left({\frac{\pi}{8}}-\pi x \right) \right. \\&\left. - 8 \cos (\pi x)+4 \cos \left(\pi \left(x+{\frac{1}{8}} \right) \right)-4 \cos \left({\frac{\pi}{8}}-\pi x \right)+3 \pi ^2-8\right ),\\ \mathcal {K}_\mathtt{D}^{M}(x) =&\, \frac{1}{8 \pi ^2} \left (128 \sin (\pi x) -64 \sin \left(\pi \left(x+{\frac{1}{8}} \right) \right)+64 \sin \left({\frac{\pi}{8}}-\pi x \right)+128 \cos (\pi x) \right. \\& \left. -64 \cos \left(\pi \left(x+{\frac{1}{8}} \right) \right)-64 \cos \left({\frac{\pi}{8}}-\pi x \right)+\pi ^2\right ), \\ \mathcal {K}_\mathtt{D}^{R}(x) =&\, -\frac{2}{\pi ^2} \left (4 \pi ^2 x^2 -7 \pi ^2 x-8 \sin (\pi x)+4 \sin \left(\pi \left(x+{\frac{1}{8}} \right) \right)-4 \sin \left({\frac{\pi}{8}}-\pi x \right) \right. \\& \left. -8 \cos (\pi x)-4 \cos \left(\pi \left(x+{\frac{1}{8}} \right) \right)+4 \cos \left({\frac{\pi}{8}}-\pi x \right)+3 \pi ^2-8\right ). \end{aligned}$$

Therefore, the computation of the right-hand side vector involves evaluation of up to ten terms at each DOF of the discretization. For the analogous case of \(s=5\), \(\mathcal {K}_\mathtt{D}^{L}\) and \(\mathcal {K}_\mathtt{D}^{R}\) each contain 38 terms, whereas \(\mathcal {K}_\mathtt{D}^{M}\) contains 22 terms. We suspect that this proliferation in the number of terms in conjunction with the increase in the number of the DOF for higher polynomial degrees is the root cause for the loss of accuracy in the lower parts of the \(p=3\) and \(p=4\) columns in Tables 2 and 3.

Table 1 History of convergence with the kernel function \(C_1\)
Table 2 History of convergence for Dirichlet BC with the kernel function \(C_{2,s}\)
Table 3 History of convergence for Neumann BC with the kernel function \(C_{2,s}\)

Asymptotic Compatibility Experiments

We carry out tests to ascertain the asymptotic compatibility of the collocation methods. For this study, we choose an exact solution \(u_0(x)\) and calculate a right-hand side function \(b_0(x)\) such that

$$\begin{aligned} -\frac{\mathrm {d}^2{u_0}}{\mathrm {d}x^2}(x) = b_0(x). \end{aligned}$$

We pick a kernel function (either \(C_1\) or \(C_{2,s}\)), a horizon size \(\delta\), and find the approximate solution \(u_{\delta ,\mathtt{h}}\) of (39) where we take \(b = b_0\). Then we compute the error \(\Vert u_{\delta ,\mathtt{h}}-u_0\Vert _{\infty }\) as \(\mathtt{h}\) approaches 0. We test the asymptotic compatibility with four different horizon functions: \(\delta = \mathtt{h}^2\), \(\delta = 3\mathtt{h}\), \(\delta = \mathtt{h}^{3/4}\), and \(\delta = \mathtt{h}^{1/2}\). Let us clarify a technical detail in the generation of the fitted grids when the horizon parameter \(\delta\) is given as a function of the grid size \(\mathtt{h}\). Suppose that we would like to generate a fitted grid with \(n+1\) nodes as described in Sect. 5 and fix a polynomial degree \(p \geqslant 1\). The set of the DOF is obtained by adding \(p-1\) internal DOF to the nodes of the grid. For adaptive grids, the necessary condition is that no DOF should be placed in a \(\delta\) distance of the boundary of the domain. Hence, the internal DOF between the first and the second nodes of the fitted grid should be placed starting at \(-1+\delta\). A similar adjustment is made at the right boundary of the domain and, hence, the internal DOF between the last and the second from the last nodes of the fitted grid are to be placed starting at \(1-\delta\). In our numerical experiments, we consider a fitted grid with uniform spacing (except the required \(\delta\) spacing around the boundary as described above). Let us denote this uniform grid spacing parameter with h. Since the size of the domain \(\Omega\) is 2, we have that

$$\begin{aligned} 2&= 2(\delta + (p-1)h) + (n-2)ph \nonumber \\&= 2\delta + (np-2)h, \end{aligned}$$
(61)

where the first term on the right-hand side of (61) accounts for the two boundary “elements” and the second term accounts for the remaining \(n-2\) bulk “elements”. For a fixed \(\delta\), we can easily solve this equation for h to determine the grid spacing. However, if \(\delta = \delta (\mathtt{h})\), as is required for asymptotic convergence studies, then we have to solve the equation

$$\begin{aligned} 2&= 2\delta (\mathtt{h}) + (np-2)h \nonumber \\&= 2\delta (ph) + (np-2)h \quad {\text {since }} \mathtt{h}= p h. \end{aligned}$$
(62)

This is, in general, a nonlinear equation and could be quite complicated depending on the choice of \(\delta\) as a function of \(\mathtt{h}\). As mentioned above, in our numerical experiments we take \(\delta = \mathtt{h}^2\), \(\delta = 3\mathtt{h}\), \(\delta = \mathtt{h}^{3/4}\), and \(\delta = \mathtt{h}^{1/2}\). Here, we show the details of this computation for \(\delta = \sqrt{\mathtt{h}} = \sqrt{ph}\). Defining the auxiliary parameter \(A = \frac{1}{2}(np-2)\), (62) can be written for this particular choice of \(\delta\) as

$$\begin{aligned} 1 = \sqrt{ph} + Ah. \end{aligned}$$
(63)

Squaring both sides of (63), we end up with the quadratic equation

$$\begin{aligned} A^2 h^2 - (2A+p)h + 1 = 0 \end{aligned}$$

whose roots are

$$\begin{aligned} h_{1,2} = \frac{2A+p \pm \sqrt{p^2+4Ap}}{2A^2}. \end{aligned}$$
(64)

By squaring (63), we have introduced an artificial root to it. Thus, we expect only one of the roots in (64) to satisfy (63). Indeed, a simple computation shows that

$$\begin{aligned} h = \frac{2A+p-\sqrt{p^2+4Ap}}{2A^2} \end{aligned}$$

is the root of (63). This is the grid spacing parameter used in generating the fitted grid.

There is a caveat in computing the approximate solution with inhomogeneous BC. Let us work out the details of this in the Dirichlet BC case. In light of (15) when testing for asymptotic compatibility, the approximate solutions satisfyFootnote 1

$$\begin{aligned} c \, \gamma \, \delta ^{-3}u_{\delta ,\mathtt{h}}^{\mathtt{D}}(\pm 1) = b_0(\pm 1). \end{aligned}$$
(65)

However, \(b_0(x) = -\frac{\mathrm {d}^2{u_0}}{\mathrm {d}x^2}(x)\) and, hence, \(b_0(\pm 1) = -\frac{\mathrm {d}^2{u_0}}{\mathrm {d}x^2}(\pm 1)\) which is not necessarily equal to \(-c \, \gamma \, \delta ^{-3}u_0(\pm 1)\). Thus, we cannot expect \(u_{\delta ,\mathtt{h}}^{\mathtt{D}}\) to converge to \(u_0\) as \(\delta \rightarrow 0\) since it does not satisfy the correct BC. Thus, we rectify\(u_{\delta ,\mathtt{h}}^{\mathtt{D}}\) by subtracting the term \(a_0 + a_1 x\) and set

$$\begin{aligned} {\widetilde{u}}_{\delta ,\mathtt{h}}^{\mathtt{D}} = u_{\delta ,\mathtt{h}}^{\mathtt{D}} - (a_0 + a_1 x). \end{aligned}$$

The constants \(a_0\) and \(a_1\) are chosen so that \({\widetilde{u}}_{\delta ,\mathtt{h}}^{\mathtt{D}}\) satisfies the correct BC, namely

$$\begin{aligned} {\widetilde{u}}_{\delta ,\mathtt{h}}^{\mathtt{D}}(\pm 1) = u_0(\pm 1). \end{aligned}$$

Using (65), these equations can be written as

$$\begin{aligned} \frac{1}{c}\cdot \frac{\delta ^3}{\gamma }\,b_0(\pm 1) - (a_0+a_1x)(\pm 1) = u_0(\pm 1). \end{aligned}$$

This results in a simple \(2\times 2\) linear system whose unique solution is

$$\begin{aligned} \left[ \begin{array}{c} a_0 \\ a_1 \\ \end{array} \right] = \left[ \begin{array}{cr} 1 &{} -1 \\ 1 &{} 1 \\ \end{array} \right] ^{-1} \left[ \begin{array}{c} \frac{1}{c}\cdot \frac{\delta ^3}{\gamma }\,b_0(-1) - u_0(-1) \\ \frac{1}{c}\cdot \frac{\delta ^3}{\gamma }\,b_0(+1) - u_0(+1) \\ \end{array} \right] . \end{aligned}$$

In the homogeneous Dirichlet BC case, we can rewrite (65) as

$$\begin{aligned} u_{\delta ,\mathtt{h}}^{\mathtt{D}}(\pm 1) = \frac{1}{c}\cdot \frac{\delta ^3}{\gamma }b_0(\pm 1) = k_{\delta } b_0(\pm 1), \end{aligned}$$

where \(k_{\delta } = {\mathcal {O}}(\delta ^2)\) due to the fact that \(c = {\mathcal {O}}(\delta )\) for both kernel functions \(C_1\) and \(C_{2,s}\). Thus, regardless of the values of \(b_0(\pm 1)\), \(u_{\delta ,\mathtt{h}}^{\mathtt{D}}(\pm 1)\) approaches 0 as \(\delta \rightarrow 0\), and hence the rectification is not necessary. However, this comes at the expense of not being able to satisfy the BC exactly in the pre-asymptotic regime. If one requires each approximate solution \(u_{\delta ,\mathtt{h}}^{\mathtt{D}}\) to satisfy the BC exactly, then the rectification is necessary.

In the Neumann BC case, by (16), when testing for the asymptotic compatibility, the approximate solutions satisfy

$$\begin{aligned} c \, \gamma \, \delta ^{-3}\frac{\mathrm {d}{u_{\delta ,\mathtt{h}}^{\mathtt{N}}}}{\mathrm {d}x}(\pm 1) = \frac{\mathrm {d}{b_0}}{\mathrm {d}x}(\pm 1). \end{aligned}$$
(66)

However, as above, \(b_0(x) = -\frac{\mathrm {d}^2{u_0}}{\mathrm {d}x^2}(x)\) and, hence, \(\frac{\mathrm {d}{b_0}}{\mathrm {d}x}(\pm 1) = -\frac{\mathrm {d}^3{u_0}}{\mathrm {d}x^3}(\pm 1)\) which is not necessarily equal to \(-c \, \gamma \, \delta ^{-3} \frac{\mathrm {d}{u_0}}{\mathrm {d}x}(\pm 1)\). So, we cannot expect \(u_{\delta ,\mathtt{h}}^{\mathtt{N}}\) to converge to \(u_0\) since it does not satisfy the correct BC. This time we rectify\(u_{\delta ,\mathtt{h}}^{\mathtt{N}}\) by subtracting the correction term \(a_0 + a_1 x + a_1 x^2\) and set

$$\begin{aligned} {\widetilde{u}}_{\delta ,\mathtt{h}}^{\mathtt{N}} = u_{\delta ,\mathtt{h}}^{\mathtt{N}} - (a_0 + a_1 x + a_2 x^2). \end{aligned}$$

The Neumann BC requires a rectification with three terms because the derivative annihilates the constant term. The constants \(a_0\), \(a_1\), and \(a_2\) are chosen so that \({\widetilde{u}}_{\delta ,\mathtt{h}}^{\mathtt{N}}\) satisfies the correct BC, namely

$$\begin{aligned} \frac{\mathrm {d}{{\widetilde{u}}_{\delta ,\mathtt{h}}^{\mathtt{N}}}}{\mathrm {d}x}(\pm 1) = \frac{\mathrm {d}{u_0}}{\mathrm {d}x}(\pm 1). \end{aligned}$$

Using (66), these equations can be written as

$$\begin{aligned} \frac{1}{c}\cdot \frac{\delta ^3}{\gamma }\,\frac{\mathrm {d}{b_0}}{\mathrm {d}x}(\pm 1) - (a_1 + 2a_2x)(\pm 1) = \frac{\mathrm {d}{u_0}}{\mathrm {d}x}(\pm 1). \end{aligned}$$
(67)

Since we have three unknowns but only two equations, in accordance with (17), we also require that

$$\begin{aligned} \int _{-1}^1 {\widetilde{u}}_{\delta ,\mathtt{h}}^{\mathtt{N}}(x)\mathop {}\!\mathrm {d}x = 0. \end{aligned}$$

Since this condition is already satisfied by \(u_{\delta ,\mathtt{h}}^{\mathtt{N}}\), this amounts to requiring that

$$\begin{aligned} \int _{-1}^1 (a_0 + a_1 x + a_2 x^2)\mathop {}\!\mathrm {d}x = 0. \end{aligned}$$
(68)

Equations (67) and (68) result in a \(3\times 3\) linear system whose unique solution is

$$\begin{aligned} \left[ \begin{array}{c} a_0 \\ a_1 \\ a_2 \\ \end{array} \right] = \left[ \begin{array}{ccr} 0 &{}1 &{}-2 \\ 0 &{}1 &{} 2 \\ 2 &{}0 &{}\frac{2}{3} \\ \end{array} \right] ^{-1} \left[ \begin{array}{c} \frac{1}{c}\cdot \frac{\delta ^3}{\gamma }\,b_0'(-1) - u_0'(-1) \\ \frac{1}{c}\cdot \frac{\delta ^3}{\gamma }\,b_0'(+1) - u_0'(+1) \\ 0 \\ \end{array} \right] . \end{aligned}$$

In Tables 4, 5, 6, 7, 8, and 9, we display the results of the numerical experiments regarding the asymptotic convergence of the collocation method. First, we note that we do not observe the asymptotic convergence for \(p=0\) and \(p=1\). More specifically, the methods seem to converge to an incorrect solution for these polynomial degrees. This is in agreement with our findings in Sect. 8 where we were not able to find a calibration constant that is independent of \(\delta\) and \(\mathtt{h}\). On the other hand, for \(p \geqslant 2\), we do obtain promising convergence results, both for the homogeneous and the inhomogeneous BC cases.

In Tables 4 and 5, we display the results for the kernel function \(C_1\). For each choice of the horizon size \(\delta\) in terms of the grid size \(\mathtt{h}\), we obtain the asymptotic convergence of \(u_\mathtt{h}^{\delta }\) to \(u_0\). The rate of this convergence changes depending on the choice of \(\delta\) as a function of \(\mathtt{h}\). We observe a higher rate of convergence when \(\delta \leqslant \mathtt{h}\) and a lower rate for \(\delta > \mathtt{h}\).

In Tables 6, 7, 8, and 9, we display analogous results for the kernel function \(C_{2,s}\). The results are similar to those of \(C_1\), in particular, we do observe the asymptotic convergence of \(u_\mathtt{h}^{\delta }\) to \(u_0\). We do not see a strict dependence of the convergence rates on the kernel order s, that is, the results for \(s=1\) and \(s=5\) are qualitatively the same. Other numerical results which we do not report here also indicate similar behavior for other values of s.

In the asymptotic convergence study, one should not expect the orders of convergence proved in Corollaries 6.6 and 6.14 because \(\delta\) does not remain fixed. In fact, we observe a convergence order of maximum 2 when \(\delta \geqslant \mathtt{h}\) and maximum 4 when \(\delta < \mathtt{h}\). More precisely, let \(\delta = {\mathcal {O}}(\mathtt{h}^\beta )\). Then we observe the order of the convergence maximum \({\mathcal {O}}(\delta ^2) = {\mathcal {O}}(\mathtt{h}^{2\beta })\). We suspect that \({\mathcal {O}}(\delta ^2)\) perturbation of \(\lambda _k(\mathcal {M}_\mathtt{BC})\) from \(\lambda (-\Delta _\mathtt{BC})\) in (11) is responsible for this behavior. A similar \({\mathcal {O}}(\delta ^2)\) convergence was reported in [24]. We also note that for the inhomogeneous Neumann BC case, when \(\delta = {\mathcal {O}}(\mathtt{h}^\beta )\), the order of the asymptotic convergence seems to be limited to \({\mathcal {O}}(\mathtt{h}^{\min \{\beta ,1\}})\). A possible research avenue would be identifying the role of this \({\mathcal {O}}(\delta ^2)\) perturbation in the order of asymptotic convergence as well as the reason for the reduction in the order of convergence for the inhomogeneous Neumann BC case.

Table 4 Asymptotic convergence for Dirichlet BC with the kernel function \(C_1\)
Table 5 Asymptotic convergence for Neumann BC with the kernel function \(C_1\)
Table 6 Asymptotic convergence for homogeneous Dirichlet BC with the kernel function \(C_{2,s}\)
Table 7 Asymptotic convergence with homogeneous Neumann BC with the kernel function \(C_{2,s}\)
Table 8 Asymptotic convergence for inhomogeneous Dirichlet BC with the kernel function \(C_{2,s}\)
Table 9 Asymptotic convergence for inhomogeneous Neumann BC with the kernel function \(C_{2,s}\)

Conclusion

We studied the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators. We proved that the methods are optimally convergent with order \(h^{p+1}\) for \(p \geqslant 0\). We carried out a calibration process via Taylor series expansions to ensure that the resulting collocation methods are asymptotically compatible. For \(p \geqslant 2\), we found that there exists a calibration constant that is independent of \(\delta\) and h such that the resulting collocation methods are asymptotically compatible. We verified these findings through a set of numerical experiments. One interesting feature about our discretization is the use of fitted grids which is required by the zero row sum property.

Since functional calculus guarantees that the nonlocal operator \(\mathcal {M}_\mathtt{BC}\) is a function of \(-\Delta _\mathtt{BC}\), for \(p \geqslant 2\), the same calibration constant becomes valid for the DOF associated not only with regular, but also with fitted basis functions. This is what makes our discretizations asymptotically compatible, a strength of functional calculus. It is remarkable that one can easily produce a wide variety of finite difference stencils as presented in Sect. 8 that can be used as an approximation to the \(-\Delta _\mathtt{BC}\) operator. One can practically construct symmetric or nonsymmetric stencils of arbitrary length by simply choosing the appropriate horizon size \(\delta\) and the order of the basis functions p. These stencils can be used for an array of computational applications that involve the \(-\Delta _\mathtt{BC}\) operator.

Notes

  1. 1.

    Here, the derivation is independent of the choice of the polynomial order p and the kernel function C, so we denote the calibration constant simply with \(\gamma\) rather than the pedantic notation \(\gamma _p^C\).

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Funding

Burak Aksoylu’s research was sponsored by the CCDC Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-16-2-0008. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

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Appendix A: the case of \(p=1\)

Appendix A: the case of \(p=1\)

For \(p=1\), we provide Expansions A1 and A2 to show that one cannot find a calibration constant \(\gamma _1\) independent of \(\delta\) and h.

  • Expansion A1 The case of p = 1, R = 1, 3-point stencil bulk.

Due to \(p=1, R=1\), one has \({\mathcal {J}}_i := \{ i-1, i, i+1 \}\); see Fig. 7a. Obtain an explicit expression of \(A_{i,j}\) in h and define

$$\begin{aligned} {\mathbf {A}}_{{\mathcal {J}}_i}&:= \left[ A_{i,i-1}, A_{i,i}, A_{i,i+1} \right] \end{aligned}$$
(A1)
$$\begin{aligned}&= \left[ -\frac{h}{2}, h, -\frac{h}{2} \right], \end{aligned}$$
(A2)
$$\begin{aligned} {\mathbf {v}}_{{\mathcal {J}}_i}&:= \left[ v(x_{i-1}), v(x_i), v(x_{i+1}) \right] = \left[ v(x_{i}-h), v(x_i), v(x_{i}+h) \right] . \end{aligned}$$
(A3)

Using Taylor series expansions,

$$\begin{aligned} v(x_i-h)&= v(x_i) +v'(x_i)(-h) + v''(x_i)\frac{(-h)^2}{2} + v'''(x_i)\frac{(-h)^3}{6}+{\mathcal {O}}(h^4), \\ v(x_i+h)&= v(x_i) +v'(x_i)h + v''(x_i)\frac{h^2}{2} + v'''(x_i)\frac{h^3}{6}+ {\mathcal {O}}(h^4), \end{aligned}$$

we obtain

$$\begin{aligned} -v(x_i-h) + 2 v(x_i) - v(x_i+h)= & {} \, -v''(x_i)h^2 + {\mathcal {O}}(h^4), \end{aligned}$$
(A4)
$$\begin{aligned} \frac{\gamma _1}{\delta ^3} \sum _{j \in \{i-1,i,i+1\}} A_{ij} v(x_j)= & {} \, \frac{\gamma _1}{\delta ^3} {\mathbf {A}}_{{\mathcal {J}}_i} \cdot {\mathbf {v}}_{{\mathcal {J}}_i} \quad \text {by } (\text{A1}) \text { and } (\text{A3}),\nonumber \\= & {} \, \frac{\gamma _1}{h^2} \left[ -\frac{1}{2}, 1 -\frac{1}{2} \right] \cdot {\mathbf {v}}_{{\mathcal {J}}_i} \quad \text {by } (\text{A2}) \text { and since } \delta =h, \nonumber \\= & {} \, \frac{\gamma _1}{2h^2} \left[ -v''(x_i) h^2 + {\mathcal {O}}(h^4) \right] \quad \text {by } (\text{A4}), \nonumber \\= & {} \, \frac{\gamma _1}{2} \left[ -v''(x_i) +{\mathcal {O}}(h^2) \right] \nonumber \\= & {} \, -v''(x_i) +{\mathcal {O}}(h^2) \quad \text {by setting } \gamma _1=2. \end{aligned}$$
  • Expansion A2 The case of p = 1, R = 2, 5-point stencil bulk.

Due to \(R=2\), one has \({\mathcal {J}}_i := \{ i-2, \cdots , i+2 \}\); see Fig. 7b. Obtain an explicit expression of \(A_{i,j}\) in h and define

$$\begin{aligned} {\mathbf {A}}_{{\mathcal {J}}_i}&:= \left[ A_{i,i-2}, \cdots , A_{i,i+2} \right] \end{aligned}$$
(A5)
$$\begin{aligned}&= \left[ -\frac{h}{2}, -h, 3h, -h, -\frac{h}{2} \right], \end{aligned}$$
(A6)
$$\begin{aligned} {\mathbf {v}}_{{\mathcal {J}}_i}&:= \left[ v(x_{i-2}), \cdots , v(x_{i+2}) \right] = \left[ v(x_i-2h), \cdots , v(x_i+2h) \right] . \end{aligned}$$
(A7)

Then,

$$\begin{aligned} \frac{\gamma _2}{\delta ^3} \sum _{j \in {\mathcal {J}}_i} A_{ij} v(x_j)&= \frac{\gamma _2}{\delta ^3} {\mathbf {A}}_{{\mathcal {J}}_i} \cdot {\mathbf {v}}_{{\mathcal {J}}_i} \quad&\text {by } (\text{A5}) \text { and } (\text{A7}), \\&= \frac{\gamma _1}{h^2} \left[ -\frac{1}{16}, -\frac{1}{8}, \frac{3}{8}, -\frac{1}{8}, -\frac{1}{16} \right] \cdot {\mathbf {v}}_{{\mathcal {J}}_i} \quad&\text {by } (\text{A6}) \text { and since } \delta =2h, \\&=\frac{3 \gamma _1}{8 h^2} \left[ -v''(x_i) h^2 + {\mathcal {O}}(h^4) \right] \quad&\text {by Taylor series expansion, }\\&= -v''(x_i) +{\mathcal {O}}(h^2) \quad&\text {by setting } \gamma _1=\frac{8}{3}. \end{aligned}$$
Fig. 7
figure7

Horizon (indicated by shaded gray) and the basis functions intersecting the horizon \(p=1\)

We end up with \(\gamma _1=2\) and \(\gamma _1=\frac{8}{3}\) in Expansions A1 and A2, respectively. Other expansions that we do not report here using various values of \(\delta\) and h lead to different values of \(\gamma _1\). The dependence of \(\gamma _1\) on \(\delta\) and h disqualifies the collocation method with \(p=1\) as an asymptotically compatible discretization for our governing operators \(\mathcal {M}_\mathtt{BC}\).

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Aksoylu, B., Celiker, F. & Gazonas, G.A. Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility. Commun. Appl. Math. Comput. 2, 261–303 (2020). https://doi.org/10.1007/s42967-019-00051-8

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Keywords

  • Nonlocal operator
  • Inhomogeneous local boundary condition
  • Nonlocal diffusion
  • Asymptotic compatibility
  • Collocation method
  • Peridynamics
  • Functional calculus

Mathematics Subject Classification

  • 65R20
  • 65M70
  • 47G10
  • 74B99