Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

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.

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

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 $\mathcal {M}_\mathtt{BC}$.