Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion

TL;DR

This paper introduces an asymptotically compatible reproducing kernel collocation method for nonlocal diffusion models, ensuring convergence to local limits and reducing computational costs through a quasi-discrete operator, validated by numerical experiments.

Contribution

It develops a novel AC RK collocation scheme for nonlocal diffusion that converges to local limits and introduces a quasi-discrete operator to lower computational costs.

Findings

The scheme converges to nonlocal and local diffusion solutions.

The quasi-discrete operator reduces the need for numerical quadrature.

Numerical experiments validate the theoretical convergence.

Abstract

Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The scheme is shown to be convergent to both nonlocal diffusion and its corresponding local limit as nonlocal interaction vanishes. The analysis is carried out on a special family of rectilinear Cartesian grids for linear RK method with designed kernel support. The key idea for the stability of the RK collocation scheme is to compare the collocation scheme with the standard Galerkin scheme which is stable. In addition, there is a large computational cost for assembling the stiffness matrix of the nonlocal problem because high order Gaussian quadrature is usually needed to evaluate the integral. We thus provide a remedy to the problem by introducing a quasi-discrete nonlocal diffusion operator for which no numerical quadrature is further needed after applying the RK collocation scheme. The quasi-discrete nonlocal diffusion operator combined with RK collocation is shown to be convergent to the correct local diffusion problem by taking the limits of nonlocal interaction and spatial resolution simultaneously. The theoretical results are then validated with numerical experiments. We additionally illustrate a connection between the proposed technique and an existing optimization based approach based on generalized moving least squares (GMLS).