A Convergent Adaptive Finite Element Algorithm for Nonlocal Diffusion and Peridynamic Models

Abstract

In this paper, we propose an adaptive finite element algorithm for the numerical solution of a class of nonlocal models which correspond to nonlocal diffusion equations and linear scalar peridynamic models with certain nonintegrable kernel functions. The convergence of the adaptive finite element algorithm is rigorously derived with the help of several basic ingredients, such as the upper bound of the estimator, the estimator reduction, and the orthogonality property. We also consider how the results are affected by the horizon parameter $\delta$ which characterizes the range of nonlocality. Numerical experiments are performed to verify our theoretical findings.