Error estimates of finite element methods for nonlocal problems using exact or approximated interaction neighborhoods

TL;DR

This paper analyzes the error between finite element solutions of nonlocal models with bounded interaction neighborhoods and the local limit, considering both exact and approximated neighborhoods, supported by numerical experiments.

Contribution

It provides asymptotic error estimates for nonlocal finite element methods approaching the local limit, including cases with polygonal approximations of interaction neighborhoods.

Findings

Error estimates for nonlocal to local convergence

Validation through numerical experiments

Comparison between exact and approximated neighborhoods

Abstract

We study the asymptotic error between the finite element solutions of nonlocal models with a bounded interaction neighborhood and the exact solution of the limiting local model. The limit corresponds to the case when the horizon parameter, the radius of the spherical nonlocal interaction neighborhood of the nonlocal model, and the mesh size simultaneously approach zero. Two important cases are discussed: one involving the original nonlocal models and the other for nonlocal models with polygonal approximations of the nonlocal interaction neighborhood. Results of numerical experiments are also reported to substantiate the theoretical studies.