Asymptotic Compatibility of the Approximate-Ball Finite Element Method for 2D Nonlocal Poisson Problem with Neumann Boundary Conditions

TL;DR

This paper analyzes the asymptotic compatibility of a finite element method for 2D nonlocal Poisson problems with Neumann boundary conditions, establishing error estimates and confirming them through numerical examples.

Contribution

It derives nonlocal Neumann boundary operators, proves their convergence to classical operators, and analyzes the asymptotic error estimates of a specific finite element discretization.

Findings

Error estimates are established for the finite element method as the nonlocal horizon parameter approaches zero.

The nonlocal Neumann boundary operators converge to classical Neumann operators.

Numerical examples confirm the theoretical asymptotic compatibility results.

Abstract

In this paper, asymptotic compatibility error estimates of a finite element discretization is presented for 2D nonlocal Poisson problems with Neumann boundary conditions. To this end, we begin with deriving two kind of nonlocal Neumann boundary operators based on nonlocal Green's identities, and establish the corresponding weak convergence to the classical Neumann operator as the horizon parameter {\delta} vanishes for general convex domains. After that, we consider the asymptotic properties (i.e. the so-called local limit) of two nonlocal Neumann boundary-value problems as {\delta} approaches zero. Finally, we analyze the asymptotical compatable error estimates of the approximate-ball-strategy finite element discretization proposed by D'Elia, Gunzburger, and Vollmann (2021), and provide numerical examples to confirm the theoretical results.