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

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.