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

Yuchen Shi, Jihong Wang, Jiwei Zhang

TL;DR

This work develops a rigorous AC framework for 2D nonlocal Poisson problems with Neumann boundary conditions, deriving nonlocal Neumann operators from nonlocal Green's identities and proving their weak convergence to the local Neumann operator as the horizon δ→0 on general convex domains. It then analyzes the local limit in both half-plane and general bounded domains, establishing second-order L^2 convergence of nonlocal Neumann solutions to the local Neumann solution after compatibility adjustments. A complete finite element analysis is provided for the approximate-ball method (Nocaps), including kernel symmetry corrections, error decompositions, and AC rates for inner and outer Neumann problems, with extensive numerical validation. The results yield a fully discrete asymptotic-compatibility theory for nonlocal Neumann problems and quantify modeling, discretization, and geometric errors, contributing to reliable simulations of nonlocal boundary phenomena on complex geometries.

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 δ 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 δ 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.

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

TL;DR

This work develops a rigorous AC framework for 2D nonlocal Poisson problems with Neumann boundary conditions, deriving nonlocal Neumann operators from nonlocal Green's identities and proving their weak convergence to the local Neumann operator as the horizon δ→0 on general convex domains. It then analyzes the local limit in both half-plane and general bounded domains, establishing second-order L^2 convergence of nonlocal Neumann solutions to the local Neumann solution after compatibility adjustments. A complete finite element analysis is provided for the approximate-ball method (Nocaps), including kernel symmetry corrections, error decompositions, and AC rates for inner and outer Neumann problems, with extensive numerical validation. The results yield a fully discrete asymptotic-compatibility theory for nonlocal Neumann problems and quantify modeling, discretization, and geometric errors, contributing to reliable simulations of nonlocal boundary phenomena on complex geometries.

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 δ 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 δ 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.

Paper Structure

This paper contains 21 sections, 25 theorems, 205 equations, 3 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Nonlocal Neumann operators
  3. Local limit in the half-plane
  4. Local limit in general bounded domains
  5. Second order Neumann boundary conditions in $L^2$ norm
  6. Neumann variational problem
  7. Compatible condition for the pure Neumann boundary problems
  8. $L^2$-norm asymptotic convergence result for the inner and outer Neumann problems
  9. Finite element analysis
  10. Preliminaries
  11. Convergence of the nonlocal solutions with polygonal approximations
  12. Finite element analysis of the inner Neumann problem (\ref{['problemsymN-']})
  13. Finite element analysis of the outer Neumann problem (\ref{['problemsymN*']})
  14. Numerical experiment
  15. Numerical validation of the errors considering the approximate ball
  16. ...and 6 more sections

Key Result

Theorem 2.2

Assume $\Omega$ is a half plane with boundary $\Gamma = \{0\}\times(-\infty, +\infty)$, if $u \in C^2(\Omega_\gamma^+\cup\Omega_\gamma^-) \cap S_N^-(\Omega)$ and $v \in C_0^1(\Omega_\gamma^+\cup\Omega_\gamma^-)$, where the size of the support of $v$ is independent of $\delta$, then it holds

Figures (3)

  • Figure 1: Schematic of the variable substitution
  • Figure 2: inscribed polygon in $B_\delta(\mathbf{x})$
  • Figure 3: Schematic when the region is $\Omega$

Theorems & Definitions (37)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.1
  • Theorem 2.3: region with $C^2$ boundary
  • Theorem 2.4: polygon
  • Remark 2.2
  • Remark 3.1
  • Lemma 3.1: Nonlocal Poincaré inequality Tadele2010ponce2004
  • Remark 3.2
  • Theorem 3.1
  • ...and 27 more