A Nonlocal diffusion model with $H^1$ convergence for Dirichlet Boundary

TL;DR

This paper introduces a novel nonlocal model for the Poisson equation with Dirichlet boundary conditions, ensuring well-posedness and first-order convergence in the $H^1$ norm, and extends to Robin boundary conditions.

Contribution

The paper develops a nonlocal approximation that handles Dirichlet boundary conditions by treating the normal derivative as an auxiliary variable, ensuring convergence and preserving key properties.

Findings

Proved well-posedness of the nonlocal model.

Established first-order convergence in $H^1$ norm.

Extended the model to Robin boundary conditions.

Abstract

In this paper, we present a nonlocal model for Poisson equation and corresponding eigenproblem with Dirichlet boundary condition. In the direct derivation of the nonlocal model, normal derivative is required which is not known for Dirichlet boundary. To overcome this difficulty, we treat the normal derivative as an auxiliary variable and derive corresponding nonlocal approximation of the boundary condition. For this specifically designed nonlocal mode, we can prove its well-posedness and convergence to the counterpart continuous model. The nonlocal model is carefully designed such that coercivity and symmetry are preserved. Based on these good properties, we can prove the nonlocal model converges with first order rate in $H^1$ norm. Our model can be naturally extended to Poisson problems with Robin boundary and corresponding eigenvalue problem.