Maximum principle preserving nonlocal diffusion model with Dirichlet boundary condition

TL;DR

This work addresses imposing Dirichlet boundary data in nonlocal diffusion while preserving the maximum principle. It introduces two nonlocal models, a first-order symmetric model with a $1/\delta$ boundary penalty and a second-order weighted model with a spatially varying weight $\mu(\mathbf{x})$, both admitting variational formulations and ensuring well-posedness. The authors establish $H^1$ convergence and, via the maximum principle, $L^\infty$ convergence rates (first order for the basic model and second order with the tuned weight), along with vanishing nonlocality results as $\delta \to 0$ for smooth solutions. The results suggest a simple, robust framework closely related to weighted nonlocal Laplacians, with potential extensions to other systems such as Stokes, and provide a foundation for accurate, boundary-aware nonlocal simulations.

Abstract

In this paper, we propose nonlocal diffusion models with Dirichlet boundary. These nonlocal diffusion models preserve the maximum principle and also have corresponding variational form. With these good properties, we can prove the well-posedness and the vanishing nonlocality convergence. Furthermore, by specifically designed weight function, we can get a nonlocal diffusion model with second order convergence which is optimal for nonlocal diffusion models.

Figures (1)

• Figure B.1: Regions for Lemma \ref{['corelemma']}

Theorems & Definitions (22)

• Proposition 2.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1
• proof
• Corollary 3.1
• Lemma 3.2