A Meshless Galerkin Method For Non-Local Diffusion Using Localized Kernel Bases

TL;DR

This paper presents a novel meshless Galerkin method using localized kernel bases for solving nonlocal diffusion problems, demonstrating well-posedness, convergence, and producing well-conditioned matrices for efficient computation.

Contribution

It introduces a new meshless approach with localized Lagrange basis functions for nonlocal diffusion, ensuring well-posedness and convergence.

Findings

The method produces well-conditioned, symmetric matrices.

Numerical results confirm convergence behavior.

The approach is applicable to both continuous and discrete formulations.

Abstract

We introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, nonlocal diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is nonconforming and uses a localized Lagrange basis that is constructed out of radial basis functions. By verifying that certain inf-sup conditions hold, we demonstrate that both the continuous and discrete problems are well-posed, and also present numerical and theoretical results for the convergence behavior of the method. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, symmetric matrix. This then is used to find the discretized solution.