Convergence analysis of Jacobi spectral collocation methods for weakly singular nonlocal diffusion equations with volume constraints

TL;DR

This paper develops and analyzes a Jacobi spectral collocation method for solving weakly singular nonlocal diffusion equations with volume constraints, demonstrating convergence and effectiveness through theoretical proofs and numerical tests.

Contribution

It introduces two-sided Jacobi spectral quadrature rules and provides a rigorous convergence analysis for nonlocal diffusion equations with singular integral operators.

Findings

The method converges in the $L^ Infty$ norm.

Numerical results verify the theoretical convergence rates.

The solution approaches the local limit as nonlocal interactions diminish.

Abstract

This paper considers efficient spectral solutions for weakly singular nonlocal diffusion equations with Dirichlet-type volume constraints. The equation we consider contains an integral operator that typically has a singularity at the midpoint of the integral domain, and the approximation of the integral operator is one of the essential difficulties in solving nonlocal equations. To overcome this problem, two-sided Jacobi spectral quadrature rules are proposed to develop a Jacobi spectral collocation method for nonlocal diffusion equations. A rigorous convergence analysis of the proposed method with the $L^\infty$ norm is presented, and we further prove that the Jacobi collocation solution converges to its corresponding local limit as nonlocal interactions vanish. Numerical examples are given to verify the theoretical results.