Schwarz Methods for Nonlocal Problems

TL;DR

This paper extends Schwarz domain decomposition methods to nonlocal problems with variable coefficients, demonstrating convergence and implementation of algorithms for nonlocal diffusion operators.

Contribution

It introduces the application of Schwarz methods to nonlocal PDEs, providing convergence analysis and implementation details for symmetric kernel-based problems.

Findings

Convergence of Schwarz methods for nonlocal problems established.

Implementation of multiplicative and additive Schwarz algorithms demonstrated.

Applicable to nonlocal diffusion operators with symmetric kernels.

Abstract

The first domain decomposition methods for partial differential equations were already developed in 1870 by H. A. Schwarz. Here we consider a nonlocal Dirichlet problem with variable coefficients, where a nonlocal diffusion operator is used. We find that domain decomposition methods like the so-called Schwarz methods seem to be a natural way to solve these nonlocal problems. In this work we show the convergence for nonlocal problems, where specific symmetric kernels are employed, and present the implementation of the multiplicative and additive Schwarz algorithms in the above mentioned nonlocal setting.