Variational Theory and Domain Decomposition for Nonlocal Problems

TL;DR

This paper develops domain decomposition methods for nonlocal operators, establishing well-posedness, spectral properties, and condition number bounds, supported by numerical experiments demonstrating effective conditioning.

Contribution

It introduces the first domain decomposition framework for nonlocal problems, including variational formulations, spectral analysis, and nonlocal Schur complement techniques.

Findings

Condition number bounds are independent of mesh size.

Nonlocal transmission conditions are equivalent to single-domain formulations.

Numerical experiments confirm favorable conditioning of the proposed methods.

Abstract

In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems, proving a nonlocal Poincar\'{e} inequality. To determine the conditioning of the discretized operator, we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix. We then introduce a nonlocal two-domain variational formulation utilizing nonlocal transmission conditions, and prove equivalence with the single-domain formulation. A nonlocal Schur complement is introduced. We establish condition number bounds for the nonlocal stiffness and Schur complement matrices. Supporting numerical experiments demonstrating the conditioning of the nonlocal one- and two-domain problems are presented.