A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions

TL;DR

This paper develops a rigorous mathematical framework for substructuring-based domain decomposition methods tailored for nonlocal problems, enabling efficient solutions by establishing equivalence between global and multi-domain formulations.

Contribution

It introduces a novel theoretical foundation for nonlocal domain decomposition, extending traditional methods to handle finite-volume interface interactions.

Findings

Proves equivalence between global and multi-domain nonlocal problems.

Provides a rigorous foundation for developing efficient solution strategies.

Establishes theoretical basis at both continuous and discrete levels.

Abstract

A rigorous mathematical framework is provided for a substructuring-based domain-decomposition approach for nonlocal problems that feature interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation problems is used in which a computational domain is subdivided into non-overlapping subdomains. In the nonlocal setting, this approach is substructuring-based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains. Key results include the equivalence between the global, single-domain nonlocal problem and its multi-domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal domain-decomposition methods.