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A rigorous mathematical framework is provided for a substructuring-baseddomain-decomposition approach for nonlocal problems that feature interactionsbetween points separated by a finite distance. Here, by substructuring it ismeant that a traditional geometric configuration for local partial differentialequation problems is used in which a computational domain is subdivided intonon-overlapping subdomains. In the nonlocal setting, this approach issubstructuring-based in the sense that those subdomains interact withneighboring domains over interface regions having finite volume, in contrast tothe local PDE setting in which interfaces are lower dimensional manifoldsseparating abutting subdomains. Key results include the equivalence between theglobal, single-domain nonlocal problem and its multi-domain reformulation, bothat the continuous and discrete levels. These results provide the rigorousfoundation necessary for the development of efficient solution strategies fornonlocal domain-decomposition methods.

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