  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.
