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

Abstract

A mathematical framework is provided for a substructuring‐based domain decomposition (DD) approach for nonlocal problems that features interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation (PDE) 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 DD methods.

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

“…We construct the stiffness matrix and right-hand side vector for each subdomain in the same manner as for the global problem, i.e., by inserting the finite element basis functions into the respective weak subdomain equations. As is the case with any DD method, the goal is to perform these steps in such a way so as to ensure that (34) the global solution obtained from the N s discretized subdomain FE systems corresponding to (26), i.e., u dd,h (x), should be the same as the solution u h of the discretized single-domain FE system (11).…”

“…Figure 3c illustrates the need to make changes to the single-domain FE grid so that the new grid does respect those common boundaries. Of course, if we define the subdomain FE discretization using the new grid of Figure 3c, there is no hope for the solution of the FE discretization of ( 26) to be the same as the solution of single-domain FE discretization (11), i.e., the goal (34) cannot be achieved. Note also that the re-meshing of Figure 3 is relatively easy to effect for Cartesian grids, but becomes a much more complex task for general grids, especially in three dimensions.…”

“…In the technical report [11], we further elucidate the equivalence between the multi-domain formulation and the single-domain problem and illustrate the application of the nonlocal DD framework. Specifically, we use the framework developed in this work along with a FETI solution approach to obtain, for a very simplified setting, illustrative numerical examples of nonlocal DD problems.…”

“…Let W h ⊂ W and W 0,h ⊂ W 0 denote finite element (FE) subspaces. Then, a FE approximation u h (x) ∈ W h of the solution u ∈ W of ( 8) is defined to be the solution of the discretized weak formulation (11) given f (x) ∈ W , g(x) ∈ W Γ , and a kernel γ(x, y),…”

“…Then, the discrete nonlocal volume-constrained problem (11), respectively (14), can be expressed in terms of the equivalent minimization problem [17,25,26,27] (16)…”