An Optimization-Based Approach for Elliptic Problems with Interfaces

Abstract

Independent meshing of subdomains in interface problems can lead to spatially noncoincident interface grids. This setting occurs both when the interface is physical, as in transmission problems, and when it results from breaking up a complex domain into simpler shapes to aid grid generation, as in mesh tying. Traditional domain decomposition enforces continuity of the states across the interface by using a Lagrange multiplier representing the interface “flux.” However, extension of this approach to noncoincident interfaces is nontrivial because of the lack of a clearly defined spatial location for the imposition of the continuity constraint. We present an alternative, optimization-based approach, which mitigates this difficulty by reversing the roles of the coupling condition and the subdomain equations. The idea is to couch the interface problem into a virtual control formulation in which the former defines the objective, the latter define the constraints, and the unknown interface flux is a virtual control, specifying Neumann boundary conditions for the subproblems. This approach has valuable computational and theoretical advantages. First, since the exact enforcement of the coupling condition on noncoincident meshes is impossible, treating it as an optimization objective rather than a constraint is better suited for this setting. For instance, the only requirement for the evaluation of the objective is linear consistency, which can be achieved by simple extension operators. Second, conservation of the flux is ensured by choosing the virtual Neumann control on each discrete interface as a pullback of a piecewise constant function defined on a common refinement of the parameterizations of these interfaces. Implementation of these pullbacks is also straightforward. Numerical studies with multiple noncoincident interface configurations confirm the accuracy and the robustness of the approach.