Coupling FEM with a Multiple-Subdomain Trefftz Method

We consider 2D electromagnetic scattering at bounded objects consisting of different, possibly inhomogeneous materials. We propose and compare three approaches to couple the finite element method (FEM) in a meshed domain encompassing material inhomogeneities and the multiple multipole program (MMP) in the unbounded complement. MMP is a Trefftz method, as it employs trial spaces composed of exact solutions of the homogeneous problem. Each of these global basis functions is anchored at a point that, if singular, is placed outside the respective domain of approximation. In the MMP domain we assume that material parameters are piecewise constant, which induces a partition: one unbounded subdomain and other bounded, but possibly very large, subdomains, each requiring its own Trefftz trial space. Coupling approaches arise from seeking stationary points of Lagrangian functionals that both enforce the variational form of the equations in the FEM domain and match the different trial functions across subdomain interfaces. Hence, on top of the transmission conditions connecting the FEM and MMP domains, one also has to impose transmission conditions between the MMP subdomains. Specifically, we consider the following coupling approaches: Least-squares-based coupling using techniques from PDE-constrained optimization.Multi-field variational formulation in the spirit of mortar finite element methods.Discontinuous Galerkin coupling between the meshed FEM domain and the single-entity MMP subdomains. We compare these approaches in a series of numerical experiments with different geometries and material parameters, including examples that exhibit triple-point singularities.