Coupling Finite Elements and Auxiliary Sources

Abstract

We propose a hybrid method to solve time-harmonic Maxwell's equations in R^3 through the Finite Element Method (FEM) in a bounded region encompassing parameter inhomogeneities, coupled with the Multiple Multipole Program (MMP) in the unbounded complement. FEM and MMP enjoy complementary capabilities. FEM requires a mesh of the computational domain of interest. This is expensive, but can treat inhomogeneous materials, shapes with corners, or other singularities. Moreover, FEM allows a purely local construction of the discrete system of equations. On the other hand, MMP belongs to the class of methods of auxiliary sources and of Trefftz methods, as it employs point sources that are exact solutions of the homogeneous equations as (global) basis functions: one only needs to evaluate them on hypersurfaces to obtain the discrete problem, and the resulting linear combination is valid in the whole domain where the equations hold, which can be unbounded. MMP performs well where the electromagnetic field is smooth, i.e. in the free space far from physical sources and material interfaces. Thus, a natural way to combine the strengths of these methods arises when one needs to simulate the electromagnetic field of components with inhomogeneous parameters or nonsmooth shapes surrounded by free space: use FEM on a mesh defined on the components and MMP in the unbounded domain outside. The boundary between the FEM and MMP domains can be nonphysical if one surrounds the components by a conforming mesh of an "airbox" also modeled by FEM. The interface conditions on the surface of the FEM domain are key to accurate coupled FEM--MMP solutions. Integrating by parts the variational form solved by FEM, surface integrals appear, through which one can impose interface conditions by substituting the ansatz of MMP. However, one interface condition (for example, the continuity of the tangential trace for Maxwell's equations) cannot be imposed in this way. To include this additional condition, we derive four methods from Lagrangian functionals that enforce both the variational form in the FEM domain and all the interface conditions between different discretizations: 1. Least-squares-based coupling using techniques from PDE-constrained optimization. This approach optimizes a functional for the additional condition, subject to a constraint expressed by the variational form of FEM. 2. Discontinuous Galerkin (DG) between the meshed FEM domain and the single-entity "MMP mesh", interpreting MMP as FEM with special basis functions. 3. Multi-field variational formulation in the spirit of mortar finite element methods, relying on the same interpretation of MMP as the DG-based coupling. 4. Coupling through the Dirichlet-to-Neumann operator: here we introduce a weak formulation of the additional condition where MMP basis functions are used as test functions. This approach is generalized to couple MMP with any numerical method based on volume meshes that fits co-chain calculus. Furthermore, to minimize the meshed region and for the sake of generalization, we assume that equation parameters are piecewise constant in the MMP domain (instead of constant everywhere). This induces a partition: we consider the case of one unbounded subdomain with other bounded, but possibly very large, subdomains, each requiring its own MMP discretization space (but without the need of meshes). Hence, on top of the interface conditions between the FEM and MMP domains, one should also impose interface conditions between the MMP subdomains themselves. We compare the FEM--MMP coupling approaches in a series of numerical experiments with different geometries and equation parameters, including examples that exhibit triple-point singularities. Convergence tests are performed for examples that cover the whole spectrum of computational electromagnetics, obtaining the expected results.