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Coupling Finite Elements and Auxiliary Sources for Maxwell's Equations

by D. Casati and R. Hiptmair and J. Smajic

(Report number 2018-13)

Abstract
The Multiple Multipole Program is a Trefftz method approximating the electromagnetic field in a domain filled with a homogeneous linear medium. MMP can easily cope with unbounded domains; yet, it cannot accommodate either inhomogeneous or nonlinear materials, situations well within the scope of the standard Finite Element Method. We propose to couple FEM and MMP to model Maxwell's equations for materials with spatially-varying properties in an unbounded domain. In some bounded parts of the domain, we use Nédélec's first family of curl-conforming elements; in the unbounded complement, multipole expansions. Several approaches are developed to couple both discretizations across the common interface: 1. Least-squares-based coupling using techniques from PDE-constrained optimization. 2. Multi-field variational formulation in the spirit of mortar finite element methods. 3. Discontinuous Galerkin coupling between the FEM mesh and the single-entity MMP subdomain. 4. Coupling by tangential components traces. We study the convergence of these approaches in a series of numerical experiments.

Keywords: finite element method, multiple multipole program, method of auxiliary sources, Trefftz method, computational electromagnetics

@Techreport{CHS18_767,
  author = {D. Casati and R. Hiptmair and J. Smajic},
  title = {Coupling Finite Elements and Auxiliary Sources for Maxwell's Equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2018-13},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2018/2018-13.pdf },
  year = {2018}
}

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