Coupling finite elements and auxiliary sources for Maxwell's equations

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 handle unbounded domains; yet, it cannot accommodate 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: Least‐squares–based coupling using techniques from PDE‐constrained optimization. Multifield variational formulation in the spirit of mortar finite element methods. Discontinuous Galerkin coupling between the FEM mesh and the single‐entity MMP subdomain. Coupling by tangential components traces. We study the convergence of these approaches in a series of numerical experiments.

Coupling finite elements and auxiliary sources for electromagnetic wave propagation

“…• M μ , M ɛ : R 3 ! ℂ 3,3 are symmetric, bounded, uniformly positive-definite matrices that correspond to an inhomogeneous, anisotropic permeability, and permittivity, respectively. We assume that both M μ (x) = μI and M ɛ (x) = ɛI 8 x ∈ R 3 / Ω ?…”

“…However, now we rely on the other main approach for imposing weak continuity on nonconforming meshes, which is the multi-field domain decomposition method. 34 For Maxwell's equations, the multi-field method aims at imposing the continuity of the tangential components trace (28) in a weak sense by means of a Lagrange multiplier λ≔γv, v∈H loc curl, Ω ð Þ, Ω⊆R 3 : ð29Þ…”

“…Existence, uniqueness, and stability of all coupling approaches are formally proven in that work, which only deals with scalar unknown functions. We offered numerical evidence for the feasibility of the coupling for Maxwell's equations (vector unknown functions) in References 3 and 4, which illustrate numerical convergence results for the magnetostatic and eddy‐current equations, respectively.…”

“…It is implicitly assumed that k ≠ 0; otherwise, if, for example, ω = 0, we would be in a magnetostatic regime. This case is discussed in Reference 3. j : ℝ 3 → ℝ 3 , with ∇ · j = 0, represents the stationary current that generates the electromagnetic field. j has compact support in Ω ⋆ . (1b) is the Silver‐Müller radiation condition ; please refer to Reference 14, p. 195, definition 6.6.…”

“…ℂ 3 electric field, into Ampère's law, 13 p. 4, (1.4c). 3 A special method of auxiliary sources is the multiple multipole program, which employs multipole expansions-see Section 2.1. 4 At first sight, one could think of combining (10a) and (10c) and impose the continuity…”