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.iii 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 as...

Trefftz co-chain calculus

“…Starting from the natural transmission conditions for differential forms, we derive the fully-discrete coupled system of equations. This generalizes our earlier approaches to couple the finite element method with Trefftz methods: see [ 15 ] for Poisson’s equation, [ 16 ] for 2D Helmholtz equation, [ 17 ] for magnetostatic Maxwell’s equations, [ 19 ] for an eddy-current model as in Sect. 1.2 , and [ 18 ] for electromagnetic wave propagation.…”

“…The former is the exponential convergence of their approximation error when the unknown possesses an analytic extension beyond the Trefftz approximation domain. 11 This is proven formally for 2D Poisson’s equation in [ 15 , p. 3, Section 2.2] and with numerical evidence for 2D Helmholtz equation and Maxwell’s equations in [ 16 ] and [ 18 ], respectively.…”

Coupling finite elements and auxiliary sources for electromagnetic wave propagation

“…For the Poisson's equation in an unbounded domain ℝ 2 / Ω ⋆ , with Ω ⋆ bounded domain, it can be proven that the approximation error in the H 1 ‐norm decreases exponentially with respect to the number of degrees of freedom of the corresponding MMP Trefftz space if the unknown possesses an analytic extension beyond the Trefftz domain, 2 p. 3, proposition 1. The proof relies on the fact that (generalized) harmonic polynomials also achieve exponential convergence in H i ‐seminorms, i = 0, …, j , j ∈ ℕ 0 , when solving 2D Poisson in a bounded domain that satisfies certain assumptions, 26 p. 61, theorem 3.2.5.…”

“…Several approaches to couple FEM and a Trefftz method for the Poisson's equation in both 2D and 3D have been discussed by the authors from the perspective of numerical analysis in Reference 2. Existence, uniqueness, and stability of all coupling approaches are formally proven in that work, which only deals with scalar unknown functions.…”

“…In particular, the mesh can be so small that it only surrounds points where the field is singular, like triple-point singularities (TPS), which emerge at the junction of three different materials 1 (Section 4.2). 2 The first equation of (1a) can be obtained by substituting B = r × u, with B : R 3 ! ℂ 3 magnetic flux density, and E = − r ϕ + ıωu, with E : R 3 !…”