Solving Maxwell’s equations using the ultra weak variational formulation

Abstract

We investigate the ultra weak variational formulation for simulating time-harmonic Maxwell problems. This study has two main goals. First, we introduce a novel derivation of the UWVF method which shows that the UWVF is an unusual version of the standard upwind discontinuous Galerkin (DG) method with a special choice of basis functions. Second, we discuss the practical implementation of an electromagnetic UWVF solver. In particular, we propose a method to avoid the conditioning problems that are known to hamper the use of the UWVF for problems in general geometries and inhomogeneous media. In addition, we show how to implement the PML in the UWVF to accurately approximate physically unbounded problems and discuss the parallelization of the UWVF. Three dimensional numerical simulations are used to examine the feasibility of the UWVF for simulating wave propagation in inhomogeneous media and scattering from complex structures.

Plane wave discontinuous Galerkin methods: Analysis of theh-version

“…Matching (2.8), (2.9), and (2.10), (2.11), we see that the original UWVF by Cessenat and Després [10] is recovered by choosing α = 1/2, β = 1/2, γ = 0, δ= 1/2. Following [16,17,23], it is also possible to show that the method by Cessenat and Després can also be recovered by writing the second order problem as a first order system, and then discretizing this system by using a discontinuous Galerkin (DG) method with flux splitting approach (classical upwind DG method). Here, we have followed a slightly different approach and cast the UWVF within the general class of DG methods presented in [8].…”

Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

“…This was noted in [15] where it was shown that the UWVF for Maxwell's equations can be derived using discontinuous Galerkin (DG) techniques and a special choice of degrees of freedom. This observation also holds for the Helmholtz equation (see also [9]).…”

“…This leads directly and elegantly to the UWVF. In [15] we showed that, in the case of Maxwell's equations, but obviously also for the Helmholtz equation written as a first order system, the UWVF results from a standard upwind discontinuous Galerkin method with a suitable choice of degrees of freedom. Here we give a third, equivalent, derivation using the techniques introduced to unify the analysis of DG methods in [2] because we wish to use methods from the analysis of DG methods to analyze the UWVF.…”

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

“…All these methods are of Trefftz type, namely, they are based on approximation spaces made of functions which are (locally) solutions to the considered PDE. We concentrate, in particular, on the UWVF, which has recently seen rapid algorithmic development and extensions; see [15,23,24,29,[34][35][36], and we would like to analyze its application to the time-harmonic Maxwell equations, considering general Trefftz approximation spaces.…”

“…For previous work on the UWVF for Maxwell, we refer to [8,17,19,34]; for different Trefftz-based approaches, we mention [20,47]. Taking cue from the UWVF and following [34], we study a class of Trefftz methods that rely on a DG formulation of the electric field-based Maxwell problem, where the divergence-free constraint is not imposed; the discrete solutions will be elementwise divergence-free, but not globally. Our analysis applies to all these methods, independently of the choice of the particular Trefftz approximation space.…”