A discontinuous Galerkin Trefftz type method for solving the two dimensional Maxwell equations
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ORIGINAL PAPER
A discontinuous Galerkin Trefftz type method for solving the two dimensional Maxwell equations Håkon Sem Fure1 • Se´bastien Pernet1 • Margot Sirdey1,2 • Se´bastien Tordeux2 Received: 27 February 2020 / Accepted: 25 July 2020 / Published online: 14 August 2020 Ó Springer Nature Switzerland AG 2020
Abstract Trefftz methods are known to be very efficient to reduce the numerical pollution when associated to plane wave basis. However, these local basis functions are not adapted to the computation of evanescent modes or corner singularities. In this article, we consider a two dimensional time-harmonic Maxwell system and we propose a formulation which allows to design an electromagnetic Trefftz formulation associated to local Galerkin basis computed thanks to an auxiliary Ne´de´lec finite element method. The results are illustrated with numerous numerical examples. The considered test cases reveal that the short range and long range propagation phenomena are both well taken into account. Keywords Trefftz method Electromagnetic wave Ne´de´lec finite element Numerical methods Transverse electric polarization Maxwell equation Mathematics Subject Classification 76M10 65N30 76M10 35Q61
1 Introduction Numerical methods like the Finite Element Method (FEM), see [3, 26, 32], and the Finite Difference Method (FDM), see [41] for example, are widely used to solve time-harmonic electromagnetic wave equations. One limitation they all face is called the pollution effect. When considering the numerical solution of a propagation phenomenon with wavenumbers This article is part of the topical collection ‘‘Waves 2019–invited papers’’ edited by Manfred Kaltenbacher and Markus Melenk. & Se´bastien Tordeux [email protected] Se´bastien Pernet [email protected] Margot Sirdey [email protected] 1
ONERA, DTIS, 2 avenue Edouard Belin, 31000 Toulouse, France
2
Universite de Pau et des Pays de l’Adour, E2S-UPPA, CNRS, INRIA, e´quipe Magique 3D, LMAP, Pau, France SN Partial Differential Equations and Applications
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SN Partial Differ. Equ. Appl. (2020) 1:23
k posed on domains with length L, the numerical accuracy deteriorates when kL becomes large. This has been highlighted in the following articles [24, 25]. This phenomenon is related to a numerical dispersion and is called numerical pollution. A detailed analysis with error estimates has been proposed in [30]. This issue is of particular importance at high frequency or on large domains where the number of degrees of freedom per wavelength should be chosen large to achieve a given accuracy. Classically, numericians resort to one or more of the following remedies, which can be combined. The first one consists in considering high-order FEM, see for example [2]. A second answer to the numerical pollution issue is Discontinuous Galerkin (DG) finite elements that are less dispersive [1]. This latter approach can be of particular interest in the context of inverse problems. The German m
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