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A singular field method for the solution of Maxwell's equations
Bonnet-Ben Dhia, Anne-Sophie
Hazard, Christophe
Lohrengel, Stéphanie
Propagation des Ondes : Étude Mathématique et Simulation (POEMS) ; Inria Saclay - Ile de France ; Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA) ; École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)
International audience
ISSN: 0036-1399
SIAM Journal on Applied Mathematics
Society for Industrial and Applied Mathematics
hal-01009853
https://hal-ensta-paris.archives-ouvertes.fr//hal-01009853
https://hal-ensta-paris.archives-ouvertes.fr//hal-01009853
SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 1999, 59 (6), pp.2028-2044. ⟨10.1137/S0036139997323383⟩
DOI: 10.1137/S0036139997323383
info:eu-repo/semantics/altIdentifier/doi/10.1137/S0036139997323383
en
info:eu-repo/semantics/article
Journal articles
It is well known that in the case of a regular domain the solution of the time-harmonic Maxwell's equations allows a discretization by means of nodal finite elements: this is achieved by solving a regularized problem similar to the vector Helmholtz equation. The present paper deals with the same problem in the case of a nonconvex polyhedron. It is shown that a nodal finite element method does not approximate in general the solution to Maxwell's equations, but actually the solution to a neighboring variational problem involving a different function space. Indeed, the solution to Maxwell's equations presents singularities near the edges and corners of the domain that cannot be approximated by Lagrange finite elements. A new method is proposed involving the decomposition of the solution field into a regular part that can be treated numerically by nodal finite elements and a singular part that has to be taken into account explicitly. This singular field method is presented in various situations such as electric and magnetic boundary conditions, inhomogeneous media, and regions with screens. Copyright © 1999 Society for Industrial and Applied Mathematics
1999