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Weighted regularization for composite materialsin electromagnetism

Published online by Cambridge University Press:  03 November 2009

Patrick Ciarlet Jr. ,
François Lefèvre ,
Stéphanie Lohrengel  and
Serge Nicaise
Patrick Ciarlet Jr.
Affiliation:
Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 32 boulevard Victor, 75739 Paris Cedex 15, France. Patrick.Ciarlet@ensta.fr
François Lefèvre
Affiliation:
Laboratoire de Mathématiques, FRE 3111, UFR Sciences exactes et naturelles, Université de Reims Champagne-Ardenne, Moulin de la Housse – B.P. 1039, 51687 Reims Cedex 2, France. francois.lefevre@univ-reims.fr; stephanie.lohrengel@univ-reims.fr
Stéphanie Lohrengel
Affiliation:
Laboratoire de Mathématiques, FRE 3111, UFR Sciences exactes et naturelles, Université de Reims Champagne-Ardenne, Moulin de la Housse – B.P. 1039, 51687 Reims Cedex 2, France. francois.lefevre@univ-reims.fr; stephanie.lohrengel@univ-reims.fr
Serge Nicaise
Affiliation:
LAMAV, Université de Valenciennes et du Hainaut Cambrésis, Le Mont Houy, 59313 Valenciennes Cedex 9, France. serge.nicaise@univ-valenciennes.fr
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Abstract

In this paper, a weighted regularization method for the time-harmonic Maxwell equationswith perfect conducting or impedance boundary condition in composite materials is presented.The computational domain Ω is the unionof polygonal or polyhedral subdomains made of different materials. As a result, the electromagnetic field presents singularitiesnear geometric singularities, which are the interior and exterior edges and corners. The variational formulation of theweighted regularized problem is given on the subspace of ${\cal H}$ ( ${\bf curl}$ ;Ω)whose fields $\textit{\textbf{ u}}$ satisfy $w^\alpha$ div ( $\varepsilon{\textit{\textbf{u}}}$ )∈L2 (Ω) and have vanishing tangential traceor tangential trace in L2 ( $\partial\Omega$ ). The weight function $w(\bf x)$ is equivalentto the distance of $\bf x$ to the geometric singularities and the minimal weight parameter αis given in terms of the singular exponents of a scalar transmission problem.A density result is proven that guarantees the approximability of the solution field by piecewise regular fields.Numerical results for the discretization of the source problemby means of Lagrange Finite Elements of type P 1 and P 2 are given onuniform and appropriately refined two-dimensional meshes.The performance of the method in the case of eigenvalue problems is addressed.

Keywords

Maxwell's equationsinterface problemsingularities of solutionsdensity resultsweighted regularization
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 1 , January 2010 , pp. 75 - 108
Copyright
© EDP Sciences, SMAI, 2009

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