In this paper, a weighted regularization method for the time-harmonic Maxwell equations with perfect conducting or impedance boundary condition in composite materials is presented. The computational domain Ω is the union of polygonal or polyhedral subdomains made of different materials. As a result, the electromagnetic field presents singularities near geometric singularities, which are the interior and exterior edges and corners. The variational formulation of the weighted regularized problem is given on the subspace of (;Ω) whose fields satisfy div ()∈ L2(Ω) and have vanishing tangential trace or tangential trace in L2(). The weight function is equivalent to the distance of to the geometric singularities and the minimal weight parameter α is 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 problem by means of Lagrange Finite Elements of type P1 and P2 are given on uniform and appropriately refined two-dimensional meshes. The performance of the method in the case of eigenvalue problems is addressed.
Keywords: Maxwell's equations, interface problem, singularities of solutions, density results, weighted regularization
@article{M2AN_2010__44_1_75_0, author = {Ciarlet Jr., Patrick and Lef\`evre, Fran\c{c}ois and Lohrengel, St\'ephanie and Nicaise, Serge}, title = {Weighted regularization for composite materials in electromagnetism}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {75--108}, publisher = {EDP-Sciences}, volume = {44}, number = {1}, year = {2010}, doi = {10.1051/m2an/2009041}, mrnumber = {2647754}, zbl = {1192.78039}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2009041/} }
TY - JOUR AU - Ciarlet Jr., Patrick AU - Lefèvre, François AU - Lohrengel, Stéphanie AU - Nicaise, Serge TI - Weighted regularization for composite materials in electromagnetism JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2010 SP - 75 EP - 108 VL - 44 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2009041/ DO - 10.1051/m2an/2009041 LA - en ID - M2AN_2010__44_1_75_0 ER -
%0 Journal Article %A Ciarlet Jr., Patrick %A Lefèvre, François %A Lohrengel, Stéphanie %A Nicaise, Serge %T Weighted regularization for composite materials in electromagnetism %J ESAIM: Modélisation mathématique et analyse numérique %D 2010 %P 75-108 %V 44 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2009041/ %R 10.1051/m2an/2009041 %G en %F M2AN_2010__44_1_75_0
Ciarlet Jr., Patrick; Lefèvre, François; Lohrengel, Stéphanie; Nicaise, Serge. Weighted regularization for composite materials in electromagnetism. ESAIM: Modélisation mathématique et analyse numérique, Volume 44 (2010) no. 1, pp. 75-108. doi : 10.1051/m2an/2009041. http://www.numdam.org/articles/10.1051/m2an/2009041/
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