Weighted regularization for composite materials in electromagnetism

Patrick Ciarlet Jr.; François Lefèvre; Stéphanie Lohrengel; Serge Nicaise

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 1, page 75-108
  • ISSN: 0764-583X

Abstract

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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 ( 𝐜𝐮𝐫𝐥 ;Ω) whose fields 𝐮 satisfy w α div ( ε 𝐮 )∈L2(Ω) and have vanishing tangential trace or tangential trace in L2( Ω ). The weight function w ( 𝐱 ) is equivalent to the distance of 𝐱 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.

How to cite

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Ciarlet Jr., Patrick, et al. "Weighted regularization for composite materials in electromagnetism." ESAIM: Mathematical Modelling and Numerical Analysis 44.1 (2010): 75-108. <http://eudml.org/doc/250791>.

@article{CiarletJr2010,
abstract = { 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 $\{\cal H\}$($\{\bf curl\}$;Ω) whose fields $\textit\{\textbf\{ u\}\}$ satisfy $w^\alpha$ div ($\varepsilon\{\textit\{\textbf\{u\}\}\}$)∈L2(Ω) and have vanishing tangential trace or tangential trace in L2($\partial\Omega$). The weight function $w(\bf x)$ is equivalent to the distance of $\bf x$ 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. },
author = {Ciarlet Jr., Patrick, Lefèvre, François, Lohrengel, Stéphanie, Nicaise, Serge},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Maxwell's equations; interface problem; singularities of solutions; density results; weighted regularization},
language = {eng},
month = {3},
number = {1},
pages = {75-108},
publisher = {EDP Sciences},
title = {Weighted regularization for composite materials in electromagnetism},
url = {http://eudml.org/doc/250791},
volume = {44},
year = {2010},
}

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: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 44
IS - 1
SP - 75
EP - 108
AB - 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 ${\cal H}$(${\bf curl}$;Ω) whose fields $\textit{\textbf{ u}}$ satisfy $w^\alpha$ div ($\varepsilon{\textit{\textbf{u}}}$)∈L2(Ω) and have vanishing tangential trace or tangential trace in L2($\partial\Omega$). The weight function $w(\bf x)$ is equivalent to the distance of $\bf x$ 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.
LA - eng
KW - Maxwell's equations; interface problem; singularities of solutions; density results; weighted regularization
UR - http://eudml.org/doc/250791
ER -

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