# 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

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topCiarlet 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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