Weighted regularization for composite materials in electromagnetism

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

ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 1, pp. 75-108.

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 $\mathscr{H}$($\mathrm{𝐜𝐮𝐫𝐥}$;Ω) whose fields $𝐮$ satisfy $w^{\alpha }$ div ($\epsilon 𝐮$)∈ L²(Ω) and have vanishing tangential trace or tangential trace in L²($\partial \Omega$). The weight function $w\left(𝐱\right)$ 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.

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