Heterogeneous Multiscale Method for the Maxwell equations with high contrast
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 1, pp. 35-61.

In this paper, we suggest a new Heterogeneous Multiscale Method (HMM) for the (time-harmonic) Maxwell scattering problem with high contrast. The method is constructed for a setting as in Bouchitté, Bourel and Felbacq [C.R. Math. Acad. Sci. Paris 347 (2009) 571–576], where the high contrast in the parameter leads to unusual effective parameters in the homogenized equation. We present a new homogenization result for this special setting, compare it to existing homogenization approaches and analyze the stability of the two-scale solution with respect to the wavenumber and the data. This includes a new stability result for solutions to time-harmonic Maxwell’s equations with matrix-valued, spatially dependent coefficients. The HMM is defined as direct discretization of the two-scale limit equation. With this approach we are able to show quasi-optimality and a priori error estimates in energy and dual norms under a resolution condition that inherits its dependence on the wavenumber from the stability constant for the analytical problem. This is the first wavenumber-explicit resolution condition for time-harmonic Maxwell’s equations. Numerical experiments confirm our theoretical convergence results.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2018064
Classification : 65N30, 65N15, 65N12, 35Q61, 78M40, 35B27
Mots-clés : Multiscale method, finite elements, homogenization, two-scale equation, Maxwell equations
Verfürth, Barbara 1

1 
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Verfürth, Barbara. Heterogeneous Multiscale Method for the Maxwell equations with high contrast. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 1, pp. 35-61. doi : 10.1051/m2an/2018064. http://www.numdam.org/articles/10.1051/m2an/2018064/

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