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.
Accepté le :
DOI : 10.1051/m2an/2018064
Keywords: Multiscale method, finite elements, homogenization, two-scale equation, Maxwell equations
Verfürth, Barbara 1
@article{M2AN_2019__53_1_35_0,
author = {Verf\"urth, Barbara},
title = {Heterogeneous {Multiscale} {Method} for the {Maxwell} equations with high contrast},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {35--61},
year = {2019},
publisher = {EDP Sciences},
volume = {53},
number = {1},
doi = {10.1051/m2an/2018064},
zbl = {1422.65415},
mrnumber = {3922817},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2018064/}
}
TY - JOUR AU - Verfürth, Barbara TI - Heterogeneous Multiscale Method for the Maxwell equations with high contrast JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 35 EP - 61 VL - 53 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2018064/ DO - 10.1051/m2an/2018064 LA - en ID - M2AN_2019__53_1_35_0 ER -
%0 Journal Article %A Verfürth, Barbara %T Heterogeneous Multiscale Method for the Maxwell equations with high contrast %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 35-61 %V 53 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2018064/ %R 10.1051/m2an/2018064 %G en %F M2AN_2019__53_1_35_0
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
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