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A new Heterogeneous Multiscale Method for time-harmonic Maxwell's equations

SIAM J. Numer. Anal., Volume 54, Number 6, page 3493--3522 - 2016
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In this paper, we suggest a new heterogeneous multiscale method (HMM) for the time-harmonic Maxwell equations in locally periodic media. The method is constructed by using a divergence-regularization in one of the cell problems. This allows us to introduce fine-scale correctors that are not subject to a cumbersome divergence-free constraint and which can hence easily be implemented. To analyze the method, we first revisit classical homogenization theory for time-harmonic Maxwell equations and derive a new homogenization result that makes use of the divergence-regularization in the two-scale homogenized equation. We then show that the HMM is equivalent to a discretization of this equation. In particular, writing both problems in a fully coupled two-scale formulation is the crucial starting point for a corresponding numerical analysis of the method. With this approach we are able to prove rigorous a priori error estimates in the H(curl)- and the $H^{-1}$-norm and we derive reliable and efficient localized residual-based a posteriori error estimates. Numerical experiments are presented to verify the a priori convergence results.

Références BibTex

  author       = {Henning, P. and Ohlberger, M. and Verf{\"u}rth, B.},
  title        = {A new Heterogeneous Multiscale Method for time-harmonic Maxwell's equations},
  journal      = {SIAM J. Numer. Anal.},
  number       = {6},
  volume       = {54},
  pages        = {3493--3522},
  year         = {2016},
  publisher    = {SIAM},
  doi          = {10.1137/15M1039225},
  url          = \{/2016/HOV16a},

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