Seminar in Numerical Analysis: Christian Stohrer (ENSTA ParisTech)
Electromagnetic phenomena can be modeled using Maxwell's equations. In particular we are interested in harmonic electromagnetic waves propagating through a highly oscillatory material such as e.g. fiber reinforced plastic. The permittivity and the permeability of such materials vary on a microscopic length scale. The use of standard edge finite elements is of limited profit, since the microscopic structure requires very refined meshes to provide satisfying approximations. This may easily result in computational costs difficult to manage. However, if one is only interested in the effective behavior of the solution and not in the microscopic details, homogenization techniques can be used to overcome these difficulties. In this talk we review first the results of analytical homogenization results for Maxwell's equations. The goal of this theory is to replace the oscillatory material with an effective one, such that the overall behavior of the solution remains unchanged. The solution of the arising equations can be solved with standard numerical methods because the effective material depends no longer on the micro scale. In the second part of the talk we propose a multiscale scheme following the framework of the finite element heterogeneous multiscale method (FE-HMM). Contrary to the discretization of the analytically homogenized equation, no effective coefficient must be precomputed beforehand. We prove that the FE-HMM solution converges to the homogenized one for periodic materials and show some numerical experiments.
This is a joint work with Sonia Fliss and Patrick Ciarlet.
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