AIMS Mathematics, 2017, 2(3): 458-478. doi: 10.3934/Math.2017.2.458

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Localized Orthogonal Decomposition for two-scale Helmholtz-typeproblems

Applied Mathematics, University of Münster, D-48149 Münster, Germany

In this paper, we present a Localized Orthogonal Decomposition (LOD) in Petrov-Galerkinformulation for a two-scale Helmholtz-type problem. The two-scale problem is, for instance, motivatedfrom the homogenization of the Helmholtz equation with high contrast, studied together with acorresponding multiscale method in a previous paper of the authors. There, an unavoidable resolutioncondition on the mesh sizes in terms of the wave number has been observed, which is known as “pollutione ect” in the finite element literature. Following ideas of Gallistl and Peterseim, we use standardfinite element functions for the trial space, whereas the test functions are enriched by solutions of subscsaleproblems (solved on a finer grid) on local patches. Provided that the oversampling parameterm, which indicates the size of the patches, is coupled logarithmically to the wave number, we obtain aquasi-optimal method under a reasonable resolution of a few degrees of freedom per wave length, thusovercoming the pollution effect. In the two-scale setting, the main challenges for the LOD lie in thecoupling of the function spaces and in the periodic boundary conditions.
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