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Localized Orthogonal Decomposition for two-scale Helmholtz-type problems

Rapport de recherchearXiv, arXiv preprint, Number 1605.03410 - may 2016
Télécharger la publication : 1605.03410.pdf [292Ko]  
In this paper, we suggest a two-scale Localized Orthogonal Decomposition (LOD) method in Petrov-Galerkin formulation for the Helmholtz equation with highly heterogeneous coefficient. The method is based on the homogenized (two-scale) formulation and approximates the numerical solution of the related Heterogeneous Multiscale Method introduced in the previous paper (A new Heterogeneous Multiscale Method for the Helmholtz equation with high contrast, arXiv:1605.03400, 2016). There, a resolution condition on the mesh sizes has been observed that is nevertheless optimal in that setting and known as ''pollution effect'' in the finite element literature. The LOD in Petrov-Galerkin formulation, following the ideas of Gallistl and Peterseim (textit{Comput. Methods Appl. Mech. Engrg.} 295:1-17, 2015), overcomes this pollution effect. Standard finite element functions are used for the trial space, whereas the test functions are enriched by solutions of subscsale problems (solved on a finer grid) on local patches. Provided that the oversampling parameter $m$, which indicates the size of the patches, is coupled logarithmically to the wave number, a reasonable resolution of a few degrees of freedom per wave length, is sufficient for the LOD to be stable and quasi-optimal.

Références BibTex

@TechReport{OV16a,
  author       = {Ohlberger, M. and Verf{\"u}rth, B.},
  title        = {Localized Orthogonal Decomposition for two-scale Helmholtz-type problems},
  institution  = {arXiv},
  number       = {1605.03410},
  month        = {may},
  year         = {2016},
  type         = {arXiv preprint},
  url          = \{/2016/OV16a},
}

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