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AIMS Mathematics

2017, Volume 2, Issue 3: 458-478. doi: 10.3934/Math.2017.2.458
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Research article

Localized Orthogonal Decomposition for two-scale Helmholtz-typeproblems

  • Applied Mathematics, University of Münster, D-48149 Münster, Germany
  • Received: 05 May 2017 Accepted: 14 August 2017 Published: 14 August 2017

  • In this paper, we present a Localized Orthogonal Decomposition (LOD) in Petrov-Galerkin formulation for a two-scale Helmholtz-type problem. The two-scale problem is, for instance, motivated from the homogenization of the Helmholtz equation with high contrast, studied together with a corresponding multiscale method in a previous paper of the authors. There, an unavoidable resolution condition on the mesh sizes in terms of the wave number has been observed, which is known as "pollution e ect" in the finite element literature. Following ideas of Gallistl and Peterseim, we use standard finite element functions 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, we obtain a quasi-optimal method under a reasonable resolution of a few degrees of freedom per wave length, thus overcoming the pollution effect. In the two-scale setting, the main challenges for the LOD lie in the coupling of the function spaces and in the periodic boundary conditions.

    Citation: Mario Ohlberger, Barbara Verfürth. Localized Orthogonal Decomposition for two-scale Helmholtz-typeproblems[J]. AIMS Mathematics, 2017, 2(3): 458-478. doi: 10.3934/Math.2017.2.458

    Related Papers:

  • In this paper, we present a Localized Orthogonal Decomposition (LOD) in Petrov-Galerkin formulation for a two-scale Helmholtz-type problem. The two-scale problem is, for instance, motivated from the homogenization of the Helmholtz equation with high contrast, studied together with a corresponding multiscale method in a previous paper of the authors. There, an unavoidable resolution condition on the mesh sizes in terms of the wave number has been observed, which is known as "pollution e ect" in the finite element literature. Following ideas of Gallistl and Peterseim, we use standard finite element functions 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, we obtain a quasi-optimal method under a reasonable resolution of a few degrees of freedom per wave length, thus overcoming the pollution effect. In the two-scale setting, the main challenges for the LOD lie in the coupling of the function spaces and in the periodic boundary conditions.


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