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Eliminating the pollution effect in Helmholtz problems by local subscale correction


Author: Daniel Peterseim
Journal: Math. Comp. 86 (2017), 1005-1036
MSC (2010): Primary 65N12, 65N15, 65N30
DOI: https://doi.org/10.1090/mcom/3156
Published electronically: August 3, 2016
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Abstract: We introduce a new Petrov-Galerkin multiscale method for the numerical approximation of the Helmholtz equation with large wave number $ \kappa $ in bounded domains in $ \mathbb{R}^d$. The discrete trial and test spaces are generated from standard mesh-based finite elements by local subscale correction in the spirit of numerical homogenization. The precomputation of the correction involves the solution of coercive cell problems on localized subdomains of size $ \ell H$, $ H$ being the mesh size and $ \ell $ being the oversampling parameter. If the mesh size and the oversampling parameter are such that $ H\kappa $ and $ \log (\kappa )/\ell $ fall below some generic constants and if the cell problems are solved sufficiently accurately on some finer scale of discretization, then the method is stable and its error is proportional to $ H$. Pollution effects are eliminated in this regime.


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Additional Information

Daniel Peterseim
Affiliation: Rheinische Friedrich-Wilhelms-Universität Bonn, Institute for Numerical Simulation, Wegelerstr. 6, 53115 Bonn, Germany
Email: peterseim@ins.uni-bonn.de

DOI: https://doi.org/10.1090/mcom/3156
Keywords: Pollution effect, finite element, multiscale method, numerical homogenization
Received by editor(s): November 27, 2014
Received by editor(s) in revised form: October 17, 2015
Published electronically: August 3, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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