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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Eliminating the pollution effect in Helmholtz problems by local subscale correction
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by Daniel Peterseim PDF
Math. Comp. 86 (2017), 1005-1036 Request permission

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
  • MR Author ID: 848711
  • Email: peterseim@ins.uni-bonn.de
  • Received by editor(s): November 27, 2014
  • Received by editor(s) in revised form: October 17, 2015
  • Published electronically: August 3, 2016
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 1005-1036
  • MSC (2010): Primary 65N12, 65N15, 65N30
  • DOI: https://doi.org/10.1090/mcom/3156
  • MathSciNet review: 3614010