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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr


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1609-9389
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Volume 13, Issue 3

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A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems

Willy Dörfler
  • Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany
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/ Stefan Sauter
Published Online: 2013-07-02 | DOI: https://doi.org/10.1515/cmam-2013-0008

Abstract.

We develop a new analysis for residual-type a posteriori error estimation for a class of highly indefinite elliptic boundary value problems by considering the Helmholtz equation at high wavenumber as our model problem. We employ a classical conforming Galerkin discretization by using hp-finite elements. In [Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp., 79 (2010), pp. 1871–1914], Melenk and Sauter introduced an hp-finite element discretization which leads to a stable and pollution-free discretization of the Helmholtz equation under a mild resolution condition which requires only degrees of freedom, where denotes the spatial dimension. In the present paper, we will introduce an a posteriori error estimator for this problem and prove its reliability and efficiency. The constants in these estimates become independent of the, possibly, high wavenumber provided the aforementioned resolution condition for stability is satisfied. We emphasize that, by using the classical theory, the constants in the a posteriori estimates would be amplified by a factor k.

Keywords: Helmholtz Equation at High Wavenumber; Stability; Convergence; hp-Finite Elements; A Posteriori Error Estimation

About the article

Published Online: 2013-07-02

Published in Print: 2013-07-01


Citation Information: Computational Methods in Applied Mathematics, Volume 13, Issue 3, Pages 333–347, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2013-0008.

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[1]
Mario Ohlberger and Barbara Verfurth
Multiscale Modeling & Simulation, 2018, Volume 16, Number 1, Page 385
[2]
S. Sauter and J. Zech
SIAM Journal on Numerical Analysis, 2015, Volume 53, Number 5, Page 2414

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