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On Stability of Discretizations of the Helmholtz Equation

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Numerical Analysis of Multiscale Problems

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 83))

Abstract

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete k-explicit stability (including k-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size h and the approximation order p are selected such that kh ∕ p is sufficiently small and p = O(logk), and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation.

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Acknowledgements

Financial support by the Vienna Science and Technology Fund (WWTF) is gratefully acknowledged.

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  1. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040, Vienna, Austria

    S. Esterhazy & J. M. Melenk

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  1. S. Esterhazy
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Correspondence to J. M. Melenk .

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Editors and Affiliations

  1. , Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom

    Ivan G. Graham

  2. , Applied & Computational Mathematics, California Institute of Technology, Pasadena, 91125, California, USA

    Thomas Y. Hou

  3. , Department of Mathematics, University of Sussex, Brighton, BN1 9QH, United Kingdom

    Omar Lakkis

  4. , Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom

    Robert Scheichl

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Esterhazy, S., Melenk, J.M. (2012). On Stability of Discretizations of the Helmholtz Equation. In: Graham, I., Hou, T., Lakkis, O., Scheichl, R. (eds) Numerical Analysis of Multiscale Problems. Lecture Notes in Computational Science and Engineering, vol 83. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22061-6_9

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