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Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions

Authors: J. M. Melenk and S. Sauter
Journal: Math. Comp. 79 (2010), 1871-1914
MSC (2010): Primary 35J05, 65N12, 65N30
Published electronically: April 27, 2010
MathSciNet review: 2684350
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Abstract: A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in ${\mathbb {R}}^{d}$, $d \in \{1,2,3\}$ is presented. General conditions on the approximation properties of the approximation space are stated that ensure quasi-optimality of the method. As an application of the general theory, a full error analysis of the classical $hp$-version of the finite element method ($hp$-FEM) is presented for the model problem where the dependence on the mesh width $h$, the approximation order $p$, and the wave number $k$ is given explicitly. In particular, it is shown that quasi-optimality is obtained under the conditions that $kh/p$ is sufficiently small and the polynomial degree $p$ is at least $O(\log k)$.

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Additional Information

J. M. Melenk
Affiliation: Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria
MR Author ID: 613978
ORCID: 0000-0001-9024-6028

S. Sauter
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstr 190, CH-8057 Zürich, Switzerland
MR Author ID: 313335

Keywords: Helmholtz equation at high wave number, stability, convergence, $hp$-finite elements.
Received by editor(s): July 15, 2008
Published electronically: April 27, 2010
Article copyright: © Copyright 2010 American Mathematical Society