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
DOI:
https://doi.org/10.1090/S0025-5718-10-02362-8
Published electronically:
April 27, 2010
MathSciNet review:
2684350
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Abstract | References | Similar Articles | Additional Information
Abstract: A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in ,
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
-version of the finite element method (
-FEM) is presented for the model problem where the dependence on the mesh width
, the approximation order
, and the wave number
is given explicitly. In particular, it is shown that quasi-optimality is obtained under the conditions that
is sufficiently small and the polynomial degree
is at least
.
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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
Email:
melenk@tuwien.ac.at
S. Sauter
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstr 190, CH-8057 Zürich, Switzerland
Email:
stas@math.uzh.ch
DOI:
https://doi.org/10.1090/S0025-5718-10-02362-8
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