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Journal of Scientific Computing

, Volume 57, Issue 3, pp 536–581 | Cite as

General DG-Methods for Highly Indefinite Helmholtz Problems

  • J. M. Melenk
  • A. Parsania
  • S. SauterEmail author
Article

Abstract

We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in \(\mathbb R ^{d}\), \(d\in \{1,2,3\}\). The theory covers conforming as well as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much weaker conditions. As an application we present the error analysis for the \(hp\)-version of the finite element method explicitly in terms of the mesh width \(h\), polynomial degree \(p\) and wavenumber \(k\). It is shown that the optimal convergence order estimate is obtained under the conditions that \(kh/\sqrt{p}\) is sufficiently small and the polynomial degree \(p\) is at least \(O(\log k)\). On regular meshes, the first condition is improved to the requirement that \(kh/p\) be sufficiently small.

Keywords

Helmholtz equation at high wavenumber Stability Convergence Discontinuous Galerkin methods Ultra-weak variational formulation Polynomial hp-finite elements 

Mathematics Subject Classification (2000)

35J05 65N12 65N30 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute for Analysis and Scientific ComputingTechnische Universität WienViennaAustria
  2. 2.Institut für MathematikUniversität Zürich8057 ZurichSwitzerland

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