A Refined Finite Element Convergence Theory for Highly Indefinite Helmholtz Problems

Abstract.

It is well known that standard h-version finite element discretisations using lowest order elements for Helmholtz' equation suffer from the following stability condition: ``The mesh width h of the finite element mesh has to satisfy k 2 h≲1'', where k denotes the wave number. This condition rules out the reliable numerical solution of Helmholtz equation in three dimensions for large wave numbers k≳50. In our paper, we will present a refined finite element theory for highly indefinite Helmholtz problems where the stability of the discretisation can be checked through an ``almost invariance'' condition. As an application, we will consider a one-dimensional finite element space for the Helmholtz equation and apply our theory to prove stability under the weakened condition hk≲1 and optimal convergence estimates.

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Correspondence to S. A. Sauter.

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Dedicated to Prof. Dr. Ivo Babuška on the occasion of his 80th birthday.

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Sauter, S.A. A Refined Finite Element Convergence Theory for Highly Indefinite Helmholtz Problems. Computing 78, 101–115 (2006). https://doi.org/10.1007/s00607-006-0177-z

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AMS Subject Classifications

  • 65N12
  • 65N15
  • 65N30

Keywords

  • Indefinite problems
  • Helmholtz equation
  • finite element methods
  • generalized FEM