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Mixed finite element discretizations of acoustic Helmholtz problems with high wavenumbers

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Abstract

We study the acoustic Helmholtz equation with impedance boundary conditions formulated in terms of velocity, and analyze the stability and convergence properties of lowest-order Raviart-Thomas finite element discretizations. We focus on the high-wavenumber regime, where such discretizations suffer from the so-called “pollution effect”, and lack stability unless the mesh is sufficiently refined. We provide wavenumber-explicit mesh refinement conditions to ensure the well-posedness and stability of discrete scheme, as well as wavenumber-explicit error estimates. Our key result is that the condition “\(k^2 h\) is sufficiently small”, where k and h respectively denote the wavenumber and the mesh size, is sufficient to ensure the stability of the scheme. We also present numerical experiments that illustrate the theory and show that the derived stability condition is actually necessary.

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Notes

  1. The result is actually stated for \(s=1\), but the general case easily follows by interpolation. See also [14].

  2. Theorem 3.2 does not explicitly treat the adjoint problem, but one easily sees that \(\varvec{\xi }\) can be equivalently defined as the unique element of \(\varvec{{\mathcal {X}}}\) such that \(b({\overline{\varvec{\xi }}},{\varvec{v}}) = ({\overline{{\varvec{q}}}},{\varvec{v}})_\varOmega \) for all \({\varvec{v}}\in \varvec{{\mathcal {X}}}\).

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Authors and Affiliations

  1. Inria Sophia Antipolis Méditerranée, 2004 Route des Lucioles, 06902, Valbonne, France

    T. Chaumont-Frelet

  2. Laboratoire J.A. Dieudonné UMR CNRS 7351, Parc Valrose, 06108, Nice, France

    T. Chaumont-Frelet

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Chaumont-Frelet, T. Mixed finite element discretizations of acoustic Helmholtz problems with high wavenumbers. Calcolo 56, 49 (2019). https://doi.org/10.1007/s10092-019-0346-z

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