On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Numerische Mathematik Year : 2021

On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation

Abstract

We propose a novel a posteriori error estimator for conforming finite element discretizations of two- and three-dimensional Helmholtz problems. The estimator is based on an equilibrated flux that is computed by solving patchwise mixed finite element problems. We show that the estimator is reliable up to a prefactor that tends to one with mesh refinement or with polynomial degree increase. We also derive a fully computable upper bound on the prefactor for several common settings of domains and boundary conditions. This leads to a guaranteed estimate without any assumption on the mesh size or the polynomial degree, though the obtained guaranteed bound may lead to large error overestimation. We next demonstrate that the estimator is locally efficient, robust in all regimes with respect to the polynomial degree, and asymptotically robust with respect to the wavenumber. Finally we present numerical experiments that illustrate our analysis and indicate that our theoretical results are sharp.
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Dates and versions

hal-02202233 , version 1 (31-07-2019)
hal-02202233 , version 2 (13-08-2020)
hal-02202233 , version 3 (04-05-2021)

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Cite

Théophile Chaumont-Frelet, Alexandre Ern, Martin Vohralík. On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation. Numerische Mathematik, 2021, 148 (3), pp.525-573. ⟨10.1007/s00211-021-01192-w⟩. ⟨hal-02202233v3⟩
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