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

Théophile Chaumont-Frelet (1, 2) , Alexandre Ern (3, 4) , Martin Vohralík (4, 3)

1 NACHOS - Numerical modeling and high performance computing for evolution problems in complex domains and heterogeneous media
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
4 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms

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 through the solve of 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 this prefactor for several common settings of domains and boundary conditions, leading to a guaranteed estimate in all wavenumber regimes without any assumption on the mesh size or the polynomial degree. We finally demonstrate that the estimator is locally efficient as soon as the mesh exhibits sufficiently many degrees of freedom per wavelength. In such a regime, the estimator is also robust with respect to the polynomial degree. We present numerical experiments that illustrate our analysis and indicate that our theoretical results are sharp.

Mots clés: Helmholtz problems; Finite element methods; A posteriori error estimates; High order methods

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

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, inPress. ⟨hal-02202233v2⟩