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

TL;DR

This paper introduces a new a posteriori error estimator for Helmholtz problems that is reliable, robust, and guarantees bounds without mesh or polynomial degree restrictions, supported by theoretical analysis and numerical tests.

Contribution

It presents a novel equilibrated flux-based error estimator for Helmholtz equations that guarantees bounds independently of mesh size and polynomial degree.

Findings

Estimator is reliable up to a prefactor approaching one with mesh refinement.

The estimator is locally efficient and robust across polynomial degrees.

Numerical experiments confirm the sharpness of theoretical results.

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.