An easily computable error estimator in space and time for the wave equation

TL;DR

This paper introduces a computationally efficient a posteriori error estimator for the wave equation, compatible with Newmark and finite element discretizations, maintaining accuracy without extra Laplacian calculations.

Contribution

It presents a simplified error estimator that reduces computational cost while preserving reliability and optimality for wave equation discretizations.

Findings

Maintains reliability and optimality on smooth solutions

Eliminates the need for Laplacian computation at each step

Offers a cheaper alternative to previous estimators

Abstract

We propose a cheaper version of \textit{a posteriori} error estimator from arXiv:1707.00057 for the linear second-order wave equation discretized by the Newmark scheme in time and by the finite element method in space. The new estimator preserves all the properties of the previous one (reliability, optimality on smooth solutions and quasi-uniform meshes) but no longer requires an extra computation of the Laplacian of the discrete solution on each time step.