On an a posteriori error analysis of mixed finite element Galerkin approximations to a second order wave equation

TL;DR

This paper develops a posteriori error estimates for mixed finite element Galerkin methods applied to second order wave equations, using mixed elliptic reconstructions and a variation of Baker's technique, under minimal regularity.

Contribution

It introduces a novel a posteriori error analysis framework for mixed finite element schemes for wave equations, including both semi-discrete and fully discrete methods.

Findings

Derived a posteriori error estimates in L-infinity(L2)-norm for semi-discrete schemes.

Established a posteriori error estimators for a first order implicit-in-time scheme.

Utilized mixed elliptic reconstructions and a variation of Baker's technique for analysis.

Abstract

In this article, a posteriori error analysis is developed for mixed finite element Galerkin approximations to a second order linear hyperbolic equation. Based on mixed elliptic reconstructions and an integration tool, which is a variation of Baker's technique introduced earlier by G. Baker ( SIAM J. Numer. Anal., 13 (1976), 564-576) in the context of a priori estimates for a second order wave equation, a posteriori error estimates of the displacement in L{\infty}(L2)-norm for the semidiscrete scheme are derived under minimal regularity. Finally, a first order implicit-in-time discrete scheme is analyzed and a posteriori error estimators are established.