A posteriori $L^\infty(L^2)$-error bounds in finite element approximation of the wave equation

TL;DR

This paper develops a posteriori error bounds in the L^ Infty(L^2)-norm for finite element approximations of the wave equation, applicable under minimal regularity assumptions for both semi-discrete and fully discrete schemes.

Contribution

It introduces new a posteriori error bounds for the wave equation using space- and time-reconstructions, extending previous techniques to weaker spatial norms.

Findings

Error bounds are valid for minimal regularity assumptions.

Bounds are derived for both semi-discrete and fully discrete schemes.

The approach relies on space- and time-reconstructions of numerical solutions.

Abstract

We address the error control of Galerkin discretization (in space) of linear second order hyperbolic problems. More specifically, we derive a posteriori error bounds in the L\infty(L2)-norm for finite element methods for the linear wave equation, under minimal regularity assumptions. The theory is developed for both the space-discrete case, as well as for an implicit fully discrete scheme. The derivation of these bounds relies crucially on carefully constructed space- and time-reconstructions of the discrete numerical solutions, in conjunction with a technique introduced by Baker (1976, SIAM J. Numer. Anal., 13) in the context of a priori error analysis of Galerkin discretization of the wave problem in weaker-than-energy spatial norms.