A posteriori finite element error estimation for second-order hyperbolic problems Original Research Article

We develop a posteriori finite element discretization error estimates for the wave equation. In one dimension, we show that the significant part of the spatial finite element error is proportional to a Lobatto polynomial and an error estimate is obtained by solving a set of either local elliptic or hyperbolic problems. In two dimensions, we show that the dichotomy principle of Babuška and Yu holds. For even-degree approximations error estimates are computed by solving a set of local elliptic or hyperbolic problems and for odd-degree approximations an error estimate is computed using jumps of solution gradients across element boundaries. This study also extends known superconvergence results for elliptic and parabolic problems [Superconvergence in Galerkin Finite Element Methods, Springer Verlag, New York, 1995] to second-order hyperbolic problems.