A posteriori error estimation for the method of lumped masses applied to second-order hyperbolic problems

Abstract

We develop a posteriori finite element discretization error estimates for the method of lumped masses applied to 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 local elliptic problems. In two dimensions, we show that the dichotomy principle of Babuška and Yu still holds. For even-degree approximations error estimates are computed by solving a set of local elliptic 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 [L.B. Whalbin, Superconvergence in Galerkin Finite Element Methods, Springer-Verlag, New York, 1995. [26]] to the method of lumped masses for second-order hyperbolic problems.