A posteriori L8(L2)-error bounds for finite element approximations to the wave equation

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 L8(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 and 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, Error estimates for finite element methods for second-order hyperbolic equations. SIAM J. Numer. Anal., 13, 564--576) in the context of a priori error analysis of Galerkin discretization of the wave problem in weaker-than-energy spatial norms.