Uniform-in-time superconvergence of the HDG methods for the acoustic wave equation

We present the first a priori error analysis of the hybridizable discontinuous Galerkin methods for the acoustic equation in the time-continuous case. We show that the velocity and the gradient converge with the optimal order of k + 1 in the L²-norm uniformly in time whenever polynomials of degree k ≥ 0 are used. Finally, we show how to take advantage of this local postprocessing to obtain an approximation to the original scalar unknown also converging with order k+2 for k ≥ 1. This puts on firm mathematical ground the numerical results obtained in J. Comput. Phys.