A priori and a posteriori error estimates of a C0-in-time method for the wave equation in second order formulation

We establish fully-discrete a priori and semi-discrete in time a posteriori error estimates for a discontinuous-continuous Galerkin discretization of the wave equation in second order formulation; the resulting method is a Petrov-Galerkin scheme based on piecewise polynomial test functions and continuous piecewise polynomial trial functions in time, respectively. Crucial tools in the a priori analysis for the fully-discrete formulation are the design of suitable projection and interpolation operators extending those used in the parabolic setting, and stability estimates based on a nonstandard choice of the test function; a priori estimates are shown, which are measured in L infinity -type norms in time. For the semi-discrete in time formulation, we exhibit reliable a posteriori error estimates for the error measured in the L infinity(L2) norm with fully explicit constants; to this aim, we design a reconstruction operator into C1 piecewise polynomials over the time grid with optimal approximation properties in terms of the polynomial degree distribution and the time steps. Numerical examples illustrate the theoretical findings.