A priori and a posteriori error estimates of a $\mathcal C^0$-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 and piecewise continuous polynomial in time test and trial spaces, 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 $L^\infty$-type norms in time. For the semi-discrete in time formulation, we exhibit reliable a posteriori error estimates for the error measured in the $L^\infty(L^2)$ norm with fully explicit constants; to this aim, we design a reconstruction operator into $\mathcal C^1$ 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.