Discontinuous Galerkin for the wave equation: a simplified a priori error analysis

Standard discontinuous Galerkin methods, based on piecewise polynomials of degree $ \qq\geq 0$, are considered for temporal semi-discretization for second order hyperbolic equations. The main goal of this paper is to present a simple and straight forward a priori error analysis of optimal order with minimal regularity requirement on the solution. Uniform norm in time error estimates are also proved for the constant and linear cases. To this end, energy identities and stability estimates of the discrete problem are proved for a slightly more general problem. These are used to prove optimal order a priori error estimates with minimal regularity requirement on the solution. The combination with the classic continuous Galerkin finite element discretization in space variable is used, to formulate a full-discrete scheme. The a priori error analysis is presented. Numerical experiments are performed to verify the theoretical rate of convergence.