Optimal Error Estimates for the Fully Discrete Interior Penalty DG Method for the Wave Equation

In [20] a symmetric interior penalty discontinuous Galerkin (DG) method was presented for the time-dependent wave equation. In particular, optimal a-priori error bounds in the energy norm and the L-norm were derived for the semi-discrete formulation. Here the error analysis is extended to the fully discrete numerical scheme, when a centered second-order finite difference approximation (“leapfrog” scheme) is used for the time discretization. For sufficiently smooth solutions, the maximal error in the L-norm error over a finite time interval converges optimally as O(h + ∆t), where p denotes the polynomial degree, h the mesh size, and ∆t the time step.