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

Authors:
Marcus J. Grote;Dominik Schötzau
Affiliations:
Department of Mathematics, University of Basel, Basel, Switzerland 4051;Mathematics Department, University of British Columbia, Vancouver, Canada V6T 1Z2
Venue:
Journal of Scientific Computing
Year:
2009

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Abstract

In Grote et al. (SIAM J. Numer. Anal., 44:2408---2431, 2006) 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 2-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 ("leap-frog" scheme) is used for the time discretization. For sufficiently smooth solutions, the maximal error in the L 2-norm error over a finite time interval converges optimally as O(h p+1+Δt 2), where p denotes the polynomial degree, h the mesh size, and Δt the time step.