, Volume 40, Issue 1-3, pp 257-272
Date: 18 Sep 2008

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

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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+1t 2), where p denotes the polynomial degree, h the mesh size, and Δt the time step.

The research of M.J. Grote was supported in part by the Swiss National Science Foundation (SNF).
The research of D. Schötzau was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).