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

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Authors and Affiliations

  1. Department of Mathematics, University of Basel, Rheinsprung 21, 4051, Basel, Switzerland

    Marcus J. Grote

  2. Mathematics Department, University of British Columbia, 121-1984 Mathematics Road, Vancouver, V6T 1Z2, Canada

    Dominik Schötzau

Authors
  1. Marcus J. Grote
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  2. Dominik Schötzau
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Correspondence to Marcus J. Grote.

Additional information

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).

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Grote, M.J., Schötzau, D. Optimal Error Estimates for the Fully Discrete Interior Penalty DG Method for the Wave Equation. J Sci Comput 40, 257–272 (2009). https://doi.org/10.1007/s10915-008-9247-z

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  • DOI: https://doi.org/10.1007/s10915-008-9247-z

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