Optimal Error Estimates for the Fully Discrete Interior Penalty DG Method for the Wave Equation
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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 L2-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 L2-norm error over a finite time interval converges optimally as O(hp+1+Δt2), where p denotes the polynomial degree, h the mesh size, and Δt the time step.
KeywordsDiscontinuous Galerkin methods Finite element methods Wave equation Interior penalty method Leap-frog scheme
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