Optimal error estimates for the fully discrete interior penalty DG method for the wave equation. (English) Zbl 1203.65182

Grote, Marcus J.; Schötzau, Dominik

J. Sci. Comput. 40, No. 1-3, 257-272 (2009).

Summary: The authors and A. Schneebeli [SIAM J. Numer. Anal. 44, No. 6, 2408–2431 (2006; Zbl 1129.65065)] 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 \(O(h p+1+\Delta t ^{2})\), where \(p\) denotes the polynomial degree, \(h\) the mesh size, and \(\Delta t\) the time step.

MSC:

65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs

Keywords:

discontinuous Galerkin methods; finite element methods; wave equation; interior penalty method; leap-frog scheme

Citations:

Zbl 1129.65065

References:

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