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On a family of discontinuous Galerkin fully-discrete schemes for the wave equation. (English) Zbl 1476.65301

Summary: In this paper, we study a family of discontinuous Galerkin (DG) fully discrete schemes for solving the second-order wave equation. The spatial variable discretization is based on an application of the DG method. The temporal variable discretization depends on a parameter \(\theta \in [0,1]\). Under suitable regularity hypotheses on the solution, optimal order error bounds are shown for the numerical schemes with \(\theta \in [\frac{1}{2},1]\), unconditionally with respect to the spatial mesh-size and the time-step, and for the numerical schemes with \(\theta \in [0,\frac{1}{2})\) where a Courant-Friedrichs-Lewy stability condition is satisfied relating the mesh-size and the time-step. The optimal order error estimates are derived for \(H^1(\Omega)\) and \(L^2(\Omega)\) norms. Simulation results are reported to provide numerical evidence of the optimal convergence orders predicted by the theory.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
49J40 Variational inequalities
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