Grote, Marcus J.; Schneebeli, Anna; Schötzau, Dominik Discontinuous Galerkin finite element method for the wave equation. (English) Zbl 1129.65065 SIAM J. Numer. Anal. 44, No. 6, 2408-2431 (2006). Summary: The symmetric interior penalty discontinuous Galerkin finite element method is presented for the numerical discretization of the second-order wave equation. The resulting stiffness matrix is symmetric positive definite, and the mass matrix is essentially diagonal; hence, the method is inherently parallel and leads to fully explicit time integration when coupled with an explicit time- stepping scheme. Optimal a priori error bounds are derived in the energy norm and the \(L^2\)-norm for the semidiscrete formulation. In particular, the error in the energy norm is shown to converge with the optimal order \({\mathcal O}(h^{\min\{s,\ell\}})\) with respect to the mesh size \(h\), the polynomial degree \(\ell\), and the regularity exponent \(s\) of the continuous solution. Under additional regularity assumptions, the \(L^2\)-error is shown to converge with the optimal order \({\mathcal O}(h^{\ell + 1})\). Numerical results confirm the expected convergence rates and illustrate the versatility of the method. Cited in 1 ReviewCited in 144 Documents MSC: 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65Y05 Parallel numerical computation 35L05 Wave equation Keywords:wave equation; acoustic waves; second-order hyperbolic problems; a priori error analysis; explicit time integration; parallel computation; symmetric interior penalty discontinuous Galerkin finite element method; numerical results; convergence Software:deal.ii PDFBibTeX XMLCite \textit{M. J. Grote} et al., SIAM J. Numer. Anal. 44, No. 6, 2408--2431 (2006; Zbl 1129.65065) Full Text: DOI