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Discontinuous Galerkin finite element method for the wave equation. (English) Zbl 1129.65065

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.

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

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