Discontinuous Galerkin Finite Element Method for the Wave Equation
Résumé
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 $O(h^{\min{s,\ell}})$ with respect to the mesh size h, the polynomial degree , 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 $O(h^{\ell +1})$. Numerical results confirm the expected convergence rates and illustrate the versatility of the method.
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