Discontinuous Galerkin Finite Element Method for the Wave Equation

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 timestepping scheme. Optimal a priori error bounds are derived in the energy norm and the L-norm for the semi-discrete formulation. In particular, the error in the energy norm is shown to converge with the optimal order O(hmin{s,`}) with respect to the mesh size h, the polynomial degree `, and the regularity exponent s of the continuous solution. Under additional regularity assumptions, the Lerror is shown to converge with the optimal order O(h). Numerical results confirm the expected convergence rates and illustrate the versatility of the method.