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 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,l)}) with respect to the mesh size h, the polynomial degree l, 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(l+1)). Numerical results confirm the expected convergence rates and illustrate the versatility of the method.