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 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, }) 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 2error is shown to converge with the optimal order O(h +1). Numerical results confirm the expected convergence rates and illustrate the versatility of the method.

In this paper, we introduce and analyze the interior penalty discontinuous Galerkin method for the numerical discretization of the indefinite time-harmonic Maxwell equations in the high-frequency regime. Based on suitable duality arguments, we derive a-priori error bounds in the energy norm and the L 2 -norm. In particular, the error in the energy norm is shown to converge with the optimal order O(h min{s, } ) with respect to the mesh size h, the polynomial degree , and the regularity exponent s of the analytical solution. Under additional regularity assumptions, the L 2 -error is shown to converge with the optimal order O(h +1 ). The theoretical results are confirmed in a series of numerical experiments.

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We consider the symmetric, interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell's equations in second-order form. In Grote et al. (2007, J. Comput. Appl. Math., 204, 375-386), optimal a priori estimates in the DG energy norm were derived, either for smooth solutions on arbitrary meshes or for low-regularity (singular) solutions on conforming, affine meshes. Here, we show that the DG methods are also optimally convergent in the L 2-norm, on tetrahedral meshes and for smooth material coefficients. The theoretical convergence rates are validated by a series of numerical experiments in two-space dimensions, which also illustrate the usefulness of the interior penalty DG method for timedependent computational electromagnetics.

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