Interior penalty method for the indefinite time-harmonic Maxwell equations

Abstract

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.

A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations

“…] also appears in discontinuous Galerkin methods for the time-harmonic Maxwell equations [16,14]. However, its role here is to ensure the consistency of the scheme (3.4) and there is no need for a penalty parameter.…”

Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system

“…Therefore the mean and the tangential jumps are defined as before by taking v − = 0. Following [19], we consider the following discontinuous Galerkin approximation of the continuous eigenvalue problem: Given a mesh T h and a polynomial degree k ≥ 1, we consider the approximation space…”

“…Note that our orthogonal decomposition is different from the one from [19,20] but is motivated by the following result proved in Proposition 4.5 of [19] (or in the Appendix of [20]):…”

“…But to our knowledge this property is not yet proved for the discontinuous Galerkin method. We here concentrate on the interior penalty method introduced in [19] (see also [24] for two-dimensional domains).…”

“…Our proof of the discrete compactness property is based on the same property for the standard Galerkin approximation proved in [26] and the use of a decomposition of the discontinuous approximation space into a continuous one and its orthogonal for an appropriate inner product similar to [19,20] (but different from the one used in these references). The discrete Friedrichs inequality follows from this discrete compactness property and a contradiction argument.…”