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

Abstract

Abstract. In this paper we prove the discrete compactness property for a discontinuous Galerkin approximation of Maxwell's system on quite general tetrahedral meshes. As a consequence, a discrete Friedrichs inequality is obtained and the convergence of the discrete eigenvalues to the continuous ones is deduced using the theory of collectively compact operators. Some numerical experiments confirm the theoretical predictions.Mathematics Subject Classification. 65N25, 65N30.

Finite element approximation of eigenvalue problems

“…Another area that deserves attention is the discontinuous Galerkin approximation of eigenvalue problems. We refer to the following papers and to the references therein: Hesthaven and Warburton (2004), Antonietti, Buffa and Perugia (2006), , Warburton and Embree (2006), Creusé and Nicaise (2006), Buffa, Houston and Perugia (2007) and Brenner, Li and Sung (2008). Nonstandard approximations, including mimetic schemes (Cangiani, Gardini and Manzini 2010), have not been discussed.…”

Approximation of the eigenvalue problem for the time harmonic Maxwell system by continuous Lagrange finite elements

“…The two hypotheses (i) and (ii) are tailored to recover some compactness from the discrete versions of (3.1). We refer to [20,29] for review papers; see also [3,11,13,21,37,31,8,9]. The bottom line is that only H(curl)-conforming edge finite elements are known to satisfy (i) and (ii).…”

“…[37,21]. Although the H 3 2 + -regularity is achieved in most Lipschitz domains, it may become quite restrictive when the permeability and permittivity fields are discontinuous.…”

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

“…For consistent DG-discretizations, these methods are spurious-free (see [16,Sec. 6], [21,27,31,48]). Other DG methods are based on "regularized" primal curl-curl formulations, with penalization of the divergence-free constraint.…”