Gauss-compatible Galerkin schemes for time-dependent Maxwell equations

International audienceIn this article we propose a unified analysis for conforming andnon-conforming finite element methods that provides a partial answer to theproblem of preserving discrete divergence constraints when computing numericalsolutions to the time-dependent Maxwell system. In particular, weformulate a compatibility condition relative to the preservation of genuinelyoscillating modes that takes the form of a generalized commuting diagram,and we show that compatible schemes satisfy convergence estimates leadingto long-time stability with respect to stationary solutions. These findings areapplied by specifying compatible formulations for several classes of Galerkinmethods, such as the usual curl-conforming finite elements and the centereddiscontinuous Galerkin (DG) scheme. We also propose a new conforming/nonconformingGalerkin (Conga) method where fully discontinuous solutions arecomputed by embedding the general structure of curl-conforming finite elementsinto larger DG spaces. In addition to naturally preserving one of theGauss laws in a strong sense, the Conga method is both spectrally correct andenergy conserving, unlike existing DG discretizations where the introductionof a dissipative penalty term is needed to avoid the presence of spurious modes