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Interior Penalty Continuous and Discontinuous Finite Element Approximations of Hyperbolic Equations

Burman, Erik ; Quarteroni, Alfio ; Stamm, Benjamin

In: Journal of Scientific Computing, 2010, vol. 43, no. 3, p. 293-312

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    Titre
    • Interior Penalty Continuous and Discontinuous Finite Element Approximations of Hyperbolic Equations
    Auteur
    • Burman, Erik. EPFL, IACS-CMCS, Station 8, 1015, Lausanne, Switzerland
    • Quarteroni, Alfio. EPFL, IACS-CMCS, Station 8, 1015, Lausanne, Switzerland
    • Stamm, Benjamin. EPFL, IACS-CMCS, Station 8, 1015, Lausanne, Switzerland
    Type de document
    • Postprint
    Langue
    • Anglais
    Publié dans
    • Journal of Scientific Computing, 2010, vol. 43, no. 3, p. 293-312. Springer US; http://www.springer-ny.com
    Autre version électronique
    Identifiant OAI-PMH
    • oai:doc.rero.ch:315371
    Summary
    In this paper we present in a unified setting the continuous and discontinuous Galerkin methods for the numerical approximation of the scalar hyperbolic equation. Both methods are stabilized by the interior penalty method, more precisely by the jump of the gradient across element faces in the continuous case whereas in the discontinuous case the stabilization of the jump of the solution and optionally of its gradient is required to achieve optimal convergence. We prove that the solution in the case of the continuous Galerkin approach can be considered as a limit of the discontinuous one when the stabilization parameter associated with the penalization of the solution jump tends to infinity. As a consequence, the limit of the numerical flux of the discontinuous method yields a numerical flux for the continuous method as well. Numerical results will highlight the theoretical results that are proven in this paper