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Interior Penalty Continuous and Discontinuous Finite Element Approximations of Hyperbolic Equations

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Abstract

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.

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Authors and Affiliations

  1. EPFL, IACS-CMCS, Station 8, 1015, Lausanne, Switzerland

    Erik Burman, Alfio Quarteroni & Benjamin Stamm

  2. Department of Mathematics, Mantell Building, University of Sussex, Falmer, Brighton, BN1 9RF, UK

    Erik Burman

  3. MOX-Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, 20133, Milano, Italy

    Alfio Quarteroni

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  1. Erik Burman
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  2. Alfio Quarteroni
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Correspondence to Benjamin Stamm.

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Burman, E., Quarteroni, A. & Stamm, B. Interior Penalty Continuous and Discontinuous Finite Element Approximations of Hyperbolic Equations. J Sci Comput 43, 293–312 (2010). https://doi.org/10.1007/s10915-008-9232-6

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  • DOI: https://doi.org/10.1007/s10915-008-9232-6

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