The Mathematical Institute, University of Oxford, Eprints Archive

Discontinuous Galerkin methods for first-order hyperbolic problems

Brezzi, Franco and Marini, Donatella and Suli, Endre (2004) Discontinuous Galerkin methods for first-order hyperbolic problems. Technical Report. Unspecified. (Submitted)

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Abstract

In this paper we consider discontinuous Galerkin (DG) finite element approximations of a model scalar linear hyperbolic equation. We show that in order to ensure continuous stabilization of the method it suffices to add a jump-penalty-term to the discretized equation. In particular, the method does not require upwinding in the usual sense. For a specific value of the penalty parameter we recover the classical discontinuous Galerkin method with upwind numerical flux function. More generally, using discontinuous piecewise polynomials of degree $k$, the familiar optimal $\mathcal{O}(h^{k+1/2})$ error estimate is proved for any value of the penalty parameter. As precisely the same jump -term is used for the purposes of stabilizing DG approximations of advection-diffusion operators, the discretization proposed here can simplify the construction of discontinuous Galerkin finite element approximations of advection-diffusion problems. Moreover, the use of the jump-stabilization makes the analysis simpler and more elegant.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1188
Deposited By:Lotti Ekert
Deposited On:18 May 2011 08:13
Last Modified:18 May 2011 08:13

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