Burman, Erik (2005) A unified analysis for conforming and nonconforming stabilised finite element methods using interior penalty. SIAM Journal on Numerical Analysis, 43 (5). pp. 2012-2033. ISSN 0036-1429

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Abstract

We discuss stabilized Galerkin approximations in a new framework, widening the scope from the usual dichotomy of the discontinuous Galerkin method on the one hand to Petrov-Galerkin methods such as the SUPG method on the other. The idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation, thus circumventing the need for a Petrov-Galerkin type choice of spaces. This is made possible by adding a higher-order penalty term giving $L^2$-control of the jumps in the gradients between adjacent elements. We consider convection-diffusion-reaction problems using piecewise linear approximations and prove optimal-order a priori error estimates for two different finite element spaces, the standard $H^1$-conforming space of piecewise linears and the nonconforming space of piecewise linear elements where the nodes are situated at the midpoint of the element sides (the Crouzeix-Raviart element). Moreover, we show how the formulation extends to discontinuous Galerkin interior penalty methods in a natural way by domain decomposition using Nitsche's method.

Item Type:	Article
Schools and Departments:	School of Mathematical and Physical Sciences > Mathematics
Depositing User:	Erik Burman
Date Deposited:	06 Feb 2012 21:12
Last Modified:	10 Jul 2012 14:19
URI:	http://sro.sussex.ac.uk/id/eprint/30222

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