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A Unified Analysis for Conforming and Nonconforming Stabilized Finite Element Methods Using Interior Penalty

Burman, E; (2005) A Unified Analysis for Conforming and Nonconforming Stabilized Finite Element Methods Using Interior Penalty. SIAM Journal on Numerical Analysis , 43 (5) pp. 2012-2033. 10.1137/S0036142903437374. Green open access

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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 and 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 of a Petrov--Galerkin-type choice of spaces. This is made possible by adding a higher-order penalty term giving L2-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 H1-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.

Type: Article
Title: A Unified Analysis for Conforming and Nonconforming Stabilized Finite Element Methods Using Interior Penalty
Open access status: An open access version is available from UCL Discovery
DOI: 10.1137/S0036142903437374
Publisher version: http://dx.doi.org/10.1137/S0036142903437374
Language: English
Additional information: Copyright © 2005 Society for Industrial and Applied Mathematics
Keywords: convection diffusion problem, interior penalty, finite element approximation, Crouzeix-Raviart element
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/1384737
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