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A unified analysis for conforming and nonconforming stabilised finite element methods using interior penalty

journal contribution
posted on 2023-06-08, 09:17 authored by Erik Burman
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.

History

Publication status

  • Published

Journal

SIAM Journal on Numerical Analysis

ISSN

0036-1429

Publisher

Society for Industrial and Applied Mathematics

Issue

5

Volume

43

Page range

2012-2033

Pages

22.0

Department affiliated with

  • Mathematics Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2012-02-06

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