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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 BurmanWe 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 AnalysisISSN
0036-1429Publisher
Society for Industrial and Applied MathematicsExternal DOI
Issue
5Volume
43Page range
2012-2033Pages
22.0Department affiliated with
- Mathematics Publications
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- No
Peer reviewed?
- Yes
Legacy Posted Date
2012-02-06Usage metrics
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