Discontinuous Galerkin and the Crouzeix-Raviart element : application to elasticity
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 1, p. 63-72

We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix-Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for which the CR element is not stable in that it does not fulfill a discrete Korn's inequality, the discontinuous framework naturally suggests the appearance of (weakly consistent) stabilization terms. Thus, a stabilized version of the CR element, which does not lock, can be used for both compressible and (nearly) incompressible elasticity. Numerical results supporting these assertions are included. The analysis directly extends to higher order elements and three spatial dimensions.

DOI : https://doi.org/10.1051/m2an:2003020
Classification:  65N30,  74B05
Keywords: Crouzeix-Raviart element, Nitsche's method, discontinuous Galerkin, incompressible elasticity
@article{M2AN_2003__37_1_63_0,
     author = {Hansbo, Peter and Larson, Mats G.},
     title = {Discontinuous Galerkin and the Crouzeix-Raviart element : application to elasticity},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {1},
     year = {2003},
     pages = {63-72},
     doi = {10.1051/m2an:2003020},
     zbl = {1137.65431},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2003__37_1_63_0}
}
Hansbo, Peter; Larson, Mats G. Discontinuous Galerkin and the Crouzeix-Raviart element : application to elasticity. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 1, pp. 63-72. doi : 10.1051/m2an:2003020. http://www.numdam.org/item/M2AN_2003__37_1_63_0/

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