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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.
@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 },
pages = {63--72},
publisher = {EDP-Sciences},
volume = {37},
number = {1},
year = {2003},
doi = {10.1051/m2an:2003020},
zbl = {1137.65431},
language = {en},
url = {https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2003020/}
}
TY - JOUR AU - Hansbo, Peter AU - Larson, Mats G. TI - Discontinuous Galerkin and the Crouzeix-Raviart element : application to elasticity JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2003 SP - 63 EP - 72 VL - 37 IS - 1 PB - EDP-Sciences UR - https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2003020/ DO - 10.1051/m2an:2003020 LA - en ID - M2AN_2003__37_1_63_0 ER -
%0 Journal Article %A Hansbo, Peter %A Larson, Mats G. %T Discontinuous Galerkin and the Crouzeix-Raviart element : application to elasticity %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2003 %P 63-72 %V 37 %N 1 %I EDP-Sciences %U https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2003020/ %R 10.1051/m2an:2003020 %G en %F 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 , Tome 37 (2003) no. 1, pp. 63-72. doi : 10.1051/m2an:2003020. https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2003020/
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