# Discontinuous Galerkin and the Crouzeix–Raviart element : application to elasticity

• Volume: 37, Issue: 1, page 63-72
• ISSN: 0764-583X

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## Abstract

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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.

## How to cite

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Hansbo, Peter, and Larson, Mats G.. "Discontinuous Galerkin and the Crouzeix–Raviart element : application to elasticity." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.1 (2003): 63-72. <http://eudml.org/doc/245520>.

@article{Hansbo2003,
abstract = {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.},
author = {Hansbo, Peter, Larson, Mats G.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Crouzeix–Raviart element; Nitsche’s method; discontinuous Galerkin; incompressible elasticity; Crouzeix-Raviart element; Nitsche's method},
language = {eng},
number = {1},
pages = {63-72},
publisher = {EDP-Sciences},
title = {Discontinuous Galerkin and the Crouzeix–Raviart element : application to elasticity},
url = {http://eudml.org/doc/245520},
volume = {37},
year = {2003},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 1
SP - 63
EP - 72
AB - 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.
LA - eng
KW - Crouzeix–Raviart element; Nitsche’s method; discontinuous Galerkin; incompressible elasticity; Crouzeix-Raviart element; Nitsche's method
UR - http://eudml.org/doc/245520
ER -

## References

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8. [8] P. Hansbo and M.G. Larson, Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Engrg. 191 (2002) 1895-1908. Zbl1098.74693
9. [9] P. Hansbo and M.G. Larson, A simple nonconforming bilinear element for the elasticity problem. Trends in Computational Structural Mechanics, W.A. Wall et al. Eds., CIMNE (2001) 317–327.
10. [10] T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, New Jersey (1987). Zbl0634.73056MR1008473
11. [11] J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9–15. Zbl0229.65079
12. [12] R. Rannacher and S. Turek, A simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differential Equations 8 (1992) 97–111. Zbl0742.76051
13. [13] F. Thomasset, Implementation of Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, New York (1981). Zbl0475.76036MR720192
14. [14] M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152–161. Zbl0384.65058
15. [15] B. Cockburn, K.E. Karniadakis and C.-W. Shu Eds., Discontinuous Galerkin Methods: Theory, Computation, and Applications. Lecture Notes Comput. Sci. Eng., Springer Verlag (1999). Zbl0935.00043MR1842160

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