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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

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topHansbo, Peter, and Larson, Mats G.. "Discontinuous Galerkin and the Crouzeix–Raviart element: Application to elasticity." ESAIM: Mathematical Modelling and Numerical Analysis 37.1 (2010): 63-72. <http://eudml.org/doc/194156>.

@article{Hansbo2010,

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},

keywords = {Crouzeix–Raviart element; Nitsche's method; discontinuous Galerkin;
incompressible elasticity.; Crouzeix-Raviart element; incompressible elasticity},

language = {eng},

month = {3},

number = {1},

pages = {63-72},

publisher = {EDP Sciences},

title = {Discontinuous Galerkin and the Crouzeix–Raviart element: Application to elasticity},

url = {http://eudml.org/doc/194156},

volume = {37},

year = {2010},

}

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

DA - 2010/3//

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; incompressible elasticity

UR - http://eudml.org/doc/194156

ER -

## References

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- M. Fortin and M. Soulie, A nonconforming piecewise quadratic finite element on triangles. Internat. J. Numer. Methods Engrg.19 (1983) 505-520.
- 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.
- 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.
- T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, New Jersey (1987).
- J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg36 (1971) 9-15.
- R. Rannacher and S. Turek, A simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differential Equations8 (1992) 97-111.
- F. Thomasset, Implementation of Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, New York (1981).
- M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal.15 (1978) 152-161.
- 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).

## Citations in EuDML Documents

top- Blanca Ayuso de Dios, Ivan Georgiev, Johannes Kraus, Ludmil Zikatanov, A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations
- Yongxing Shen, Adrian J. Lew, A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity
- Yongxing Shen, Adrian J. Lew, A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity
- Yongxing Shen, Adrian J. Lew, A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity

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