# A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity∗

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

- Volume: 46, Issue: 5, page 1003-1028
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topShen, Yongxing, and Lew, Adrian J.. "A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.5 (2012): 1003-1028. <http://eudml.org/doc/277848>.

@article{Shen2012,

abstract = {We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for
nearly and perfectly incompressible linear elasticity. These mixed methods allow the
choice of polynomials of any order k ≥ 1 for the approximation of the
displacement field, and of order k or k − 1 for the
pressure space, and are stable for any positive value of the stabilization parameter. We
prove the optimal convergence of the displacement and stress fields in both cases, with
error estimates that are independent of the value of the Poisson’s ratio. These estimates
demonstrate that these methods are locking-free. To this end, we prove the corresponding
inf-sup condition, which for the equal-order case, requires a construction to establish
the surjectivity of the space of discrete divergences on the pressure space. In the
particular case of near incompressibility and equal-order approximation of the
displacement and pressure fields, the mixed method is equivalent to a displacement method
proposed earlier by Lew et al. [Appel. Math. Res. express
3 (2004) 73–106]. The absence of locking of this displacement
method then follows directly from that of the mixed method, including the uniform error
estimate for the stress with respect to the Poisson’s ratio. We showcase the performance
of these methods through numerical examples, which show that locking may appear if
Dirichlet boundary conditions are imposed strongly rather than weakly, as we do here.},

author = {Shen, Yongxing, Lew, Adrian J.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Discontinuous Galerkin; locking; mixed method; inf-sup condition; discontinuous Galerkin},

language = {eng},

month = {2},

number = {5},

pages = {1003-1028},

publisher = {EDP Sciences},

title = {A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity∗},

url = {http://eudml.org/doc/277848},

volume = {46},

year = {2012},

}

TY - JOUR

AU - Shen, Yongxing

AU - Lew, Adrian J.

TI - A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity∗

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2012/2//

PB - EDP Sciences

VL - 46

IS - 5

SP - 1003

EP - 1028

AB - We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for
nearly and perfectly incompressible linear elasticity. These mixed methods allow the
choice of polynomials of any order k ≥ 1 for the approximation of the
displacement field, and of order k or k − 1 for the
pressure space, and are stable for any positive value of the stabilization parameter. We
prove the optimal convergence of the displacement and stress fields in both cases, with
error estimates that are independent of the value of the Poisson’s ratio. These estimates
demonstrate that these methods are locking-free. To this end, we prove the corresponding
inf-sup condition, which for the equal-order case, requires a construction to establish
the surjectivity of the space of discrete divergences on the pressure space. In the
particular case of near incompressibility and equal-order approximation of the
displacement and pressure fields, the mixed method is equivalent to a displacement method
proposed earlier by Lew et al. [Appel. Math. Res. express
3 (2004) 73–106]. The absence of locking of this displacement
method then follows directly from that of the mixed method, including the uniform error
estimate for the stress with respect to the Poisson’s ratio. We showcase the performance
of these methods through numerical examples, which show that locking may appear if
Dirichlet boundary conditions are imposed strongly rather than weakly, as we do here.

LA - eng

KW - Discontinuous Galerkin; locking; mixed method; inf-sup condition; discontinuous Galerkin

UR - http://eudml.org/doc/277848

ER -

## References

top- D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo21 (1984) 337–344. Zbl0593.76039
- D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal.39 (2002) 1749–1779. Zbl1008.65080
- F. Bassi and S. Rebay, A High-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys.131 (1997) 267–279. Zbl0871.76040
- R. Becker, E. Burman and P. Hansbo, A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Engrg.198 (2009) 3352–3360. Zbl1230.74169
- M. Bercovier and O.A. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math.33 (1977) 211–224. Zbl0423.65058
- S.C. Brenner, Korn’s inequalities for piecewise H1 vector fields. Math. Comp.73 (2003) 1067–1087. Zbl1055.65118
- S.C. Brenner, Poincaré-Friedrichs inequalities for piecewise H1 functions. SIAM J. Numer. Anal.41 (2003) 306–324. Zbl1045.65100
- S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 3th edition, Springer (2008). Zbl1135.65042
- S.C. Brenner and L.-Y. Sung, Linear finite element methods for planar linear elasticity. Math. Comp.59 (1992) 321–338. Zbl0766.73060
- F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO Anal. Numér.8 (1974) 129–151. Zbl0338.90047
- F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer Series in Computational Mathematics, Springer-Verlag, New York (1991). Zbl0788.73002
- F. Brezzi, J. DouglasJr., and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math.47 (1985) 217–235. Zbl0599.65072
- F. Brezzi, J. DouglasJr., R. Durán and M. Fortin, Mixed finite elements for second order elliptic problems in three variables. Numer. Math.51 (1987) 237–250.
- F. Brezzi, G. Manzini, D. Marini, P. Pietra and A. Russo, Discontinuous galerkin approximations for elliptic problems. Numer. Methods Partial Differential Equations (2000) 365–378. Zbl0957.65099
- F. Brezzi, T.J.R. Hughes, L.D. Marini and A. Masud, Mixed discontinuous Galerkin methods for Darcy flow. J. Sci. Comput.22, 23 (2005) 119–145. Zbl1103.76031
- J. Carrero, B. Cockburn and D. Schötzau, Hybridized globally divergence-free LDG methods. Part I : the Stokes problem. Math. Comp.75 (2005) 533–563. Zbl1087.76061
- P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). Zbl0383.65058
- B. Cockburn, G. Kanschat, D. Schötzau and C. Schwab, Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal.40 (2002) 319–343. Zbl1032.65127
- B. Cockburn, D. Schötzau and J. Wang, Discontinuous Galerkin methods for incompressible elastic materials. Comput. Methods Appl. Mech. Engrg.195 (2006) 3184–3204. Zbl1128.74041
- M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Sér. Rouge7 (1973) 33–75. Zbl0302.65087
- M. Fortin, An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér.11 (1977) 341–354. Zbl0373.65055
- V. Girault, B. Rivière and M.F. Wheeler, A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems. Math. Comp.74 (2005) 53–84. Zbl1057.35029
- 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
- P. Hansbo and M.G. Larson, Discontinuous Galerkin and the Crouzeix-Raviart element : Application to elasticity. ESAIM : M2AN37 (2003) 63–72. Zbl1137.65431
- P. Hansbo and M.G. Larson, Piecewise divergence-free discontinuous Galerkin methods for Stokes flow. Comm. Num. Methods Engrg.24 (2008) 355–366. Zbl1138.76046
- F. Hecht, Construction d’une base de fonctions P1 non conforme à divergence nulle dans R3. RAIRO Anal. Numér.15 (1981) 119–150. Zbl0471.76028
- P. Hood and C. Taylor, Numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids1 (1973) 1–28. Zbl0328.76020
- P. Hood and C. Taylor, Navier-Stokes equations using mixed interpolation. Finite Element Methods in Flow Problems, edited by J.T. Oden. UAH Press, Huntsville, Alabama (1974).
- R. Kouhia and R. Stenberg, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg.124 (1995) 195–212. Zbl1067.74578
- A. Lew, P. Neff, D. Sulsky and M. Ortiz, Optimal BV estimates for a discontinuous Galerkin method for linear elasticity. Appl. Math. Res. express3 (2004) 73–106. Zbl1115.74021
- N.C. Nguyen, J. Peraire and B. Cockburn, A hybridizable discontinuous Galerkin method for Stokes flow. Comput. Methods Appl. Mech. Engrg.199 (2010) 582–597. Zbl1227.76036
- B. Rivière and V. Girault, Discontinuous finite element methods for incompressible flows on subdomains with non-matching interfaces. Comput. Methods Appl. Mech. Engrg.195 (2006) 3274–3292. Zbl1121.76038
- D. Schötzau, C. Schwab and A. Toselli, Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal.40 (2003) 2171–2194. Zbl1055.76032
- L.R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modélisation Mathématique et Analyse Numérique19 (1985) 111–143. Zbl0608.65013
- S.-C. Soon, B. Cockburn and H.K. Stolarski, A hybridizable discontinuous Galerkin method for linear elasticity. Int. J. Numer. Methods Engrg.80 (2009), 1058–1092. Zbl1176.74196
- A. Ten Eyck, and A. Lew, Discontinuous Galerkin methods for non-linear elasticity. Int. J. Numer. Methods Engrg.67 (2006) 1204–1243. Zbl1113.74068
- A. Ten Eyck, F. Celiker and A. Lew, Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity : Analytical estimates. Comput. Methods Appl. Mech. Engrg.197 (2008) 2989–3000. Zbl1194.74390
- F. Thomasset, Implementation of finite element methods for Navier-Stokes equations. Springer-Verlag, New York (1981). Zbl0475.76036
- J.P. Whiteley, Discontinuous Galerkin finite element methods for incompressible non-linear elasticity, Comput. Methods Appl. Mech. Engrg.198 (2009) 3464–3478. Zbl1230.74200
- T.P. Wihler, Locking-free DGFEM for elasticity problems in polygons. IMA J. Numer. Anal.24 (2004) 45–75. Zbl1057.74046

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.