Geodesic
A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 5, pp. 1003-1028.

Voir la notice de l'article provenant de la source Numdam

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.

DOI : 10.1051/m2an/2011046
Classification : 65N30, 65N12, 65N15
Keywords: discontinuous Galerkin, locking, mixed method, inf-sup condition 
@article{M2AN_2012__46_5_1003_0,
     author = {Shen, Yongxing and Lew, Adrian J.},
     title = {A family of discontinuous {Galerkin} mixed methods for nearly and perfectly incompressible elasticity},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1003--1028},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {5},
     year = {2012},
     doi = {10.1051/m2an/2011046},
     mrnumber = {2916370},
     zbl = {1267.74116},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2011046/}
}
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 
PY  - 2012
SP  - 1003
EP  - 1028
VL  - 46
IS  - 5
PB  - EDP-Sciences
UR  - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2011046/
DO  - 10.1051/m2an/2011046
LA  - en
ID  - M2AN_2012__46_5_1003_0
ER  - 
%0 Journal Article
%A Shen, Yongxing
%A Lew, Adrian J.
%T A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2012
%P 1003-1028
%V 46
%N 5
%I EDP-Sciences
%U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2011046/
%R 10.1051/m2an/2011046
%G en
%F M2AN_2012__46_5_1003_0
Shen, Yongxing; Lew, Adrian J. A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 5, pp. 1003-1028. doi : 10.1051/m2an/2011046. http://geodesic.mathdoc.fr/articles/10.1051/m2an/2011046/

[1] D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo 21 (1984) 337-344. | Zbl | MR

[2] 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. | Zbl | MR

[3] 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. | Zbl | MR

[4] 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. | Zbl | MR

[5] 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. | Zbl | MR

[6] S.C. Brenner, Korn's inequalities for piecewise H1 vector fields. Math. Comp. 73 (2003) 1067-1087. | Zbl | MR

[7] S.C. Brenner, Poincaré-Friedrichs inequalities for piecewise H1 functions. SIAM J. Numer. Anal. 41 (2003) 306-324. | Zbl | MR

[8] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 3th edition, Springer (2008). | Zbl | MR

[9] S.C. Brenner and L.-Y. Sung, Linear finite element methods for planar linear elasticity. Math. Comp. 59 (1992) 321-338. | Zbl | MR

[10] F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO Anal. Numér. 8 (1974) 129-151. | Zbl | MR | mathdoc-id

[11] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer Series in Computational Mathematics, Springer-Verlag, New York (1991). | Zbl | MR

[12] F. Brezzi, J. Douglas Jr., and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217-235. | Zbl | MR

[13] F. Brezzi, J. Douglas Jr., R. Durán and M. Fortin, Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237-250. | Zbl | MR

[14] 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. | Zbl | MR

[15] 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. | Zbl | MR

[16] J. Carrero, B. Cockburn and D. Schötzau, Hybridized globally divergence-free LDG methods. Part I : the Stokes problem. Math. Comp. 75 (2005) 533-563. | Zbl | MR

[17] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | Zbl | MR

[18] 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. | Zbl | MR

[19] B. Cockburn, D. Schötzau and J. Wang, Discontinuous Galerkin methods for incompressible elastic materials. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3184-3204. | Zbl | MR

[20] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Sér. Rouge 7 (1973) 33-75. | Zbl | MR | mathdoc-id

[21] M. Fortin, An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér. 11 (1977) 341-354. | Zbl | MR | mathdoc-id

[22] 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. | Zbl | MR

[23] 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. | Zbl | MR

[24] P. Hansbo and M.G. Larson, Discontinuous Galerkin and the Crouzeix-Raviart element : Application to elasticity. ESAIM : M2AN 37 (2003) 63-72. | Zbl | MR | mathdoc-id

[25] P. Hansbo and M.G. Larson, Piecewise divergence-free discontinuous Galerkin methods for Stokes flow. Comm. Num. Methods Engrg. 24 (2008) 355-366. | Zbl | MR

[26] F. Hecht, Construction d'une base de fonctions P1 non conforme à divergence nulle dans R3. RAIRO Anal. Numér. 15 (1981) 119-150. | Zbl | MR | mathdoc-id

[27] P. Hood and C. Taylor, Numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1 (1973) 1-28. | Zbl | MR

[28] 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).

[29] 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. | Zbl | MR

[30] A. Lew, P. Neff, D. Sulsky and M. Ortiz, Optimal BV estimates for a discontinuous Galerkin method for linear elasticity. Appl. Math. Res. express 3 (2004) 73-106. | Zbl | MR

[31] 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. | Zbl | MR

[32] 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. | Zbl | MR

[33] D. Schötzau, C. Schwab and A. Toselli, Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal. 40 (2003) 2171-2194. | Zbl | MR

[34] 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 Numérique 19 (1985) 111-143. | Zbl | MR | mathdoc-id

[35] 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. | Zbl | MR

[36] A. Ten Eyck, and A. Lew, Discontinuous Galerkin methods for non-linear elasticity. Int. J. Numer. Methods Engrg. 67 (2006) 1204-1243. | Zbl | MR

[37] 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. | Zbl | MR

[38] F. Thomasset, Implementation of finite element methods for Navier-Stokes equations. Springer-Verlag, New York (1981). | Zbl | MR

[39] J.P. Whiteley, Discontinuous Galerkin finite element methods for incompressible non-linear elasticity, Comput. Methods Appl. Mech. Engrg. 198 (2009) 3464-3478. | Zbl

[40] T.P. Wihler, Locking-free DGFEM for elasticity problems in polygons. IMA J. Numer. Anal. 24 (2004) 45-75. | Zbl | MR

Cité par Sources :