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.

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://www.numdam.org/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://www.numdam.org/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://www.numdam.org/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 , Volume 46 (2012) no. 5, pp. 1003-1028. doi : 10.1051/m2an/2011046. http://www.numdam.org/articles/10.1051/m2an/2011046/

[1] A stable finite element for the Stokes equations. Calcolo 21 (1984) 337-344. | MR | Zbl

, and ,[2] Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749-1779. | MR | Zbl

, , and ,[3] 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. | MR | Zbl

and ,[4] A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3352-3360. | MR | Zbl

, and ,[5] Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math. 33 (1977) 211-224. | MR | Zbl

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

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

,[8] The mathematical theory of finite element methods, 3th edition, Springer (2008). | MR | Zbl

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

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

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

and ,[12] Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217-235. | MR | Zbl

, , and ,[13] Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237-250. | MR | Zbl

, , and ,[14] Discontinuous galerkin approximations for elliptic problems. Numer. Methods Partial Differential Equations (2000) 365-378. | MR | Zbl

, , , and ,[15] Mixed discontinuous Galerkin methods for Darcy flow. J. Sci. Comput. 22, 23 (2005) 119-145. | MR | Zbl

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

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

,[18] Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40 (2002) 319-343. | MR | Zbl

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

, and ,[20] Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Sér. Rouge 7 (1973) 33-75. | Numdam | MR | Zbl

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

,[22] A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems. Math. Comp. 74 (2005) 53-84. | MR | Zbl

, and ,[23] Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche's method. Comput. Methods Appl. Mech. Engrg. 191 (2002) 1895-1908. | MR | Zbl

and ,[24] Discontinuous Galerkin and the Crouzeix-Raviart element : Application to elasticity. ESAIM : M2AN 37 (2003) 63-72. | Numdam | MR | Zbl

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

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

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

and ,[28] Navier-Stokes equations using mixed interpolation. Finite Element Methods in Flow Problems, edited by J.T. Oden. UAH Press, Huntsville, Alabama (1974).

and ,[29] A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg. 124 (1995) 195-212. | MR | Zbl

and ,[30] Optimal BV estimates for a discontinuous Galerkin method for linear elasticity. Appl. Math. Res. express 3 (2004) 73-106. | MR | Zbl

, , and ,[31] A hybridizable discontinuous Galerkin method for Stokes flow. Comput. Methods Appl. Mech. Engrg. 199 (2010) 582-597. | MR | Zbl

, and ,[32] Discontinuous finite element methods for incompressible flows on subdomains with non-matching interfaces. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3274-3292. | MR | Zbl

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

, and ,[34] 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. | Numdam | MR | Zbl

and ,[35] A hybridizable discontinuous Galerkin method for linear elasticity. Int. J. Numer. Methods Engrg. 80 (2009), 1058-1092. | MR | Zbl

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

, and ,[37] Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity : Analytical estimates. Comput. Methods Appl. Mech. Engrg. 197 (2008) 2989-3000. | MR | Zbl

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

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

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

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