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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.
@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
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