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A family of discontinuous Galerkin mixed methods for nearly andperfectly incompressible elasticity∗

Published online by Cambridge University Press:  13 February 2012

Yongxing Shen  and
Adrian J. Lew
Yongxing Shen
Affiliation:
Laboratori de Càlcul Numèric, Universitat Politècnica de Catalunya (UPC BarcelonaTech), Barcelona, Spain. [email protected]
Adrian J. Lew
Affiliation:
Department of Mechanical Engineering, Stanford University, Stanford, 94305-4040 California, USA; [email protected]
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Abstract

We introduce a family of mixed discontinuous Galerkin (DG) finite element methods fornearly and perfectly incompressible linear elasticity. These mixed methods allow thechoice of polynomials of any order k ≥ 1 for the approximation of thedisplacement field, and of order k or k − 1 for thepressure space, and are stable for any positive value of the stabilization parameter. Weprove the optimal convergence of the displacement and stress fields in both cases, witherror estimates that are independent of the value of the Poisson’s ratio. These estimatesdemonstrate that these methods are locking-free. To this end, we prove the correspondinginf-sup condition, which for the equal-order case, requires a construction to establishthe surjectivity of the space of discrete divergences on the pressure space. In theparticular case of near incompressibility and equal-order approximation of thedisplacement and pressure fields, the mixed method is equivalent to a displacement methodproposed earlier by Lew et al. [Appel. Math. Res. express3 (2004) 73–106]. The absence of locking of this displacementmethod then follows directly from that of the mixed method, including the uniform errorestimate for the stress with respect to the Poisson’s ratio. We showcase the performanceof these methods through numerical examples, which show that locking may appear ifDirichlet boundary conditions are imposed strongly rather than weakly, as we do here.

Keywords

Discontinuous Galerkinlockingmixed methodinf-sup condition
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 5 , September 2012 , pp. 1003 - 1028
Copyright
© EDP Sciences, SMAI, 2012

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