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

¹ Laboratori de Càlcul Numèric, Universitat Politècnica de Catalunya (UPC BarcelonaTech), Barcelona, Spain
yongxing.shen@upc.edu
² Department of Mechanical Engineering, Stanford University, Stanford, 94305-4040 California, USA
lewa@stanford.edu

Received: 22 January 2011
Revised: 12 June 2011

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.