IMA Journal of Numerical Analysis (2005) Page 1 of 17 doi: 10.1093/imanum/dri017 Inf-sup stable finite element pairs based on dual meshes and bases for nearly incompressible elasticity BISHNU P. LAMICHHANE University of Aston, Aston Triangle, Birmingham, B4 7ET, UK. We consider finite element methods based on simplices to solve the problem of nearly incompressible elasticity. Two different approaches based respectively on dual meshes and dual bases are presented, where in both approaches pressure is discontinuous and can be statically condensed out from the sys- tem. These novel approaches lead to displacement-based low order finite element methods for nearly incompressible elasticity based on rigorous mathematical framework. Numerical results are provided to demonstrate the efficiency of the approach. Keywords: mixed finite elements, nodal average pressure, nearly incompressible elasticity, dual bases, dual meshes. AMS Subject Classification: 65F30, 65N15, 74B10 1. Introduction Low order finite elements based on quadrilaterals, hexahedra or simplices exhibit a poor performance when applied to a nearly incompressible elasticity problems. This poor performance is well-known to be the locking effect, that means, they do not converge uniformly with respect to the Lam´ e parameter λ . There are many approaches to overcome this difficulty. One obvious remedy for such problem is to work with higher order finite elements. For example, in Scott & Vogelius (1985), it is shown that working with the h-version finite elements of order higher than three on a class of triangular meshes locking in linear elasticity can completely be avoided. On the other hand, in Babuˇ ska & Suri (1992a), it has been shown that the h-version can never be fully free of locking in rectangular meshes no matter how higher-order finite elements are used in the sense that suboptimal convergence rates are observed. For the mathematical analysis of the locking effect, we refer to Babuˇ ska & Suri (1992a,b). An another approach is related to working with mixed methods. The linear elasticity problem can be formulated as a mixed formulation in many different ways, see Brezzi & Fortin (1991); Braess (1996, 2001); Arnold & Winther (2002). The general approach in these mixed formulations is to introduce extra variables leading to a saddle point problem. The essential point is to prove that the method is robust for the limiting problem, which is the Stokes problem. Treating displacement and the ’mean pressure’ as two independent field variables, one of the mixed formulations is given by Herrmann (1965), which is also known as the reduced form of Hellinger-Reissner principle. The Herrmann principle has also been extended to nonlinear hyperelastic problems in Reissner & Atluri (1989); van den Bogert et al. (1991); de Souza et al. (1996); Piltner & Taylor (1999). Other mixed formulations for nonlinear elasticity are introduced in Gatica & Stephan (2002); Gatica et al. (2007), see also Carstensen & Funken (2001); Brink & Stephan (2001); Carstensen et al. (2005). The approaches based on the bilinear or trilinear displacement and piecewise constant pressure are often used to overcome the volumetric locking in nearly incompressible elasticity, see Hughes (1987); Brezzi & Fortin (1991); Braess (2001). Separation IMA Journal of Numerical Analysis c  Institute of Mathematics and its Applications 2005; all rights reserved.