Convergence in the incompressible limit of finite element approximations based on the Hu-Washizu formulation

The classical Hu–Washizu mixed formulation for plane problems in elasticity is examined afresh, with the emphasis on behavior in the incompressible limit. The classical continuous problem is embedded in a family of Hu–Washizu problems parametrized by a scalar α for which corresponds to the classical formulation, with λ and μ being the Lamé parameters. Uniform well- posedness in the incompressible limit of the continuous problem is established for α ≠ − 1. Finite element approximations are based on the choice of piecewise bilinear approximations for the displacements on quadrilateral meshes. Conditions for uniform convergence are made explicit. These conditions are shown to be met by particular choices of bases for stresses and strains, and include bases that are well known, as well as newly constructed bases. Though a discrete version of the spherical part of the stress exhibits checkerboard modes, it is shown that a λ-independent a priori error estimate for the displacement can be established. Furthermore, a λ-independent estimate is established for the post-processed stress. The theoretical results are explored further through selected numerical examples.