A stable mixed finite element method for nearly incompressible linear elastostatics

We present a new, stable, mixed finite element (FE) method for linear elastostatics of nearly incompressible solids. The method is the automatic variationally stable FE method of Calo et al., in which we consider a Petrov–Galerkin weak formulation where the stress and displacement variables are in the space H(div) and H1, respectively. This allows us to employ a fully conforming FE discretization for any elastic solid using classical FE subspaces of H(div) and H1. Hence, the resulting FE approximation yields both continuous stresses and displacements. To ensure stability of the method, we employ the philosophy of the discontinuous Petrov–Galerkin method of Demkowicz and Gopalakrishnan and use optimal test spaces. Thus, the resulting FE discretization is stable even as the Poisson's ratio ν→0.5, and the system of linear algebraic equations is symmetric and positive definite. Our method also comes with a built‐in a posteriori error estimator as well as indicators which are used to drive mesh adaptive refinements. We present several numerical verifications of our method including comparisons to existing FE technologies.