  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 (AVS-FE) method of Calo, Romkes and Valseth, in which
we consider a Petrov-Galerkin weak formulation where the stress and
displacement variables are in the space H(div)xH1, 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 (DPG) method of Demkowicz and Gopalakrishnan and
use optimal test spaces. Thus, the resulting FE discretization is stable even
as the Poisson ratio approaches 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 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.
