A hybrid-mixed method for elasticity

Abstract

This work presents a family of stable finite element methods for two-and three-dimensional linear elasticity models. The weak form posed on the skeleton of the partition is a byproduct of the primal hybridization of the elasticity problem. The unknowns are the piecewise rigid body modes and the Lagrange multipliers used to relax the continuity of displacements. They characterize the exact displacement through a direct sum of rigid body modes and solutions to local elasticity problems with Neumann boundary conditions driven by the multipliers. The local problems define basis functions which are in a one-to-one correspondence with the basis of the subspace of Lagrange multipliers used to discretize the problem. Under the assumption that such a basis is available exactly, we prove that the underlying method is well posed, and the stress and the displacement are super-convergent in natural norms driven by (high-order) interpolating multipliers. Also, a local post-processing computation yields strongly symmetric stress which is in local equilibrium and possesses continuous traction on faces. A face-based a posteriori estimator is shown to be locally efficient and reliable with respect to the natural norms of the error. Next, we propose a second level of discretization to approximate the basis functions. A two-level numerical analysis establishes sufficient conditions under which the well-posedness and super-convergent properties of the one-level method is preserved.