Normal-normal continuous symmetric stresses in mixed finite element elasticity

Abstract:

The classical continuous mixed formulation of linear elasticity with pointwise symmetric stresses allows for a conforming finite element discretization with piecewise polynomials of degree at least three. Symmetric stress approximations of lower polynomial order are only possible when their $\operatorname {div}$-conformity is weakened to the continuity of normal-normal components. In two dimensions, this condition is meant pointwise along edges for piecewise polynomials, but a corresponding characterization for general piecewise $H(\operatorname {div})$ tensors has been elusive.

We introduce such a space and establish a continuous mixed formulation of linear planar elasticity with pointwise symmetric stresses that have, in a distributional sense, continuous normal-normal components across the edges of a shape-regular triangulation. The displacement is split into an $L_2$ field and a tangential trace on the skeleton of the mesh. The well-posedness of the new mixed formulation follows with a duality lemma relating the normal-normal continuous stresses with the tangential traces of displacements.

For this new formulation we present a lowest-order conforming discretization. Stresses are approximated by piecewise quadratic symmetric tensors, whereas displacements are discretized by piecewise linear polynomials. The tangential displacement trace acts as a Lagrange multiplier and guarantees global $\operatorname {div}$-conformity in the limit as the mesh-size tends to zero. We prove locking-free, quasi-optimal convergence of our scheme and illustrate this with numerical examples.