Normal-normal continuous symmetric stresses in mixed finite element elasticity
Carsten Carstensen and Norbert Heuer
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 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(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 L2 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 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.