Normal-normal continuous symmetric stresses in mixed finite element elasticity
Carsten Carstensen and Norbert Heuer
Math. Comp. 94 (354), 1571–1602, 2025.
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.