Summary: RECTANGULAR MIXED ELEMENTS FOR ELASTICITY WITH
WEAKLY IMPOSED SYMMETRY CONDITION
Abstract. We present new rectangular mixed finite elements for linear elasticity.
The approach is based on a modification of the Hellinger-Reissner functional in
which the symmetry of the stress field is enforced weakly through the introduction
of a Lagrange multiplier. The elements are analogues of the lowest order elements
described in Arnold, Falk and Winther [ Mixed finite element methods for linear
elasticity with weakly imposed symmetry. Mathematics of Computation 76 (2007),
pp. 16991723]. Piecewise constants are used to approximate the displacement and
the rotation. The first order BDM elements are used to approximate each row of
the stress field.
The theory of elasticity is used to predict the response of a material to applied forces.
The unknowns in the equations are the stress field, a symmetric matrix field which
encodes the internal forces and the displacement, a vector field. For various reasons,
mixed finite elements where one approximates both the stress and displacement are
the methods of choice. One seeks the stress in the space of symmetric matrix fields
with components square integrable and with divergence, taken row-wise, also square
integrable. The displacement is sought in the space of square integrable vector fields.