  We develop a multipoint stress mixed finite element method for linear
elasticity with weak stress symmetry on quadrilateral grids, which can be
reduced to a symmetric and positive definite cell centered system. The method
is developed on simplicial grids in [4]. The method utilizes the lowest order
Brezzi-Douglas-Marini finite element spaces for the stress and the trapezoidal
quadrature rule in order to localize the interaction of degrees of freedom,
which allows for local stress elimination around each vertex. We develop two
variants of the method. The first uses a piecewise constant rotation and
results in a cell-centered system for displacement and rotation. The second
uses a continuous piecewise bilinear rotation and trapezoidal quadrature rule
for the asymmetry bilinear form. This allows for further elimination of the
rotation, resulting in a cell-centered system for the displacement only.
Stability and error analysis is performed for both methods. First-order
convergence is established for all variables in their natural norms. A duality
argument is employed to prove second order superconvergence of the displacement
at the cell centers. Numerical results are presented in confirmation of the
theory.
