  We present a mixed finite element method for a five-field formulation of the
Biot system of poroelasticity that reduces to a cell-centered
pressure-displacement system on simplicial and quadrilateral grids. A mixed
stress-displacement-rotation formulation for elasticity with weak stress
symmetry is coupled with a mixed velocity-pressure Darcy formulation. The
spatial discretization is based on combining the multipoint stress mixed finite
element (MSMFE) method for elasticity and the multipoint flux mixed finite
element (MFMFE) method for Darcy flow. It uses the lowest order
Brezzi-Douglas-Marini mixed finite element spaces for the poroelastic stress
and Darcy velocity, piecewise constant displacement and pressure, and
continuous piecewise linear or bilinear rotation. A vertex quadrature rule is
applied to the velocity, stress, and stress-rotation bilinear forms, which
block-diagonalizes the corresponding matrices and allows for local velocity,
stress, and rotation elimination. This leads to a cell-centered
positive-definite system for pressure and displacement at each time step. We
perform error analysis for the semidiscrete and fully discrete formulations,
establishing first order convergence for all variables in their natural norms.
The numerical tests confirm the theoretical convergence rates and illustrate
the locking-free property of the method.
