  We introduce a new cell-centered finite volume discretization for elasticity
with weakly enforced symmetry of the stress tensor. The method is motivated by
the need for robust discretization methods for deformation and flow in porous
media, and falls in the category of multi-point stress approximations (MPSA).
By enforcing symmetry weakly, the resulting method has additional flexibility
beyond previous MPSA methods. This allows for a construction of a method which
is applicable to all grid types, and in particular the method amends a crucial
shortcoming in previous MPSA methods for simplex grids.
  By formulating the method as a discrete variational problem, we prove
convergence of the new method for a wide range of problems, with conditions
that can be verified at the time of discretization. We present the first set of
comprehensive numerical tests for the MPSA methods in three dimensions,
covering Cartesian and simplex grids, with both heterogeneous and nearly
incompressible media. The tests show that the new method consistently is second
order convergent in displacement, despite being lowest order, with a rate that
mostly is between 1 and 2 for stresses. The results further show that the new
method is more robust and computationally cheaper than previous MPSA methods.
