  We develop a class of mixed virtual volume methods for elliptic problems on
polygonal/polyhedral grids. Unlike the mixed virtual element methods introduced
in \cite{brezzi2014basic,da2016mixed}, our methods are reduced to symmetric,
positive definite problems for the primary variable without using Lagrangian
multipliers.
  We start from the usual way of changing the given equation into a mixed
system using the Darcy's law, $\bu=-{\cal
  K} \nabla p$. By integrating the system of equations with some judiciously
chosen test spaces on each element, we define new mixed virtual volume methods
of all orders. We show that these new schemes are equivalent to the
nonconforming virtual element methods for the primal variable $p$.
  Once the primary variable is computed solving the symmetric, positive
definite system, all the degrees of freedom for the Darcy velocity are locally
computed. Also, the $L^2$-projection onto the polynomial space is easy to
compute. Hence our work opens an easy way to compute Darcy velocity on the
polygonal/polyhedral grids. For the lowest order case, we give a formula to
compute a Raviart-Thomas space like representation which satisfies the
conservation law.
  An optimal error analysis is carried out and numerical results are presented
which support the theory.
