Mixed virtual volume methods for elliptic problems

We develop a class of mixed virtual volume methods for elliptic problems on polygonal/polyhedral grids. Unlike the mixed virtual element methods introduced in [22], [13], 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, u=−K∇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.