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.

翻译：我们开发了一种混合虚拟体积方法, 用于多边形/ 波利希拉格格网格上的椭圆体问题。 与在\ cite{brezzi2014Basic,da2016 mixed} 中引入的混合虚拟元素方法不同, 我们的方法在不使用 Lagrangian 乘数的情况下被降低为对称性, 对主变量来说是肯定的。 我们从通常使用达西法律将给定方程式转换成混合系统的方法开始, $\bu=- lical K}\nabla p$。 通过将方程式系统与每个元素中一些明智选择的测试空间结合起来, 我们定义了新的混合虚拟体积方法。 我们显示这些新法方法相当于原始值变量的不兼容性虚拟元素方法 $p$。 一旦主要变量计算了对称性、 肯定的系统, 达西速度的所有自由度都是本地计算的。 另外, 将 $L% 2 投到多诺米空间 空间 比较一些测试空间的测试空间空间, 因此我们的工作打开了一种轻松的方法, 来对达基理论化的理论代表数数数数式的公式, 。 将使得我们进行了一个最高级的公式分析。