Vertex-centered control-volume mimetic finite difference methods

We develop a new class of vertex-centered control-volume mimetic finite difference methods on polytopal meshes for second order elliptic equations. The schemes are based on a mixed velocity-pressure formulation. The pressure is constant on dual mesh control-volumes constructed around the primary mesh vertices. The normal velocity is constant on the faces of the control-volumes, resulting in local mass conservation over the control-volumes. We consider both symmetric velocity integration rules constructed over the control-volumes, as well as non-symmetric quadrature rules constructed over sub-volumes obtained by the intersection of primary and dual elements. The latter choice allows for explicit gradient construction and local multipoint flux elimination within the primary elements, resulting in a positive definite vertex-centered pressure system. On simplicial, quadrilateral or hexahedral meshes, these local flux methods are closely related, and in some cases equivalent, to the classical vertex-centered control-volume finite element methods based on piecewise polynomial finite element basis functions for the pressure. The mimetic finite difference framework is utilized to analyze the well posedness and accuracy of the proposed methods. We establish first order convergence for the pressure and the velocity in the discrete mimetic norms, as well as second order pressure superconvergence in the case of symmetric quadrature rules. A series of numerical experiments illustrates the convergence properties of the methods on problems with varying degree of anisotropy, heterogeneity, and grid complexity in two and three dimensions.