Superconvergence for Control‐Volume Mixed Finite Element Methods on Rectangular Grids

We consider control-volume mixed finite element methods for the approximation of second-order elliptic problems on rectangular grids. These methods associate control volumes (covolumes) with the vector variable as well as the scalar, obtaining local algebraic representation of the vector equation (e.g., Darcy’s law) as well as the scalar equation (e.g., conservation of mass). We establish $O(h^2)$ superconvergence for both the scalar variable in a discrete $L^2$-norm and the vector variable in a discrete $H({\rm div})$-norm. The analysis exploits a relationship between control-volume mixed finite element methods and the lowest order Raviart-Thomas mixed finite element methods.