A New Class of Higher Order Mixed Finite Volume Methods for Elliptic Problems

We introduce a new class of higher order mixed finite volume methods for elliptic problems. 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 finite volume methods of higher order. We show that these new schemes are equivalent to the nonconforming finite element spaces used to define them. The Darcy velocity can be locally recovered from the solution of nonconforming finite element method. Hence our work opens a way to make higher order mixed method more practicable. Threedimensional extensions to parallelepiped elements are also presented. This work can be viewed as a generalization of earlier works which were devoted to reducing the (lowest order) mixed finite element method to a corresponding nonconforming finite element method. An optimal error analysis is carried out and numerical results are presented which confirm the theory.