Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes

We consider the lowest-order Raviart–Thomas mixed finite element method for second-order elliptic problems on simplicial meshes in two and three space dimensions. This method produces saddle-point problems for scalar and flux unknowns. We show how to easily and locally eliminate the flux unknowns, which implies the equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. The matrix of the final linear system is sparse, positive definite for a large class of problems, but in general nonsymmetric. We next show that these ideas also apply to mixed and upwind-mixed finite element discretizations of nonlinear parabolic convection–diffusion–reaction problems. Besides the theoretical relationship between the two methods, the results allow for important computational savings in the mixed finite element method, which we finally illustrate on a set of numerical experiments