EQUIVALENCE BETWEEN LOWEST-ORDER MIXED FINITE ELEMENT AND MULTI-POINT FINITE VOLUME METHODS. DERIVATION, PROPERTIES, AND NUMERICAL EXPERIMENTS

We consider the lowest-order Raviart-Thomas mixed finite element method for elliptic diffusion problems on simplicial meshes in two or three space dimensions. This method produces saddle-point problems for scalar and flux unknowns. We show how to easily eliminate the flux unknowns, which implies an 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-reaction-diffusion problems. We finally present a set of numerical experiments confirming important computational savings while using the equivalent finite volume form of the lowest-order mixed finite element method.