From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensions

The link between Mixed Finite Element (MFE) and Finite Volume (FV) methods applied to elliptic partial di erential equations has been investigated by many authors. Recently, a FV formulation of the mixed approach has been developed. This approach was restricted to 2D problems with a scalar for the parameter used to calculate uxes from the state variable gradient. This new approach is extended to 2D problems with a full parameter tensor and to 3D problems. The objective of this new formulation is to reduce the total number of unknowns while keeping the same accuracy. This is achieved by de ning one new variable per element. For the 2D case with full parameter tensor, this new formulation exists for any kind of triangulation. It allows the reduction of the number of unknowns to the number of elements instead of the number of edges. No additional assumptions are required concerning the averaging of the parameter in heterogeneous domains. For 3D problems, we demonstrate that the new formulation cannot exist for a general 3D tetrahedral discretization, unlike in the 2D problem. However, it does exist when the tetrahedrons are regular, or deduced from rectangular parallelepipeds, and allows reduction of the number of unknowns. Numerical experiments and comparisons between both formulations in 2D show the e ciency of the new formulation.