A SECOND ORDER MIMETIC DISCRETIZATION OF THE DIFFUSION PROBLEM

Abstract. The Mimetic Discretization Method (often called Mimetic Finite Difference method in the literature) can be understood, developed and analyzed as a generalization of the finite element method [1]. Such generalization allows (1) for polygonal/polyhedral meshes with non convex elements and hanging nodes and (2) for a more flexible discretization of the operators involved in the problem. The idea behind the approach is to express the operators and bilinear forms directly as a function of the discrete degrees of freedom, without build-ing and integrating the related basis functions. Clearly, the construction must satisfy certain fundamental properties of stability and consistency. In the present talk we show, as an example, a second order mimetic discretization of the diffusion problem in mixed form (in Rd), following the results of [2, 3]. The method adopts d degrees of freedom for every face for the fluxes and one degree of freedom for every element for the pressures. After describing the scheme and the fundamental conditions mentioned above, we show the convergence properties of the method, including a post-processed scalar variable. Finally, we present a set of numerical tests in full agreement with the theory. If time allows, in the end also a local a posteriori error estimator for the above method, which is shown to be reliable and efficient, will be briefly presented [4]