Mat-1.3656 Seminar on numerical analysis and computational science

Monday, Jan 19, 2008, room U322 at 14.15, Eirola & Stenberg

Lourenco Beirao Da Veiga, Università degli Studi di Milano
A posteriori error estimates for the Mimetic Finite Difference method

 
The Mimetic Finite Difference (MFD) method can be interpreted as a finite element scheme where the basis functions related to the discrete degrees of freedom are not explicitly defined. As a consequence, the operators and other quantities appearing in the problem must be approximated by discrete counterparts that satisfy finite dimensional analogs of some fundamental property. This approach allows for a greater flexibility of the mesh and the possibility to mimic intrinsic properties of the differential problem under study. In particular,general polyhedral (or polygonal in 2 dimensions) meshes, even with non convex and non matching elements, can be adopted. This flexibility makes the MFD method a very appealing ground for the application of mesh adaptivity.

The present talk is divided in two parts. The first part is devoted to the introduction of the Mimetic Finite Difference method for the diffusion problem in mixed form, presented from the standpoint of classical finite elements. The construction of the method and some fundamental theoretical results are shown. In the second part we derive local a posteriori error estimates for the MFD scheme. The error estimator is shown to be both reliable and efficient with respect to an energy type norm involving a post-processed pressure. Finally, the error indicator is combined with an adaptive process and a set of numerical tests is presented.