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.