Numerical Analysis Seminars: Mimetic Finite difference Methods
14 March 2008 14:15 in CM105
Mimetic Finite Difference (MFD) methods are relatively new numerical techniques that have already been applied to the solution of problems in continuum mechanics, electromagnetics, gas dynamics and linear diffusion. They may be classified as standing in between Mixed Finite Element Methods and Finite Volumes. The idea behind this new discretisation technique is to define discrete operators by imposing that the essential properties of the underlying differential operators are preserved. For instance, when applied to linear diffusion problems written in mixed form, the discrete (mimetic) differential operators are defined imposing the Green's formula with respect to some discrete scalar products. In this way, conservation laws and solution symmetries are embedded in the method. Another crucial property of MFD is that very general polyhedral mesh elements can be handled, allowing for non-convex, degenerate polyhedrons, and even polyhedrons with curved faces. The flexibility in the mesh design gives an obvious advantage in the treatment of complex solution domains and heterogeneous materials. Moreover, allowing non-matching, non-convex mixed types of elements facilitates adaptive mesh refinement, particularly in the coarsening phase, making it a completely local process. The talk will overview the definition and features of MFD concentrating on linear diffusion problems. We shall demonstrate through extensive numerical examples the flexibility of the method, and present our recent analysis on the method's superconvergence properties and their use in a-posteriori error estimation. Finally, we shall present the first a priori analysis of the method applied to the solution of steady convection-diffusion problems.
Contact David.Bourne@durham.ac.uk for more information