Compatible Discretizations, Covolume Algorithms and Differential Forms
Tuesday, May 11, 2004 - 11:00am - 11:50am
Roy Nicolaides (Carnegie-Mellon University)
Compatible discretizations have become prominent during the last few years, although they have been under development for at least 15 years. In the finite element setting they are exemplified by edge and related elements. Less well known are mimetic and covolume discretizations which may be considered as generalized finite difference techniques. In most compatible discretizations there are good analogs of exact sequence diagrams, ensuring the existence of analogs of vector identities that are valuable for obtaining good error estimates and reliable numerical results. The covolume approach uses complementary volumes to achieve compatible discretizations. The complementary volumes are typically tetrahedra and their corresponding Voronoi polyhedra. Use of these dual meshes is what distinguishes the covolume methodology from other compatible discretizations. This talk will begin with a review of the basic covolume methodology and use it to illustrate the main ideas of compatible discretization, exact sequences of spaces and so forth. Following that it will be shown how the covolume technique can be used to discretize differential forms. There is a remarkable parallel between certain operations on differential forms and the basic constructs appearing in covolume approximations. These will be discussed along with applications to partial differential equations on manifolds.