Variational and Geometric Aspects of Compatible Discretizations

Wednesday, May 12, 2004 - 9:30am - 10:20am
Keller 3-180
Pavel Bochev (Sandia National Laboratories)
PDE models arise in virtually all fields of science and engineering. Their compatible discretizations are finite dimensional models of the physical process that are stable and provide accurate and physically meaningful solutions.

Variational principles take advantage of the intrinsic connections between the structure of many PDEs and optimization problems to identify their compatible discretizations.

Differential complexes provide another tool to encode the structure of a PDE. Differential forms represent global quantities rather than fields, and provide a model for the way we observe the physical process. The idea that differential forms can and should be used to develop compatible (mimetic) discretizations started to permeate computational sciences approximately two decades ago and led to fundamental advances in computational electromagnetics.

Since then, geometrical approaches to discretization have enjoyed a steady and ever increasing interest and appreciation in computational sciences. The goal of this lecture is two-fold. I will show how variational and geometric techniques can complement each other in the quest for accurate and stable discretizations by providing tools for the analysis and the design of compatible models. In doing so I will retrace the key steps that have led to our modern understanding of these connections. To illustrate, as well as to compare and contrast variational identifications of compatibility, I will use the Kelvin and the Dirichlet principles. Factorization diagrams will reveal the geometrical structure of the problem and form the basis for the design of compatible discretizations. Then, with the help of the grid decomposition property and the commuting diagram property I will show the fundamental links between the two approaches.

I will conclude with examples of alternative discrete models that circumvent the rigid structural constraints imposed by compatibility and talk about the advantages and the perils associated with their use.