Differential Complexes and Stability of Finite Element Methods

Tuesday, May 11, 2004 - 9:30am - 10:20am
Keller 3-180
Douglas Arnold (University of Minnesota, Twin Cities)
Many of the partial differential equations of mathematical physics are related to differential complexes which determine their structure and well-posedness. Many successful finite element discretizations of these problems can best be understood as arising from piecewise polynomial subcomplexes. The stability of these methods is obtainted by relating the discrete subcomplex to the continuous differential complex via a commuting diagram. The best known case is the de Rham complex, which underlies both electromagnetic and diffusion problems. In this case there are a large number of possible piecewise polynomial subcomplexes of each order. These can be presented systematically using the Koszul complex. The elasticity equations are related to another differential complex which can be related to the de Rham complex through a subtle homological construction. This has lead to recent progress in the design of stable mixed finite elements for elasticity in two and three dimensions.