Computational Electromagnetism and Whitney Forms

Wednesday, May 12, 2004 - 11:00am - 11:50am
Alain Bossavit (Laboratoire de Génie Electrique de Paris)
This talk will give an overview of the use of Whitney forms in electromagnetism, from 1980 to now, and of desirable developments.

Edge elements (not to be known by that name until about 1986) allowed to solve eddy current problems in dimension 3, a notorious conundrum during the 70's. Within ten years, edge element discretization, Galerkin style, became the established method in computational electromagnetism, (CEM) while efforts to answer the question why edge elements? slowly fostered familiarity with Whitney forms and cohomology in the CEM community. Once the central place of cohomology and commutative diagrams was acknowledged, ideas about mimetic discretizations, equivalent network methods, etc., could develop in the 90's, and the order of the day, now, is to found an appropriate discrete exterior calculus -- which holds promises far beyond electromagnetics. Lots of questions remain, however: Convergence issues, Whitney forms on non-simplicial meshes, in particular, will be addressed. (Some new results about the geometric interpretation of degrees of freedom for higher-degree Whitney forms will be presented.) As for the future, the demands of engineering about coupled problems (such as magneto-elasticity, MHD, etc.) will promote a better understanding of the differential-geometric structures which underlie both electromagnetism and continuum mechanics.