Departmental Colloquium
Douglas Arnold
University of Minnesota
Finite element exterior calculus: where scientific computing meets algebraic topology
Abstract: This talk will discuss a substantial interplay of algebraic
topology with numerical analysis which has developed over the
last decade. During this period, de Rham cohomology and the Hodge
theory of Riemannian manifolds have come to play a crucial role
in the development and understanding of computational algorithms
for the solution of problems in partial differential equations.
Hodge theory is at the foundation of the well-posedness of many
important problems in partial differential equations, but only
recently has it been understood how the stable numerical solution
of PDE problems often depends on capturing the correct topological
structures at the discrete level. This may be accomplished by
constructing subcomplexes of the de Rham complex which consist
of finite element differential forms. These latter have become
a key technology in scientific computation, but first appeared
in algebraic topology in the works of Whitney and Sullivan half a
century ago. Interactions of topology and numerical analysis are
not just applications of the former to the latter, but also occur
in the reverse direction. Recently, the theory of superconvergence
developed for finite element methods was used to settle a question
concerning the combinatorial codifferential posed by Dodziuk and
Patodi in 1976.
Friday August 29, 2014 at 3:00 PM in SEO 636