January 21 - May 6, 2014
FEEC is a confluence of two streams of research, one emanating
from numerical analysis and scientific computation, the other
from topology and geometry. FEEC is based on the connection of
various fundamental partial differential equations to specific
differential complexes. FEEC steps back from the problem of
discretizing the PDE to the problem of discretizing the associated
complex. It captures key structures of the continuous complex
(homology and Hodge theory) at the discrete level and relates
the discrete and continuous structures, in order to obtain
accurate discretizations of the PDE. In this way FEEC has led
to breakthroughs in both numerical algorithms for some PDE that
had resisted previous efforts, and to a better understanding of
the behavior of many numerical approximations.
Although, FEEC was designed to apply topological tools to numerical
analysis, the reverse has happened as well. Recently tools from
finite element analysis were applied to resolve a conjecture
in topology from the 1970s by Dodziuk and Patodi concerning the
combinatorial codifferential.
The course will present FEEC and its applications with an aim
to being self-contained for a broad mathematical audience.
In particular a background in numerical analysis is not expected.