Finite element exterior calculus (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 (PDEs) 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 PDEs 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, aiming to be self-contained for a broad mathematical audience. In particular, a background in numerical analysis is not expected.