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Finite Element Exterior Calculus
January 21 - May 6, 2014

Douglas N. Arnold (University of Minnesota, Twin Cities)

Math 8994

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.

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