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Fall 2010 Special Course

School of Mathematics, University of Minnesota

President, Society for Industrial and Applied Mathematics (SIAM)

Poster

Any young (or not so young) mathematician who spends the time to master this paper will have tools that will be useful for his or her entire career.

— Math Reviews (referring to [1])

This will be the mandatory reference for many years to come.

— Roland Glowinski (referring to [2])

Abstract: This course will be a self-contained overview of the
Finite Element Exterior Calculus (FEEC) aimed at researchers
and
graduate students with an interest in numerical analysis of
PDE.
FEEC is a theoretical approach to the design and understanding
of discretizations for a wide variety of systems of partial
differential equations. It brings to bear tools and structures
from geometry and topology to develop and analyze numerical
methods which are compatible with the structures which underlie
the well-posedness of the PDE problem being posed. In FEEC,
many finite element spaces are revealed as spaces of piecewise
polynomial differential forms. These spaces relate to each
other through a structure known as a Hilbert complex, which
plays
a similar role in FEEC as the standard Hilbert space theory of
Galerkin methods. The FEEC viewpoint greatly clarifies and
unifies
the theory of stable finite element methods, especially in
mixed
finite element formualtions, and has enabled the development of
previously elusive stable mixed finite elements for elasticity.
Other applications include elliptic systems, electromagnetism,
elliptic eigenvalue problems, and preconditioners.

Prerequisites: A basic familiarity with finite element methods
and functional analysis (Hilbert spaces) is expected. All the
necessary geometry and topology will be included in the course.

References: The course will basically cover the material in
these two long papers:

- Finite element exterior calculus, homological techniques, and applications. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther. Acta Numer., 15:1-155, 2006.
- Finite element exterior calculus: from Hodge theory to numerical stability. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther. Bull. Amer. Math. Soc. (N.S.), 47:281-354, 2010.