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Simulating Our Complex World: Modeling, Computation and Analysis

Fall 2010 Special Course

Douglas N. Arnold

   McKnight Presidential Professor of Mathematics
   School of Mathematics, University of Minnesota
   President, Society for Industrial and Applied Mathematics (SIAM)


Finite Element Exterior Calculus

Thursdays, 11:15-12:15, Lind Hall 305, starting September 30 (except during workshops)
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:
  1. Finite element exterior calculus, homological techniques, and applications. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther. Acta Numer., 15:1-155, 2006.

  2. 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.

Lecture Videos:

9/30/2010 Lecture (flv)
10/07/2010 Lecture (flv)
10/14/2010 Lecture (flv)
10/28/2010 Lecture (flv)
11/11/2010 Lecture (flv)
11/18/2010 Lecture (flv)
12/9/2010 Lecture (flv)
12/16/2010 Lecture (flv)