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Finite element exterior calculus, homological techniques, and applications

Published online by Cambridge University Press:  16 May 2006

Douglas N. Arnold
Affiliation:
Institute for Mathematics and its Applications and School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA E-mail: arnold@ima.umn.edu
Richard S. Falk
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA E-mail: falk@math.rutgers.edu
Ragnar Winther
Affiliation:
Centre of Mathematics for Applications and Department of Informatics, University of Oslo, PO Box 1053, 0316 Oslo, Norway E-mail: ragnar.winther@cma.uio.no

Abstract

Finite element exterior calculus is an approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are revealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Laplacian, Maxwell’s equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.

Type
Research Article
Copyright
2006 Cambridge University Press

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