Acta Numerica

Finite element exterior calculus, homological techniques, and applications

Douglas N. Arnold a1, Richard S. Falk a2 and Ragnar Winther a3
a1 Institute for Mathematics and its Applications and School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA E-mail:
a2 Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA E-mail:
a3 Centre of Mathematics for Applications and Department of Informatics, University of Oslo, PO Box 1053, 0316 Oslo, Norway E-mail:

Article author query
arnold dn   [Google Scholar] 
falk rs   [Google Scholar] 
winther r   [Google Scholar] 


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.

(Published Online May 16 2006)

Dedicated to Carme, Rena, and Rita