Title:         Finite element exterior calculus, homological techniques, and applications

Authors:       Douglas N. Arnold, Richard S. Falk, and Ragnar Winther

Source:        Acta Numerica 15 (2006), pp. 1-155.

Status:        Published

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.

Keywords:      finite element, exterior calculus, mixed method, Hodge, de Rham

Subj. class.:  65N30, 58A10, 58A12, 58A14, 65N12, 65N15, 65N25, 65N55, 74S05, 78M20

URL:           http://www.math.rutgers.edu/~falk/papers/acta.pdf