Finite element exterior calculus is a new theoretical approach to the
design and understanding of 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 spaces
connect to each other in discrete subcomplexes of elliptic differential
complexes, which are themselves connected to the continuous elliptic
complex through projections which commute with the complex
differential. This structure relates directly to the stability of
discretization methods based on the finite element spaces.
Applications include elliptic systems, electromagnetism, elasticity,
elliptic eigenvalue problems, and preconditioners.