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Finite Element Exterior Calculus

IMA Director Doug Arnold co-authored a chapter in the latest Acta Numerica:
"In this paper we survey and develop the finite element exterior calculus, a new theoretical 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. . . Thus the main theme of the paper is the development of finite element subcomplexes of certain elliptic differential complexes and cochain projections onto them, and their implications and applications in numerical PDEs."
Douglas N. Arnold, Richard S. Falk and Ragnar Winther. "Finite element exterior calculus, homological techniques, and applications." Acta Numerica (2006), pp. 1–155.