Finite element exterior calculus, homological techniques, and applications

TLDR

Finite element exterior calculus is an approach to designing and understanding finite element discretizations for a wide variety of systems of partial differential equations. It employs tools from differential geometry, algebraic topology, and homological algebra to develop discretizations that respect the geometric, topological, and algebraic structures underlying well‑posedness, and it is applied to discretize the Hodge Laplacian, Maxwell’s equations, elasticity, elliptic eigenvalue problems, and to construct preconditioners. The framework reveals that many finite element spaces are piecewise polynomial differential forms linked in discrete subcomplexes of elliptic complexes, with commuting projections to the continuous complex, and it is applied to discretize the Hodge Laplacian, Maxwell’s equations, elasticity, elliptic eigenvalue problems, and to design preconditioners.

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.