2006/05/01 by Douglas N. Arnold, Richard S. Falk, Ragnar Winther · 5 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Numerical methods in engineering #Electromagnetic Simulation and Numerical Methods #Mathematics #Finite element method #Vector calculus #Extended finite element method #Mixed finite element method #Differential algebraic geometry #Discretization #Piecewise #Algebra over a field #Differential form #Spectral element method #Mathematical analysis #Differential equation #Pure mathematics #Differential algebraic equation #Ordinary differential equation
paper · doi:10.1017/s0962492906210018
openalex publication_date 2006/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.