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Finite element exterior calculus, homological techniques, and applications. (English) Zbl 1185.65204
Summary: 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 partial differential equation 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.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
58J10 Differential complexes
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
74B05 Classical linear elasticity
35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35Q61 Maxwell equations
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