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.

##### MSC:

65N30 | Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) |

58J10 | Differential complexes; elliptic complexes |

65N25 | Numerical methods for eigenvalue problems (BVP of PDE) |

65F08 | Preconditioners for iterative methods |

74B05 | Classical linear elasticity |

35P15 | Estimation of eigenvalues and upper and lower bounds for PD operators |

35J05 | Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation |

35Q61 | Maxwell equations |