Finite element exterior calculus, homological techniques, and applications.

*(English)*Zbl 1185.65204Summary: 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 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 |