## Compatible Spatial Discretizations (Google eBook)Douglas N. Arnold, Pavel B. Bochev, Richard B. Lehoucq, Roy A. Nicolaides, Mikhail Shashkov This IMA Volume in Mathematics and its Apphcations COMPATIBLE SPATIAL DISCRETIZATIONS contains papers presented at a highly successful IMA Hot Topics Work shop: Compatible Spatial Discretizations for Partial Differential Equations. The event which was held on May 11-15, 2004 was organized by Douglas N. Arnold (IMA, University of Minnesota), Pavel B. Bochev (Computa tional Mathematics and Algorithms Department, Sandia National Labora tories), Richard B. Lehoucq (Computational Mathematics and Algorithms Department, Sandia National Laboratories), Roy A. Nicolaides (Depart ment of Mathematical Sciences, Carnegie-Mellon University), and Mikhail Shashkov (MS-B284, Group T-7, Theoretical Division, Los Alamos Na tional Laboratory). We are grateful to all participants and organizers for making this a very productive and stimulating meeting, and we would like to thank the organizers for their role in editing this proceeding. We take this opportunity to thank the National Science Foundation for its support of the IMA and the Department of Energy for providing additional funds to support this workshop. Series Editors Douglas N. Arnold, Director of the IMA Arnd Scheel, Deputy Director of the IMA PREFACE In May 2004 over 80 mathematicians and engineers gathered in Min neapolis for a "hot topics" IMA workshop to talk, argue and conjecture about compatibility of spatial discretizations for Partial Differential Equa tions. We define compatible, or mimetic, spatial discretizations as those that inherit or mimic fundamental properties of the PDE such as topology, conservation, symmetries, and positivity structures and maximum princi ples. |

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### Contents

1 | |

DIFFERENTIAL COMPLEXES AND STABILITY OF FINITE ELEMENT METHODS I THE DE RHAM COMPLEX | 23 |

THE ELASTICITY COMPLEX | 47 |

ON THE ROLE OF INVOLUTIONS IN THE DISCONTINUOUS GALERKIN DISCRETIZATION OF MAXWELL AND MAGNETOHYDRODYN... | 69 |

PRINCIPLES OF MIMETIC DISCRETIZATIONS OF DIFFERENTIAL OPERATORS | 89 |

COMPATIBLE DISCRETIZATIONS FOR EIGENVALUE PROBLEMS | 121 |

CONJUGATED BUBNOVGALERKIN INFINITE ELEMENT FOR MAXWELL EQUATIONS | 143 |

COVOLUME DISCRETIZATION OF DIFFERENTIAL FORMS | 161 |

MIMETIC RECONSTRUCTION OF VECTORS | 173 |

A CELLCENTERED FINITE DIFFERENCE METHOD ON QUADRILATERALS | 189 |

DEVELOPMENT AND APPLICATION OF COMPATIBLE DISCRETIZATIONS OF MAXWELLS EQUATIONS | 209 |

LIST OF WORKSHOP PARTICIPANTS | 235 |

### Common terms and phrases

2-forms adjoint algebraic analysis Applications approximation basis functions bilinear boundary conditions cell center cochains coefficients commuting diagram compatible discretizations component Computational conservation convergence rates corresponding covolume curl defined degrees of freedom denote Department of Mathematics derived differential forms dimensions discontinuous Galerkin domain dual eigenmodes eigenvalue problem elasticity complex electromagnetics elliptic entropy error exterior exterior derivative faces fc-forms finite difference finite element methods finite element spaces finite volume flux Galerkin method H(div inner product integral involution Jn Jn L2 norm Laplacian Lemma linear magnetic Math matrix Maxwell Maxwell's equations mesh MHD system mimetic mixed finite element models MPFA norm normal velocities numerical obtained PDEs piecewise polynomial pressure reconstruction operator Rham complex satisfied scalar Shashkov Sobolev space solution stability Stokes theorem subspace symmetric term theorem theory tion variables vector field vertex wave wedge product