## Compatible Spatial Discretizations (Google eBook)The IMA Hot Topics workshop on compatible spatialdiscretizations was held May 11-15, 2004 at the University of Minnesota. The purpose of the workshop was to bring together scientists at the forefront of the research in the numerical solution of PDEs to discuss recent advances and novel applications of geometrical and homological approaches to discretization. This volume contains original contributions based on the material presented at the workshop. A unique feature of the collection is the inclusion of work that is representative of the recent developments in compatible discretizations across a wide spectrum of disciplines in computational science. Compatible spatial discretizations are those that inherit or mimic fundamental properties of the PDE such as topology, conservation, symmetries, and positivity structures and maximum principles. The papers in the volume offer a snapshot of the current trends and developments in compatible spatial discretizations. The reader will find valuable insights on spatial compatibility from several different perspectives and important examples of applications compatible discretizations in computational electromagnetics, geosciences, linear elasticity, eigenvalue approximations and MHD. The contributions collected in this volume will help to elucidate relations between different methods and concepts and to generally advance our understanding of compatible spatial discretizations for PDEs. Abstracts and presentation slides from the workshop can be accessed at http://www.ima.umn.edu/talks/workshops/5-11-15.2004/. |

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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