Compatible Spatial Discretizations - The IMA Volumes in Mathematics and its Applications 2006 edition

Marc Notes: IMA Hot Topics Workshop held May 11-15, 2004.; Based on pre-pub. info. only. Due: 07/06. Table of Contents: Numerical Convergence of the MPFA O-Method for General Quadrilateral Grids in Two and Three Dimensions.- Differential Complexes and Stability of Finite Element Methods I. The de Rham Complex.- Defferential Complexes and Stability of Finite Element Methods II: The Elasticity Complex.- On the Role of Involutions in the Discontinuous Galerkin Discretization of Maxwell and Magnetohydrodynamic Systems.- Principles of Mimetic Discretizations of Differential Operators.- Compatible Discretizations for Eigenvalue Problems.- Conjugated Bubnov-Galerkin Infinite Element for Maxwell Equations.- Covolume Discretization of Differential Forms.- Mimetic Reconstruction of Vectors.- A Cell-Centered Finite Difference Method on Quadrilaterals.- Development and Application of Compatible Discretizations of Maxwell s Equations."Jacket Description/Back: 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/. Publisher Marketing: 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."