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Computational methods to approximate the solution of differential equations play a crucial role in science, engineering, mathematics, and technology. The key processes that govern the physical world-wave propagation, thermodynamics, fluid flow, solid deformation, electricity and magnetism, quantum mechanics, general relativity, and many more-are described by differential equations. We depend on numerical methods for the ability to simulate, explore, predict, and control systems involving these processes.The finite element exterior calculus, or FEEC, is a powerful new theoretical approach to the design and understanding of numerical methods to solve partial differential equations (PDEs). The methods derived with FEEC preserve crucial geometric and topological structures underlying the equations and are among the most successful examples of structure-preserving methods in numerical PDEs. This volume aims to help numerical analysts master the fundamentals of FEEC, including the geometrical and functional analysis preliminaries, quickly and in one place. It is also accessible to mathematicians and students of mathematics from areas other than numerical analysis who are interested in understanding how techniques from geometry and topology play a role in numerical PDEs. [Publisher]

Differential equations --- Finite element method. --- Calculus of variations. --- Équations différentielles --- Éléments finis, Méthode des. --- Calcul des variations. --- Numerical solutions. --- Solutions numériques.

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Symmetries in various forms pervade mathematics and physics. Globally, there are the symmetries of a homogenous space induced by the action of a Lie group. Locally, there are the infinitesimal symmetries induced by differential operators, including not only those of first order but of higher order too. This three-week summer program considered the symmetries preserving various natural geometric structures. Often these structures are themselves derived from partial differential equations whilst their symmetries turn out to be contrained by overdetermined systems. This leads to further topics including separation of variables, conserved quantities, superintegrability, parabolic geometry, represantation theory, the Bernstein-Gelfand-Gelfand complex, finite element schemes, exterior differential systems and moving frames. There are two parts to the Proceedings. The articles in the first part are expository but all contain significant new material. The articles in the second part are concerned with original research. All articles were thoroughly refereed and the range of interrelated work ensures that this will be an extremely useful collection. These Proceedings are dedicated to the memory of Thomas P. Branson who played a leading role in the conception and organization of this Summer Program but did not live to see its realization. .

Mathematics. --- Partial Differential Equations. --- Applications of Mathematics. --- Differential equations, partial. --- Mathématiques --- Differential equations, Partial. --- Differential equations, Partial --- Symmetry (Mathematics) --- Global analysis (Mathematics) --- Geometry, Differential --- Calculus --- Geometry --- Mathematics --- Physical Sciences & Mathematics --- Invariance (Mathematics) --- Partial differential equations. --- Applied mathematics. --- Engineering mathematics. --- Group theory --- Automorphisms --- Math --- Science --- Partial differential equations --- Engineering --- Engineering analysis --- Mathematical analysis

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Mathematics --- toegepaste wiskunde

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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/.

Partial differential equations --- Numerical analysis --- Mathematics --- differentiaalvergelijkingen --- toegepaste wiskunde --- numerieke analyse