Adaptive Finite Elements In The Discretization Of Parabolic Problems
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Adaptive Finite Elements In The Discretization Of Parabolic Problems
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Author : Christian A. Möller
language : en
Publisher: Logos Verlag Berlin GmbH
Release Date : 2011
Adaptive Finite Elements In The Discretization Of Parabolic Problems written by Christian A. Möller and has been published by Logos Verlag Berlin GmbH this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.
Adaptivity is a crucial tool in state-of-the-art scientific computing. However, its theoretical foundations are only understood partially and are subject of current research. This self-contained work provides theoretical basics on partial differential equations and finite element discretizations before focusing on adaptive finite element methods for time dependent problems. In this context, aspects of temporal adaptivity and error control are considered in particular. Based on the gained insights, a specific adaptive algorithm is designed and analyzed thoroughly. Most importantly, it is proven that the presented adaptive method terminates within any demanded error tolerance. Moreover, the developed algorithm is analyzed from a numerical point of view and its performance is compared to well-known standard methods. Finally, it is applied to the real-life problem of concrete carbonation, where two different discretizations are compared.
Least Squares Finite Element Methods
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Author : Pavel B. Bochev
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-04-28
Least Squares Finite Element Methods written by Pavel B. Bochev and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-04-28 with Mathematics categories.
Since their emergence in the early 1950s, ?nite element methods have become one of the most versatile and powerful methodologies for the approximate numerical solution of partial differential equations. At the time of their inception, ?nite e- ment methods were viewed primarily as a tool for solving problems in structural analysis. However, it did not take long to discover that ?nite element methods could be applied with equal success to problems in other engineering and scienti?c ?elds. Today, ?nite element methods are also in common use, and indeed are often the method of choice, for incompressible ?uid ?ow, heat transfer, electromagnetics, and advection-diffusion-reaction problems, just to name a few. Given the early conn- tion between ?nite element methods and problems engendered by energy minimi- tion principles, it is not surprising that the ?rst mathematical analyses of ?nite e- ment methods were given in the environment of the classical Rayleigh–Ritz setting. Yet again, using the fertile soil provided by functional analysis in Hilbert spaces, it did not take long for the rigorous analysis of ?nite element methods to be extended to many other settings. Today, ?nite element methods are unsurpassed with respect to their level of theoretical maturity.
Numerical Solution Of Partial Differential Equations By The Finite Element Method
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Author : Claes Johnson
language : en
Publisher: Courier Corporation
Release Date : 2012-05-23
Numerical Solution Of Partial Differential Equations By The Finite Element Method written by Claes Johnson and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-05-23 with Mathematics categories.
An accessible introduction to the finite element method for solving numeric problems, this volume offers the keys to an important technique in computational mathematics. Suitable for advanced undergraduate and graduate courses, it outlines clear connections with applications and considers numerous examples from a variety of science- and engineering-related specialties.This text encompasses all varieties of the basic linear partial differential equations, including elliptic, parabolic and hyperbolic problems, as well as stationary and time-dependent problems. Additional topics include finite element methods for integral equations, an introduction to nonlinear problems, and considerations of unique developments of finite element techniques related to parabolic problems, including methods for automatic time step control. The relevant mathematics are expressed in non-technical terms whenever possible, in the interests of keeping the treatment accessible to a majority of students.
Galerkin Finite Element Methods For Parabolic Problems
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Author : Vidar Thomee
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17
Galerkin Finite Element Methods For Parabolic Problems written by Vidar Thomee and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.
My purpose in this monograph is to present an essentially self-contained account of the mathematical theory of Galerkin finite element methods as applied to parabolic partial differential equations. The emphases and selection of topics reflects my own involvement in the field over the past 25 years, and my ambition has been to stress ideas and methods of analysis rather than to describe the most general and farreaching results possible. Since the formulation and analysis of Galerkin finite element methods for parabolic problems are generally based on ideas and results from the corresponding theory for stationary elliptic problems, such material is often included in the presentation. The basis of this work is my earlier text entitled Galerkin Finite Element Methods for Parabolic Problems, Springer Lecture Notes in Mathematics, No. 1054, from 1984. This has been out of print for several years, and I have felt a need and been encouraged by colleagues and friends to publish an updated version. In doing so I have included most of the contents of the 14 chapters of the earlier work in an updated and revised form, and added four new chapters, on semigroup methods, on multistep schemes, on incomplete iterative solution of the linear algebraic systems at the time levels, and on semilinear equations. The old chapters on fully discrete methods have been reworked by first treating the time discretization of an abstract differential equation in a Hilbert space setting, and the chapter on the discontinuous Galerkin method has been completely rewritten.
Advanced Finite Element Methods With Applications
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Author : Thomas Apel
language : en
Publisher: Springer
Release Date : 2019-06-28
Advanced Finite Element Methods With Applications written by Thomas Apel and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-06-28 with Mathematics categories.
Finite element methods are the most popular methods for solving partial differential equations numerically, and despite having a history of more than 50 years, there is still active research on their analysis, application and extension. This book features overview papers and original research articles from participants of the 30th Chemnitz Finite Element Symposium, which itself has a 40-year history. Covering topics including numerical methods for equations with fractional partial derivatives; isogeometric analysis and other novel discretization methods, like space-time finite elements and boundary elements; analysis of a posteriori error estimates and adaptive methods; enhancement of efficient solvers of the resulting systems of equations, discretization methods for partial differential equations on surfaces; and methods adapted to applications in solid and fluid mechanics, it offers readers insights into the latest results.
Fast Solution Of Discretized Optimization Problems
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Author : Karl-Heinz Hoffmann
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
Fast Solution Of Discretized Optimization Problems written by Karl-Heinz Hoffmann and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Differential equations - partial as well as ordinary - are one of the main tools for the modeling of real world application problems. Pursuing the ultimate aim of influencing these systems in a desired way, one is confronted with the task of optimizing discretized models. This volume contains selected papers presented at the International Work shop on "Fast Solution of Discretized Optimization Problems", which took place at the Weierstrass Institute for Applied Analysis and Stochastics in Berlin from May 08 until May 12, 2000. The conference was attended by 59 scientists from 10 countries. The scientific program consisted of 8 invited lectures presented by H. G. Bock (IWR Heidelberg) M. Heinkenschloss (Rice University, Houston) K. Kunisch (University of Graz) U. Langer (University Linz) B. Mohammadi (University of Montpellier) J. Petersson (University of Linkoping) E. Sachs (University of Trier) F. Troltzsch (Technical University of Chemnitz) and 28 contributed talks. The aim of this workshop was to foster the exchange of ideas between the still comparatively separated disciplines of nonlinear optimiza tion on the one side and numerical methods for differential equations on the other side. This is necessary for the successful solution of various current optimization problems in practical applications (shape optimization, topology optimization, pro cess optimization . . . ). Therefore the organizing committee as well as the speakers have come from both these communities.
Numerical Methods For Elliptic And Parabolic Partial Differential Equations
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Author : Peter Knabner
language : en
Publisher: Springer Science & Business Media
Release Date : 2003-06-26
Numerical Methods For Elliptic And Parabolic Partial Differential Equations written by Peter Knabner and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-06-26 with Mathematics categories.
This text provides an application oriented introduction to the numerical methods for partial differential equations. It covers finite difference, finite element, and finite volume methods, interweaving theory and applications throughout. The book examines modern topics such as adaptive methods, multilevel methods, and methods for convection-dominated problems and includes detailed illustrations and extensive exercises.
Advanced Numerical Approximation Of Nonlinear Hyperbolic Equations
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Author : B. Cockburn
language : en
Publisher: Springer
Release Date : 2006-11-14
Advanced Numerical Approximation Of Nonlinear Hyperbolic Equations written by B. Cockburn and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.
This volume contains the texts of the four series of lectures presented by B.Cockburn, C.Johnson, C.W. Shu and E.Tadmor at a C.I.M.E. Summer School. It is aimed at providing a comprehensive and up-to-date presentation of numerical methods which are nowadays used to solve nonlinear partial differential equations of hyperbolic type, developing shock discontinuities. The most effective methodologies in the framework of finite elements, finite differences, finite volumes spectral methods and kinetic methods, are addressed, in particular high-order shock capturing techniques, discontinuous Galerkin methods, adaptive techniques based upon a-posteriori error analysis.
Trends In Differential Equations And Applications
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Author : Francisco Ortegón Gallego
language : en
Publisher: Springer
Release Date : 2016-06-09
Trends In Differential Equations And Applications written by Francisco Ortegón Gallego and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-06-09 with Mathematics categories.
This work collects the most important results presented at the Congress on Differential Equations and Applications/Congress on Applied Mathematics (CEDYA/CMA) in Cádiz (Spain) in 2015. It supports further research in differential equations, numerical analysis, mechanics, control and optimization. In particular, it helps readers gain an overview of specific problems of interest in the current mathematical research related to different branches of applied mathematics. This includes the analysis of nonlinear partial differential equations, exact solutions techniques for ordinary differential equations, numerical analysis and numerical simulation of some models arising in experimental sciences and engineering, control and optimization, and also trending topics on numerical linear Algebra, dynamical systems, and applied mathematics for Industry. This volume is mainly addressed to any researcher interested in the applications of mathematics, especially in any subject mentioned above. It may be also useful to PhD students in applied mathematics, engineering or experimental sciences.
Adaptive Finite Elements In Linear And Nonlinear Solid And Structural Mechanics
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Author : Erwin Stein
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-04-02
Adaptive Finite Elements In Linear And Nonlinear Solid And Structural Mechanics written by Erwin Stein and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-04-02 with Computers categories.
This course with 6 lecturers intends to present a systematic survey of recent re search results of well-known scientists on error-controlled adaptive finite element methods in solid and structural mechanics with emphasis to problem-dependent concepts for adaptivity, error analysis as well as h- and p-adaptive refinement techniques including meshing and remeshing. Challenging applications are of equal importance, including elastic and elastoplastic deformations of solids, con tact problems and thin-walled structures. Some major topics should be pointed out, namely: (i) The growing importance of goal-oriented and local error estimates for quan tities of interest—in comparison with global error estimates—based on dual finite element solutions; (a) The importance of the p-version of the finite element method in conjunction with parameter-dependent hierarchical approximations of the mathematical model, for example in boundary layers of elastic plates; (Hi) The choice of problem-oriented error measures in suitable norms, consider ing residual, averaging and hierarchical error estimates in conjunction with the efficiency of the associated adaptive computations; (iv) The importance of implicit local postprocessing with enhanced test spaces in order to get constant-free, i. e. absolute-not only relative-discretizati- error estimates; (v) The coupling of error-controlled adaptive discretizations and the mathemat ical modeling in related subdomains, such as boundary layers. The main goals of adaptivity are reliability and efficiency, combined with in sight and access to controls which are independent of the applied discretization methods. By these efforts, new paradigms in Computational Mechanics should be realized, namely verifications and even validations ofengineering models.