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Adaptive Finite Element Methods For Parabolic Partial Differential Equations
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Book Synopsis Adaptive Finite Element Methods for Parabolic Partial Differential Equations by : J. E. Flaherty
Download or read book Adaptive Finite Element Methods for Parabolic Partial Differential Equations written by J. E. Flaherty and published by . This book was released on 1983 with total page 23 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors discuss a finite element method for solving initial-boundary value problems for vector systems of partial differential equations in one space dimension and time. The method automatically adjusts the computational mesh as the solution evolves in time so as to approximately minimize the local discretization error. They are thus able to calculate accurate solutions with fewer elements than would be necessary with a uniform mesh. This overall method contains two distinct steps: a solution step and a mesh selection step. They solve the partial differential equations using a finite element-Galerkin method on trapezoidal space-time-elements with either piecewise linear or cubic Hermits polynomial approximations. A variety of mesh selection strategies are discussed and analyzed. Results are presented for several computational examples.
Book Synopsis Adaptive Finite Elements in the Discretization of Parabolic Problems by : Christian A. Möller
Download or read book Adaptive Finite Elements in the Discretization of Parabolic Problems written by Christian A. Möller and published by Logos Verlag Berlin GmbH. This book was released on 2011 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis A Local Adaptive Finite Element Method for Solving One- and Two-dimensional Systems of Parabolic Partial Differential Equations by : Peter Kimball Moore
Download or read book A Local Adaptive Finite Element Method for Solving One- and Two-dimensional Systems of Parabolic Partial Differential Equations written by Peter Kimball Moore and published by . This book was released on 1988 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Galerkin Finite Element Methods for Parabolic Problems by : Vidar Thomee
Download or read book Galerkin Finite Element Methods for Parabolic Problems written by Vidar Thomee and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Adaptive Finite Element Methods for Parabolic Systems in One- and Two-Space Dimensions by : Slimane Adjerid
Download or read book Adaptive Finite Element Methods for Parabolic Systems in One- and Two-Space Dimensions written by Slimane Adjerid and published by . This book was released on 1987 with total page 39 pages. Available in PDF, EPUB and Kindle. Book excerpt: Adaptive finite element methods are given for solving initial boundary value problems for vector systems of parabolic partial differential equations in one- and two-space dimensions. One-dimension systems are discretized using piecewise linear finite element approximations in space and a backward difference code for stiff ordinary differential systems in time. A spatial error estimate is calculated using piecewise quadratic approximations that employ nodal superconvergence to increase computational efficiency. This error estimate is used to move and refine the finite element mesh in order to equidistribute a measure of the total spatial error and to satisfy a prescribed error tolerance. Ordinary differential equations for the spatial error estimate and the mesh motion are integrated in time using the same backward difference software that is used to determine the finite element solution. Two-dimension systems are discretized using piecewise bilinear finite element approximations in space and backward difference software in time. A spatial error estimate is calculated using piecewise cubic approximations that take advantage of nodal superconvergence. This error estimate is used to locally refine a stationary finite element mesh in order to satisfy a prescribed spatial error tolerance.
Book Synopsis Finite Element Methods and Their Applications by : Zhangxin Chen
Download or read book Finite Element Methods and Their Applications written by Zhangxin Chen and published by Springer Science & Business Media. This book was released on 2005-06-23 with total page 415 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduce every concept in the simplest setting and to maintain a level of treatment that is as rigorous as possible without being unnecessarily abstract. Contains unique recent developments of various finite elements such as nonconforming, mixed, discontinuous, characteristic, and adaptive finite elements, along with their applications. Describes unique recent applications of finite element methods to important fields such as multiphase flows in porous media and semiconductor modelling. Treats the three major types of partial differential equations, i.e., elliptic, parabolic, and hyperbolic equations.
Book Synopsis A Local Adaptive Finite Element Method for Solving One- and Two-dimensional Systems of Parabolic Partial Differential Equations by : Peter K. Moore
Download or read book A Local Adaptive Finite Element Method for Solving One- and Two-dimensional Systems of Parabolic Partial Differential Equations written by Peter K. Moore and published by . This book was released on 1988 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems by : Jens Lang
Download or read book Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems written by Jens Lang and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nowadays there is an increasing emphasis on all aspects of adaptively gener ating a grid that evolves with the solution of a PDE. Another challenge is to develop efficient higher-order one-step integration methods which can handle very stiff equations and which allow us to accommodate a spatial grid in each time step without any specific difficulties. In this monograph a combination of both error-controlled grid refinement and one-step methods of Rosenbrock-type is presented. It is my intention to impart the beauty and complexity found in the theoretical investigation of the adaptive algorithm proposed here, in its realization and in solving non-trivial complex problems. I hope that this method will find many more interesting applications. Berlin-Dahlem, May 2000 Jens Lang Acknowledgements I have looked forward to writing this section since it is a pleasure for me to thank all friends who made this work possible and provided valuable input. I would like to express my gratitude to Peter Deuflhard for giving me the oppor tunity to work in the field of Scientific Computing. I have benefited immensly from his help to get the right perspectives, and from his continuous encourage ment and support over several years. He certainly will forgive me the use of Rosenbrock methods rather than extrapolation methods to integrate in time.
Book Synopsis Adaptive Finite Element Methods for Differential Equations by : Wolfgang Bangerth
Download or read book Adaptive Finite Element Methods for Differential Equations written by Wolfgang Bangerth and published by Birkhäuser. This book was released on 2013-11-11 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: These Lecture Notes have been compiled from the material presented by the second author in a lecture series ('Nachdiplomvorlesung') at the Department of Mathematics of the ETH Zurich during the summer term 2002. Concepts of 'self adaptivity' in the numerical solution of differential equations are discussed with emphasis on Galerkin finite element methods. The key issues are a posteriori er ror estimation and automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method (or shortly D WR method) for goal-oriented error estimation is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. 'Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. The basics of the DWR method and various of its applications are described in the following survey articles: R. Rannacher [114], Error control in finite element computations. In: Proc. of Summer School Error Control and Adaptivity in Scientific Computing (H. Bulgak and C. Zenger, eds), pp. 247-278. Kluwer Academic Publishers, 1998. M. Braack and R. Rannacher [42], Adaptive finite element methods for low Mach-number flows with chemical reactions.
Book Synopsis Adaptive Computational Methods for Partial Differential Equations by : Ivo Babushka
Download or read book Adaptive Computational Methods for Partial Differential Equations written by Ivo Babushka and published by SIAM. This book was released on 1983-01-01 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: List of participants; Elliptic equations; Parabolic equations; Hyperbolic equations.
Book Synopsis An Adaptive Local Hpr-refinement Finite Element Method for Parabolic Partial Differential Equations by : Yun J. Wang
Download or read book An Adaptive Local Hpr-refinement Finite Element Method for Parabolic Partial Differential Equations written by Yun J. Wang and published by . This book was released on 1991 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Adaptive Finite Element Method I: Solution Algorithm and Computational Examples by :
Download or read book Adaptive Finite Element Method I: Solution Algorithm and Computational Examples written by and published by . This book was released on 1994 with total page 57 pages. Available in PDF, EPUB and Kindle. Book excerpt: An adaptive finite element method is developed to solve initial boundary value problems for vector systems of parabolic partial differential equations in one space dimension and time. The differential equations are discretized in space using piecewise linear finite element approximations. Superconvergence properties and quadratic polynomials are used to derive a computation ally inexpensive approximation to the spatial component of the error. This technique is coupled with time integration schemes of successively higher orders to obtain an approximation of the temporal and total discretization errors. These approximate errors are used to control an adaptive mesh refinement strategy. Refinement is performed in space, time, or both space and time depending on the dominant component of the error estimate. A computer code coupling this refinement strategy and stable mesh movement has been written and applied to a number of problems. These computations confirm that proper mesh movement can reduce the computational efforts associated with mesh refinement.
Book Synopsis Galerkin Finite Element Methods for Parabolic Problems by : Vidar Thomée
Download or read book Galerkin Finite Element Methods for Parabolic Problems written by Vidar Thomée and published by Springer Science & Business Media. This book was released on 2010 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Advanced Finite Element Methods with Applications by : Thomas Apel
Download or read book Advanced Finite Element Methods with Applications written by Thomas Apel and published by Springer. This book was released on 2019-06-28 with total page 428 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Adaptive Refinement Methods for Nonlinear Parabolic Partial Differential Equations by : M. Bietermman
Download or read book Adaptive Refinement Methods for Nonlinear Parabolic Partial Differential Equations written by M. Bietermman and published by . This book was released on 1984 with total page 35 pages. Available in PDF, EPUB and Kindle. Book excerpt: This document considers two adaptive finite element techniques for parabolic partial differential equations (PDEs) that are based on using error estimates to control mesh refinement. One technique is a method of lines approach that uses a Galerkin method to discretize the PDEs in space and implicit multi-step integration in time. Spatial elements are added and deleted in regions of high and low error and are all advanced with the same sequence of varying time steps. The second technique is a local refinement method that uses Galerkin approximations in both space and time. Fine grids of space-time elements are added to coarser grids and the problem is recursively solved in regions of high error. (Author).
Book Synopsis Least-Squares Finite Element Methods by : Pavel B. Bochev
Download or read book Least-Squares Finite Element Methods written by Pavel B. Bochev and published by Springer Science & Business Media. This book was released on 2009-04-28 with total page 669 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since their emergence, finite element methods have taken a place as one of the most versatile and powerful methodologies for the approximate numerical solution of Partial Differential Equations. These methods are used in incompressible fluid flow, heat, transfer, and other problems. This book provides researchers and practitioners with a concise guide to the theory and practice of least-square finite element methods, their strengths and weaknesses, established successes, and open problems.
Book Synopsis The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations by : A. K. Aziz
Download or read book The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations written by A. K. Aziz and published by Academic Press. This book was released on 2014-05-10 with total page 814 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations is a collection of papers presented at the 1972 Symposium by the same title, held at the University of Maryland, Baltimore County Campus. This symposium relates considerable numerical analysis involved in research in both theoretical and practical aspects of the finite element method. This text is organized into three parts encompassing 34 chapters. Part I focuses on the mathematical foundations of the finite element method, including papers on theory of approximation, variational principles, the problems of perturbations, and the eigenvalue problem. Part II covers a large number of important results of both a theoretical and a practical nature. This part discusses the piecewise analytic interpolation and approximation of triangulated polygons; the Patch test for convergence of finite elements; solutions for Dirichlet problems; variational crimes in the field; and superconvergence result for the approximate solution of the heat equation by a collocation method. Part III explores the many practical aspects of finite element method. This book will be of great value to mathematicians, engineers, and physicists.
