An Analysis of a Finite Element Method for Convection-Diffusion Problems. Part II. A Posteriori Error Estimates and Adaptivity PDF Download

Author: W. G. Szymczak
Publisher:
ISBN:
Category :
Languages : en
Pages : 51

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Book Description
A posteriori error estimates are derived for the finite element method presented in Part I. These estimates are proven to have the property that the effectivity index theta = (error estimate/true error) converges to one as the maximum mesh size goes to zero. An adaptive mesh refinement strategy is based on equilibriating local error indicators whose sum comprises the global error estimate. Numerical results show that theta is nearly one even on coarse meshes, and that optimal meshes are created by the adaptive procedure. The successful solution of a non linear problem-modelling flow through an expanding duct, makes evident the robustness of the method. (Author).

Author: W. G. Szymczak
Publisher:
ISBN:
Category :
Languages : en
Pages : 61

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A detailed analysis is performed for a finite element method applied to the general one-dimensional convection diffusion problem. Piecewise polynomials are used for the trial space. The test space is formed by locally projecting L-spline basis functions onto upwinded polynomials. The error is measured in the LP mesh dependent norm. The method is proven to be quasi-optimal (yielding nearly the best approximation from the trial space), provided that the input data is piecewise smooth. This assumption is usually observed in practice. These results are used to establish a posteriori error estimates and an adaptive mesh refinement strategy in Part II of this series (35). (Author).

Author: Rüdiger Verfürth
Publisher: Oxford University Press
ISBN: 0199679428
Category : Mathematics
Languages : en
Pages : 414

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Book Description
A posteriori error estimation techniques are fundamental to the efficient numerical solution of PDEs arising in physical and technical applications. This book gives a unified approach to these techniques and guides graduate students, researchers, and practitioners towards understanding, applying and developing self-adaptive discretization methods.

Author: Johannes Neher
Publisher: Logos Verlag Berlin GmbH
ISBN: 3832530886
Category : Mathematics
Languages : en
Pages : 106

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There is a variety of finite element based methods applicable to the discretization of second order elliptic boundary value problems in mixed form. However, it is expensive to solve the resulting discrete linear system due to its size and its algebraic structure. Hybridization serves as a tool to circumvent these difficulties. Furthermore hybridization is an elegant concept to establish connections among various finite element methods. In this work connections between the methods and their hybridized counterparts are established after showing the link between three different formulations of the elliptic model problem. The main part of the work contains the development of a reliable a posteriori error estimator, which is applicable to all of the methods above. This estimator is the key ingredient of an adaptive numerical approximation of the original boundary value problem. Finally, a number of numerical tests is discussed in order to exhibit the performance of the adaptive hybridized methods.

Author:
Publisher:
ISBN:
Category : Aeronautics
Languages : en
Pages : 1572

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Author: Sergey I. Repin
Publisher: Walter de Gruyter
ISBN: 3110203049
Category : Mathematics
Languages : en
Pages : 329

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This book deals with the reliable verification of the accuracy of approximate solutions which is one of the central problems in modern applied analysis. After giving an overview of the methods developed for models based on partial differential equations, the author derives computable a posteriori error estimates by using methods of the theory of partial differential equations and functional analysis. These estimates are applicable to approximate solutions computed by various methods.

Author: Johannes Kraus
Publisher: Walter de Gruyter
ISBN: 3110927098
Category : Mathematics
Languages : en
Pages : 241

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Book Description
This book contains four survey papers related to different topics in computational mechanics, in particular (1) novel discretization and solver techniques in mechanics and (2) inverse, control, and optimization problems in mechanics. These topics were considered in lectures, seminars, tutorials, and workshops at the Special Semester on Computational Mechanics held at the Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, in December 2005.

Author: Markus Kirkilionis
Publisher: Springer Science & Business Media
ISBN: 3662052814
Category : Mathematics
Languages : en
Pages : 427

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Book Description
Applied mathematics is a central connecting link between scientific observations and their theoretical interpretation. Nonlinear analysis has surely contributed major developments which nowadays shape the face of applied mathematics. At the beginning of the millennium, all sciences are expanding at increased speed. Technological, ecological, economical and medical problem solving is a central issue of every modern society. Mathematical models help to expose fundamental structures hidden in these problems and serve as unifying tools to deepen our understanding. What are the new challenges applied mathematics has to face with the increased diversity of scientific problems? In which direction should the classical tools of nonlinear analysis be developed further? How do new available technologies influence the development of the field? How can problems be solved which have been beyond reach in former times? It is the aim of this book to explore new developments in the field by way of discussion of selected topics from nonlinear analysis.

Author: Hans-Joachim Bungartz
Publisher: Springer Nature
ISBN: 3030813622
Category : Mathematics
Languages : en
Pages : 268

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Sparse grids are a popular tool for the numerical treatment of high-dimensional problems. Where classical numerical discretization schemes fail in more than three or four dimensions, sparse grids, in their different flavors, are frequently the method of choice. This volume of LNCSE presents selected papers from the proceedings of the fifth workshop on sparse grids and applications, and demonstrates once again the importance of this numerical discretization scheme. The articles present recent advances in the numerical analysis of sparse grids in connection with a range of applications including uncertainty quantification, plasma physics simulations, and computational chemistry, to name but a few.

Author: Wolfgang Bangerth
Publisher: Springer Science & Business Media
ISBN: 9783764370091
Category : Mathematics
Languages : en
Pages : 222

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Book Description
The key issues are a posteriori error estimation and it automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual 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. At the end of each chapter some exercises are posed in order to assist the interested reader in better understanding the concepts presented. Solutions and accompanying remarks are given in the Appendix.