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A Posteriori Error Estimation For Standard Finite Element Analysis
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Book Synopsis A Posteriori Error Estimation in Finite Element Analysis by : Mark Ainsworth
Download or read book A Posteriori Error Estimation in Finite Element Analysis written by Mark Ainsworth and published by John Wiley & Sons. This book was released on 2011-09-28 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: An up-to-date, one-stop reference-complete with applications This volume presents the most up-to-date information available on aposteriori error estimation for finite element approximation inmechanics and mathematics. It emphasizes methods for ellipticboundary value problems and includes applications to incompressibleflow and nonlinear problems. Recent years have seen an explosion in the study of a posteriorierror estimators due to their remarkable influence on improvingboth accuracy and reliability in scientific computing. In an effortto provide an accessible source, the authors have sought to presentkey ideas and common principles on a sound mathematicalfooting. Topics covered in this timely reference include: * Implicit and explicit a posteriori error estimators * Recovery-based error estimators * Estimators, indicators, and hierarchic bases * The equilibrated residual method * Methodology for the comparison of estimators * Estimation of errors in quantities of interest A Posteriori Error Estimation in Finite Element Analysis is a lucidand convenient resource for researchers in almost any field offinite element methods, and for applied mathematicians andengineers who have an interest in error estimation and/or finiteelements.
Book Synopsis A Posteriori Error Estimation for Standard Finite Element Analysis by : Pedro Díez
Download or read book A Posteriori Error Estimation for Standard Finite Element Analysis written by Pedro Díez and published by . This book was released on 1995 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Posteriori Error Estimation Techniques for Finite Element Methods by : Rüdiger Verfürth
Download or read book A Posteriori Error Estimation Techniques for Finite Element Methods written by Rüdiger Verfürth and published by OUP Oxford. This book was released on 2013-04-18 with total page 573 pages. Available in PDF, EPUB and Kindle. Book excerpt: Self-adaptive discretization methods are now an indispensable tool for the numerical solution of partial differential equations that arise from physical and technical applications. The aim is to obtain a numerical solution within a prescribed tolerance using a minimal amount of work. The main tools in achieving this goal are a posteriori error estimates which give global and local information on the error of the numerical solution and which can easily be computed from the given numerical solution and the data of the differential equation. This book reviews the most frequently used a posteriori error estimation techniques and applies them to a broad class of linear and nonlinear elliptic and parabolic equations. Although there are various approaches to adaptivity and a posteriori error estimation, they are all based on a few common principles. The main aim of the book is to elaborate these basic principles and to give guidelines for developing adaptive schemes for new problems. Chapters 1 and 2 are quite elementary and present various error indicators and their use for mesh adaptation in the framework of a simple model problem. The basic principles are introduced using a minimal amount of notations and techniques providing a complete overview for the non-specialist. Chapters 4-6 on the other hand are more advanced and present a posteriori error estimates within a general framework using the technical tools collected in Chapter 3. Most sections close with a bibliographical remark which indicates the historical development and hints at further results.
Book Synopsis A posteriori error estimation in finite element analysis by : Mark Ainsworth
Download or read book A posteriori error estimation in finite element analysis written by Mark Ainsworth and published by . This book was released on 1996 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Advances in a Posteriori Error Estimation on Anisotropic Finite Element Discretizations by : Gerd Kunert
Download or read book Advances in a Posteriori Error Estimation on Anisotropic Finite Element Discretizations written by Gerd Kunert and published by Logos Verlag Berlin. This book was released on 2003 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Certain classes of partial differential equations generically give rise to solutions with strong directional features, e.g. with boundary layers. Such solutions are called anisotropic. Their discretization by means of the finite element method (for example) can favourably employ so-called anisotropic meshes. These meshes are characterized by stretched, anisotropic finite elements with a (very) large stretching ratio. The widespread use of computer simulation leads to an increasing demand for semi- or fully automatic solution procedures. Within such self-adaptive algorithms, a posteriori error estimators form an indispensable ingredient for quality control. They are well understood for standard, isotropic discretizations. The knowledge about a posteriori error estimation on anisotropic meshes is much less mature. During the last decade the foundation and basic principles have been proposed, discussed and established, mostly for the Poisson problem. This monograph summarises some of the recent advances in anisotropic error estimation for more challenging problems. Emphasis is given to the contributions of the author. In Chapter 3 the investigation starts with singularly perturbed reaction diffusion problems which frequently lead to solutions with boundary layers. This problem class often arises when simplifying more complex models. Chapter 4 treats singularly perturbed convection diffusion problems, i.e. the convection is dominating. The solution structure is more intricate, and often features boundary layer and/or interior layer solutions. Chapter 5 is devoted to the Stokes equations. Flow problems generically give rise to anisotropic solutions (e.g. with edge singularities or containing layers). The Stokes equations often serve as a simplified or linearised model. In all three chapters, the main results consist in error estimators and corresponding error bounds that are robust with respect to the mesh anisotropy, as far as possible. Finally Chapter 6 addresses the robustness of a posteriori error estimation with respect to the mesh anisotropy.In particular the relation between anisotropic mesh construction and error estimation is investigated. This thesis presents the philosophy of anisotropic error estimation as well as the main results and the definitions required. Proofs and technical details are omitted; instead the key ideas are explained.The compact style of presentation aims at practitioners in particular by providing easily accessible error estimators and error bounds. Further insight is readily possible through the references.
Book Synopsis A Posteriori Error Estimation in the Finite Element Method by : Mark Ainsworth
Download or read book A Posteriori Error Estimation in the Finite Element Method written by Mark Ainsworth and published by . This book was released on 1989 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Posteriori Error Estimation for the Finite Element Method Via Local Averaging by : Varis Carey
Download or read book A Posteriori Error Estimation for the Finite Element Method Via Local Averaging written by Varis Carey and published by . This book was released on 2005 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A-posteriori error estimation for finite element and generalized finite element method by : Ivo Babuska
Download or read book A-posteriori error estimation for finite element and generalized finite element method written by Ivo Babuska and published by . This book was released on 1998 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Feedback Finite Element Method and Some Basic Properties of the A Posteriori Error Estimation by : Ivo Babuska
Download or read book A Feedback Finite Element Method and Some Basic Properties of the A Posteriori Error Estimation written by Ivo Babuska and published by . This book was released on 1984 with total page 68 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Posteriori Error Estimation Techniques for Finite Element Methods by : Rüdiger Verfürth
Download or read book A Posteriori Error Estimation Techniques for Finite Element Methods written by Rüdiger Verfürth and published by Oxford University Press. This book was released on 2013-04-18 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Refined a Posteriori Error Estimation for Classical and Pressure-robust Stokes Finite Element Methods by : Philip Lukas Lederer
Download or read book Refined a Posteriori Error Estimation for Classical and Pressure-robust Stokes Finite Element Methods written by Philip Lukas Lederer and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure-independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the L2-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier-Stokes equations. The novel error estimators only take the curl of the righthand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressure-dominant situations. This is also confirmed by some numerical examples with the novel pressure-robust modifications of the Taylor-Hood and mini finite element methods.
Download or read book Finite Elements written by Ivo Babuska and published by . This book was released on 2010-11-04 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: Most of the many books on finite elements are devoted either to mathematical theory or to engineering applications, but not to both. This book presents computed numbers which not only illustrate the theory but can only be analysed using the theory. This approach, both dual and interacting between theory and computation makes this book unique.
Book Synopsis A-posteriori Error Estimates for the Finite Element Method by : University of Maryland at College Park. Computer Science Center
Download or read book A-posteriori Error Estimates for the Finite Element Method written by University of Maryland at College Park. Computer Science Center and published by . This book was released on 1977 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Multiscale a Posteriori Error Estimation and Mesh Adaptivity for Reliable Finite Element Analysis by : H. Ahmed Elsheikh
Download or read book Multiscale a Posteriori Error Estimation and Mesh Adaptivity for Reliable Finite Element Analysis written by H. Ahmed Elsheikh and published by . This book was released on 2007 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Posteriori Error Estimates for Surface Finite Element Methods by : Fernando Faustino Camacho
Download or read book A Posteriori Error Estimates for Surface Finite Element Methods written by Fernando Faustino Camacho and published by . This book was released on 2014 with total page 62 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Posteriori Error Estimation for Finite Element Approximations of Fractional Laplacian Problems and Applications to Poro-elasticity by : Raphaël Bulle
Download or read book A Posteriori Error Estimation for Finite Element Approximations of Fractional Laplacian Problems and Applications to Poro-elasticity written by Raphaël Bulle and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This manuscript is concerned with a posteriori error estimation for the finiteelement discretization of standard and fractional partial differential equationsas well as an application of fractional calculus to the modeling of thehuman meniscus by poro-elasticity equations.In the introduction, we give an overview of the literature about a posteriori errorestimation for finite element methods and about adaptive mesh refinement methods.We also review the literature about fractional partial differential equationsand Caputo's fractional derivative with anomalous diffusion applications.We emphasize on the state-of-the-art of the Bank-Weiser estimator and of aposteriori error estimation for the spectral fractional Laplacian.The rest of the manuscript is organized as follows.The Chapter 1 is concerned with a proof of the reliability of theBank-Weiser estimator for three-dimensional problems discretized with linearLagrange finite elements. This result is an extension of a previous result fromthe literature.In Chapter 2 we present a numerical study of the Bank-Weiserestimator.We provide a novel implementation of the estimator in the FEniCS finiteelement software and working in parallel.We apply our code to a variety of elliptic equations, several differenttwo-dimensional Poisson problems and a three-dimensional linear elasticityproblem.In particular, we use our implementation into an adaptive mesh refinement method anda goal-oriented error estimation method.In addition we provide convergence studies for these methods as well as atimescale study of our error estimation method when performed in parallel.In Chapter 3 we derive a novel a posteriori estimator for theL2 error induced by the finite element discretization of the fractionalLaplacian operator.We provide an implementation of our method in the FEniCS finite elementsoftware.We apply our estimator to an adaptive refinement method for two-dimensional andthree-dimensional fractional Poisson equations.In addition, we provide numerical results on the convergence of this method.In Chapter 4 we present new theoretical results on theconvergence of a rational approximation method with consequences on theapproximation of fractional norms and a priori error estimation of the semi-discretization of the spectral fractional Laplacian.Finally, in Chapter 5 we provide an application of fractionalcalculus to the study of the human meniscus via poro-elasticity equations and the Caputo derivative.
Book Synopsis A Posteriori Error Estimates for the Finite Element Method by : Erwin Pier-Ribbert
Download or read book A Posteriori Error Estimates for the Finite Element Method written by Erwin Pier-Ribbert and published by . This book was released on 1989 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt:
