Mathematical Foundations Of Finite Elements And Iterative Solvers

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Mathematical Foundations of Finite Elements and Iterative Solvers

Author : SCI085000
Publisher : SIAM
ISBN 13 : 1611977096
Total Pages : 186 pages
Book Rating : 4.6/5 (119 download)

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Book Synopsis Mathematical Foundations of Finite Elements and Iterative Solvers by : SCI085000

Download or read book Mathematical Foundations of Finite Elements and Iterative Solvers written by SCI085000 and published by SIAM. This book was released on 2022-06-27 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt: “This book combines an updated look, at an advanced level, of the mathematical theory of the finite element method (including some important recent developments), and a presentation of many of the standard iterative methods for the numerical solution of the linear system of equations that results from finite element discretization, including saddle point problems arising from mixed finite element approximation. For the reader with some prior background in the subject, this text clarifies the importance of the essential ideas and provides a deeper understanding of how the basic concepts fit together.” — Richard S. Falk, Rutgers University “Students of applied mathematics, engineering, and science will welcome this insightful and carefully crafted introduction to the mathematics of finite elements and to algorithms for iterative solvers. Concise, descriptive, and entertaining, the text covers all of the key mathematical ideas and concepts dealing with finite element approximations of problems in mechanics and physics governed by partial differential equations while interweaving basic concepts on Sobolev spaces and basic theorems of functional analysis presented in an effective tutorial style.” — J. Tinsley Oden, The University of Texas at Austin This textbook describes the mathematical principles of the finite element method, a technique that turns a (linear) partial differential equation into a discrete linear system, often amenable to fast linear algebra. Reflecting the author’s decade of experience in the field, Mathematical Foundations of Finite Elements and Iterative Solvers examines the crucial interplay between analysis, discretization, and computations in modern numerical analysis; furthermore, it recounts historical developments leading to current state-of-the-art techniques. While self-contained, this textbook provides a clear and in-depth discussion of several topics, including elliptic problems, continuous Galerkin methods, iterative solvers, advection-diffusion problems, and saddle point problems. Accessible to readers with a beginning background in functional analysis and linear algebra, this text can be used in graduate-level courses on advanced numerical analysis, data science, numerical optimization, and approximation theory. Professionals in numerical analysis and finite element methods will also find the book of interest.

Finite Elements and Fast Iterative Solvers

Author : Howard C. Elman
Publisher : Oxford University Press
ISBN 13 : 0199678790
Total Pages : 495 pages
Book Rating : 4.1/5 (996 download)

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Book Synopsis Finite Elements and Fast Iterative Solvers by : Howard C. Elman

Download or read book Finite Elements and Fast Iterative Solvers written by Howard C. Elman and published by Oxford University Press. This book was released on 2014 with total page 495 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a description of why and how to do Scientific Computing for fundamental models of fluid flow. It contains introduction, motivation, analysis, and algorithms and is closely tied to freely available MATLAB codes that implement the methods described. The focus is on finite element approximation methods and fast iterative solution methods for the consequent linear(ized) systems arising in important problems that model incompressible fluid flow. The problems addressed are the Poisson equation, Convection-Diffusion problem, Stokes problem and Navier-Stokes problem, including new material on time-dependent problems and models of multi-physics. The corresponding iterative algebra based on preconditioned Krylov subspace and multigrid techniques is for symmetric and positive definite, nonsymmetric positive definite, symmetric indefinite and nonsymmetric indefinite matrix systems respectively. For each problem and associated solvers there is a description of how to compute together with theoretical analysis that guides the choice of approaches and describes what happens in practice in the many illustrative numerical results throughout the book (computed with the freely downloadable IFISS software). All of the numerical results should be reproducible by readers who have access to MATLAB and there is considerable scope for experimentation in the "computational laboratory" provided by the software. Developments in the field since the first edition was published have been represented in three new chapters covering optimization with PDE constraints (Chapter 5); solution of unsteady Navier-Stokes equations (Chapter 10); solution of models of buoyancy-driven flow (Chapter 11). Each chapter has many theoretical problems and practical computer exercises that involve the use of the IFISS software. This book is suitable as an introduction to iterative linear solvers or more generally as a model of Scientific Computing at an advanced undergraduate or beginning graduate level.

Finite Elements and Fast Iterative Solvers

Author : Howard Elman
Publisher : OUP Oxford
ISBN 13 : 0191667927
Total Pages : 495 pages
Book Rating : 4.1/5 (916 download)

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Book Synopsis Finite Elements and Fast Iterative Solvers by : Howard Elman

Download or read book Finite Elements and Fast Iterative Solvers written by Howard Elman and published by OUP Oxford. This book was released on 2014-06-19 with total page 495 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a description of why and how to do Scientific Computing for fundamental models of fluid flow. It contains introduction, motivation, analysis, and algorithms and is closely tied to freely available MATLAB codes that implement the methods described. The focus is on finite element approximation methods and fast iterative solution methods for the consequent linear(ized) systems arising in important problems that model incompressible fluid flow. The problems addressed are the Poisson equation, Convection-Diffusion problem, Stokes problem and Navier-Stokes problem, including new material on time-dependent problems and models of multi-physics. The corresponding iterative algebra based on preconditioned Krylov subspace and multigrid techniques is for symmetric and positive definite, nonsymmetric positive definite, symmetric indefinite and nonsymmetric indefinite matrix systems respectively. For each problem and associated solvers there is a description of how to compute together with theoretical analysis that guides the choice of approaches and describes what happens in practice in the many illustrative numerical results throughout the book (computed with the freely downloadable IFISS software). All of the numerical results should be reproducible by readers who have access to MATLAB and there is considerable scope for experimentation in the "computational laboratory " provided by the software. Developments in the field since the first edition was published have been represented in three new chapters covering optimization with PDE constraints (Chapter 5); solution of unsteady Navier-Stokes equations (Chapter 10); solution of models of buoyancy-driven flow (Chapter 11). Each chapter has many theoretical problems and practical computer exercises that involve the use of the IFISS software. This book is suitable as an introduction to iterative linear solvers or more generally as a model of Scientific Computing at an advanced undergraduate or beginning graduate level.

The Mathematics of Finite Elements and Applications

Author : J. R. Whiteman
Publisher : Academic Press
ISBN 13 : 1483268845
Total Pages : 535 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis The Mathematics of Finite Elements and Applications by : J. R. Whiteman

Download or read book The Mathematics of Finite Elements and Applications written by J. R. Whiteman and published by Academic Press. This book was released on 2014-05-10 with total page 535 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Mathematics of Finite Elements and Applications provides information pertinent to the mathematics of finite elements, applications, algorithms, and computational techniques. This book discusses the developments in the mathematics of finite elements. Organized into 32 chapters, this book begins with an overview of the basis of the finite element process as a general approximation tool. This text then examines the methods for obtaining bounds on the errors in finite element solutions to two-dimensional elliptic boundary value problems defined on simply connected polygonal regions. Other chapters consider the practical implementation of the Galerkin and the Rayleigh–Ritz methods to equations of importance to physics and engineering. This book discusses as well a fundamental investigation into the problem of convergence in the finite element method. The final chapter deals with an algorithm that is applicable to the analysis of arbitrary plane stress or plane strain configurations. This book is a valuable resource for numerical analysts, mathematical physicist, applied mathematicians, computer scientists, and engineers.

The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations

Author : A. K. Aziz
Publisher :
ISBN 13 :
Total Pages : 797 pages
Book Rating : 4.:/5 (911 download)

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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 . This book was released on 1972 with total page 797 pages. Available in PDF, EPUB and Kindle. Book excerpt:

An Introduction to the Mathematical Theory of Finite Elements

Author : J. T. Oden
Publisher : Courier Corporation
ISBN 13 : 0486142213
Total Pages : 450 pages
Book Rating : 4.4/5 (861 download)

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Book Synopsis An Introduction to the Mathematical Theory of Finite Elements by : J. T. Oden

Download or read book An Introduction to the Mathematical Theory of Finite Elements written by J. T. Oden and published by Courier Corporation. This book was released on 2012-05-23 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introduction to the theory of Sobolev spaces and Hilbert space methods in partial differential equations is geared toward readers of modest mathematical backgrounds. It offers coherent, accessible demonstrations of the use of these techniques in developing the foundations of the theory of finite element approximations. J. T. Oden is Director of the Institute for Computational Engineering & Sciences (ICES) at the University of Texas at Austin, and J. N. Reddy is a Professor of Engineering at Texas A&M University. They developed this essentially self-contained text from their seminars and courses for students with diverse educational backgrounds. Their effective presentation begins with introductory accounts of the theory of distributions, Sobolev spaces, intermediate spaces and duality, the theory of elliptic equations, and variational boundary value problems. The second half of the text explores the theory of finite element interpolation, finite element methods for elliptic equations, and finite element methods for initial boundary value problems. Detailed proofs of the major theorems appear throughout the text, in addition to numerous examples.

Finite Element Solution of Boundary Value Problems

Author : O. Axelsson
Publisher : Academic Press
ISBN 13 : 1483260569
Total Pages : 453 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Finite Element Solution of Boundary Value Problems by : O. Axelsson

Download or read book Finite Element Solution of Boundary Value Problems written by O. Axelsson and published by Academic Press. This book was released on 2014-05-10 with total page 453 pages. Available in PDF, EPUB and Kindle. Book excerpt: Finite Element Solution of Boundary Value Problems: Theory and Computation provides an introduction to both the theoretical and computational aspects of the finite element method for solving boundary value problems for partial differential equations. This book is composed of seven chapters and begins with surveys of the two kinds of preconditioning techniques, one based on the symmetric successive overrelaxation iterative method for solving a system of equations and a form of incomplete factorization. The subsequent chapters deal with the concepts from functional analysis of boundary value problems. These topics are followed by discussions of the Ritz method, which minimizes the quadratic functional associated with a given boundary value problem over some finite-dimensional subspace of the original space of functions. Other chapters are devoted to direct methods, including Gaussian elimination and related methods, for solving a system of linear algebraic equations. The final chapter continues the analysis of preconditioned conjugate gradient methods, concentrating on applications to finite element problems. This chapter also looks into the techniques for reducing rounding errors in the iterative solution of finite element equations. This book will be of value to advanced undergraduates and graduates in the areas of numerical analysis, mathematics, and computer science, as well as for theoretically inclined workers in engineering and the physical sciences.

Finite Elements

Author : Dietrich Braess
Publisher : Cambridge University Press
ISBN 13 : 9780521011952
Total Pages : 374 pages
Book Rating : 4.0/5 (119 download)

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Book Synopsis Finite Elements by : Dietrich Braess

Download or read book Finite Elements written by Dietrich Braess and published by Cambridge University Press. This book was released on 2001-04-12 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This definitive introduction to finite element methods has been updated thoroughly for this third edition, which features important new material for both research and application of the finite element method. The discussion of saddle point problems is a highlight of the book and has been elaborated to include many more non-standard applications. The chapter on applications in elasticity now contains a complete discussion of locking phenomena." "Graduate students who do not necessarily have any particular background in differential equations, but require an introduction to finite element methods, will find the text invaluable. Specifically, the chapter on finite elements in solid mechanics provides a bridge between mathematics and engineering."--BOOK JACKET.

The Mathematical Theory of Finite Element Methods

Author : Susanne Brenner
Publisher : Springer Science & Business Media
ISBN 13 : 1475736584
Total Pages : 369 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis The Mathematical Theory of Finite Element Methods by : Susanne Brenner

Download or read book The Mathematical Theory of Finite Element Methods written by Susanne Brenner and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 369 pages. Available in PDF, EPUB and Kindle. Book excerpt: A rigorous and thorough mathematical introduction to the subject; A clear and concise treatment of modern fast solution techniques such as multigrid and domain decomposition algorithms; Second edition contains two new chapters, as well as many new exercises; Previous edition sold over 3000 copies worldwide

The Finite Element Method

Author : Patrick Ciarlet
Publisher : John Wiley & Sons
ISBN 13 : 1786307685
Total Pages : 404 pages
Book Rating : 4.7/5 (863 download)

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Book Synopsis The Finite Element Method by : Patrick Ciarlet

Download or read book The Finite Element Method written by Patrick Ciarlet and published by John Wiley & Sons. This book was released on 2023-08-29 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: The finite element method, which emerged in the 1950s to deal with structural mechanics problems, has since undergone continuous development. Using partial differential equation models, it is now present in such fields of application as mechanics, physics, chemistry, economics, finance and biology. It is also used in most scientific computing software, and many engineers become adept at using it in their modeling and numerical simulation activities. This book presents all the essential elements of the finite element method in a progressive and didactic way: the theoretical foundations, practical considerations of implementation, algorithms, as well as numerical illustrations created in MATLAB. Original exercises with detailed answers are provided at the end of each chapter.

Advanced Finite Element Methods with Applications

Author : Thomas Apel
Publisher : Springer
ISBN 13 : 3030142442
Total Pages : 428 pages
Book Rating : 4.0/5 (31 download)

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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.

The Finite Element Method for Elliptic Problems

Author : Philippe G. Ciarlet
Publisher : SIAM
ISBN 13 : 9780898719208
Total Pages : 553 pages
Book Rating : 4.7/5 (192 download)

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Book Synopsis The Finite Element Method for Elliptic Problems by : Philippe G. Ciarlet

Download or read book The Finite Element Method for Elliptic Problems written by Philippe G. Ciarlet and published by SIAM. This book was released on 2002-01-01 with total page 553 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Finite Element Method for Elliptic Problems is the only book available that analyzes in depth the mathematical foundations of the finite element method. It is a valuable reference and introduction to current research on the numerical analysis of the finite element method, as well as a working textbook for graduate courses in numerical analysis. It includes many useful figures, and there are many exercises of varying difficulty. Although nearly 25 years have passed since this book was first published, the majority of its content remains up-to-date. Chapters 1 through 6, which cover the basic error estimates for elliptic problems, are still the best available sources for material on this topic. The material covered in Chapters 7 and 8, however, has undergone considerable progress in terms of new applications of the finite element method; therefore, the author provides, in the Preface to the Classics Edition, a bibliography of recent texts that complement the classic material in these chapters. Audience: this book is particularly useful to graduate students, researchers, and engineers using finite element methods. The reader should have knowledge of analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces. Other than these basics, the book is mathematically self-contained.

Mathematical Theory of Finite Elements

Author : Leszek F. Demkowicz
Publisher : SIAM
ISBN 13 : 1611977738
Total Pages : 217 pages
Book Rating : 4.6/5 (119 download)

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Book Synopsis Mathematical Theory of Finite Elements by : Leszek F. Demkowicz

Download or read book Mathematical Theory of Finite Elements written by Leszek F. Demkowicz and published by SIAM. This book was released on 2023-09-22 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book discusses the foundations of the mathematical theory of finite element methods. The focus is on two subjects: the concept of discrete stability, and the theory of conforming elements forming the exact sequence. Both coercive and noncoercive problems are discussed.. Following the historical path of development, the author covers the Ritz and Galerkin methods to Mikhlin’s theory, followed by the Lax–Milgram theorem and Cea’s lemma to the Babuska theorem and Brezzi’s theory. He finishes with an introduction to the discontinuous Petrov–Galerkin (DPG) method with optimal test functions. Based on the author’s personal lecture notes for a popular version of his graduate course on mathematical theory of finite elements, the book includes a unique exposition of the concept of discrete stability and the means to guarantee it, a coherent presentation of finite elements forming the exact grad-curl-div sequence, and an introduction to the DPG method. Intended for graduate students in computational science, engineering, and mathematics programs, Mathematical Theory of Finite Elements is also appropriate for graduate mathematics and mathematically oriented engineering students. Instructors will find the book useful for courses in real analysis, functional analysis, energy (Sobolev) spaces, and Hilbert space methods for PDEs.

Finite Element Methods for Engineering Sciences

Author : Joel Chaskalovic
Publisher : Springer Science & Business Media
ISBN 13 : 3540763430
Total Pages : 261 pages
Book Rating : 4.5/5 (47 download)

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Book Synopsis Finite Element Methods for Engineering Sciences by : Joel Chaskalovic

Download or read book Finite Element Methods for Engineering Sciences written by Joel Chaskalovic and published by Springer Science & Business Media. This book was released on 2008-09-16 with total page 261 pages. Available in PDF, EPUB and Kindle. Book excerpt: This self-tutorial offers a concise yet thorough grounding in the mathematics necessary for successfully applying FEMs to practical problems in science and engineering. The unique approach first summarizes and outlines the finite-element mathematics in general and then, in the second and major part, formulates problem examples that clearly demonstrate the techniques of functional analysis via numerous and diverse exercises. The solutions of the problems are given directly afterwards. Using this approach, the author motivates and encourages the reader to actively acquire the knowledge of finite-element methods instead of passively absorbing the material, as in most standard textbooks. The enlarged English-language edition, based on the original French, also contains a chapter on the approximation steps derived from the description of nature with differential equations and then applied to the specific model to be used. Furthermore, an introduction to tensor calculus using distribution theory offers further insight for readers with different mathematical backgrounds.

Finite Element Methods and Their Applications

Author : Zhangxin Chen
Publisher : Springer Science & Business Media
ISBN 13 : 3540240780
Total Pages : 415 pages
Book Rating : 4.5/5 (42 download)

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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.

The Mathematical Basis of Finite Element Methods with Applications to Partial Differential Equations

Author : Institute of Mathematics and Its Applications
Publisher : Oxford University Press, USA
ISBN 13 :
Total Pages : 208 pages
Book Rating : 4.:/5 (45 download)

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Book Synopsis The Mathematical Basis of Finite Element Methods with Applications to Partial Differential Equations by : Institute of Mathematics and Its Applications

Download or read book The Mathematical Basis of Finite Element Methods with Applications to Partial Differential Equations written by Institute of Mathematics and Its Applications and published by Oxford University Press, USA. This book was released on 1984 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: Combining theoretical insights with practical applications, this stimulating collection provides a state-of-the-art survey of the finite element method, one of the most powerful tools available for the solution of physical problems. Written by leading experts, this volume consider such topics as parabolic Galerkin methods, nonconforming elements, the treatment of singularities in elliptic boundary value problems, and conforming methods for self-adjount elliptic problems. This will be an invaluable basic reference for computational mathematicians and engineers who use finite element methods in academic or industrial research.

Advanced Finite Element Methods and Applications

Author : Thomas Apel
Publisher : Springer Science & Business Media
ISBN 13 : 3642303161
Total Pages : 380 pages
Book Rating : 4.6/5 (423 download)

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Book Synopsis Advanced Finite Element Methods and Applications by : Thomas Apel

Download or read book Advanced Finite Element Methods and Applications written by Thomas Apel and published by Springer Science & Business Media. This book was released on 2012-07-16 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume on some recent aspects of finite element methods and their applications is dedicated to Ulrich Langer and Arnd Meyer on the occasion of their 60th birthdays in 2012. Their work combines the numerical analysis of finite element algorithms, their efficient implementation on state of the art hardware architectures, and the collaboration with engineers and practitioners. In this spirit, this volume contains contributions of former students and collaborators indicating the broad range of their interests in the theory and application of finite element methods. Topics cover the analysis of domain decomposition and multilevel methods, including hp finite elements, hybrid discontinuous Galerkin methods, and the coupling of finite and boundary element methods; the efficient solution of eigenvalue problems related to partial differential equations with applications in electrical engineering and optics; and the solution of direct and inverse field problems in solid mechanics.