Partial Differential Equations And The Finite Element Method

Author : Pavel Ŝolín
Publisher : John Wiley & Sons
Page : 505 pages
File Size : 49,5 Mb
Release : 2005-12-16
Category : Mathematics
ISBN : 9780471764090

Get Book

Partial Differential Equations and the Finite Element Method by Pavel Ŝolín Pdf

A systematic introduction to partial differential equations and modern finite element methods for their efficient numerical solution Partial Differential Equations and the Finite Element Method provides a much-needed, clear, and systematic introduction to modern theory of partial differential equations (PDEs) and finite element methods (FEM). Both nodal and hierachic concepts of the FEM are examined. Reflecting the growing complexity and multiscale nature of current engineering and scientific problems, the author emphasizes higher-order finite element methods such as the spectral or hp-FEM. A solid introduction to the theory of PDEs and FEM contained in Chapters 1-4 serves as the core and foundation of the publication. Chapter 5 is devoted to modern higher-order methods for the numerical solution of ordinary differential equations (ODEs) that arise in the semidiscretization of time-dependent PDEs by the Method of Lines (MOL). Chapter 6 discusses fourth-order PDEs rooted in the bending of elastic beams and plates and approximates their solution by means of higher-order Hermite and Argyris elements. Finally, Chapter 7 introduces the reader to various PDEs governing computational electromagnetics and describes their finite element approximation, including modern higher-order edge elements for Maxwell's equations. The understanding of many theoretical and practical aspects of both PDEs and FEM requires a solid knowledge of linear algebra and elementary functional analysis, such as functions and linear operators in the Lebesgue, Hilbert, and Sobolev spaces. These topics are discussed with the help of many illustrative examples in Appendix A, which is provided as a service for those readers who need to gain the necessary background or require a refresher tutorial. Appendix B presents several finite element computations rooted in practical engineering problems and demonstrates the benefits of using higher-order FEM. Numerous finite element algorithms are written out in detail alongside implementation discussions. Exercises, including many that involve programming the FEM, are designed to assist the reader in solving typical problems in engineering and science. Specifically designed as a coursebook, this student-tested publication is geared to upper-level undergraduates and graduate students in all disciplines of computational engineeringand science. It is also a practical problem-solving reference for researchers, engineers, and physicists.

Author : Claes Johnson
Publisher : Courier Corporation
Page : 290 pages
File Size : 48,6 Mb
Release : 2012-05-23
Category : Mathematics
ISBN : 9780486131597

Get Book

Numerical Solution of Partial Differential Equations by the Finite Element Method by Claes Johnson Pdf

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.

Author : A. J. Davies
Publisher : Oxford University Press
Page : 308 pages
File Size : 50,7 Mb
Release : 2011-09-08
Category : Mathematics
ISBN : 9780199609130

Get Book

The Finite Element Method by A. J. Davies Pdf

An introduction to the application of the finite element method to the solution of boundary and initial-value problems posed in terms of partial differential equations. Contains worked examples throughout and each chapter has a set of exercises with detailed solutions.

Author : Anders Logg,Kent-Andre Mardal,Garth Wells
Publisher : Springer Science & Business Media
Page : 731 pages
File Size : 42,8 Mb
Release : 2012-02-24
Category : Computers
ISBN : 9783642230998

Get Book

Automated Solution of Differential Equations by the Finite Element Method by Anders Logg,Kent-Andre Mardal,Garth Wells Pdf

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Following are chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Author : A. K. Aziz
Publisher : Academic Press
Page : 796 pages
File Size : 49,5 Mb
Release : 2014-05-10
Category : Technology & Engineering
ISBN : 9781483267982

Get Book

The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations by A. K. Aziz Pdf

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.

Author : Mats G. Larson,Fredrik Bengzon
Publisher : Springer Science & Business Media
Page : 403 pages
File Size : 47,6 Mb
Release : 2013-01-13
Category : Computers
ISBN : 9783642332876

Get Book

The Finite Element Method: Theory, Implementation, and Applications by Mats G. Larson,Fredrik Bengzon Pdf

This book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations. In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB is and its PDE-Toolbox. We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.​

Author : Mohammad Asadzadeh
Publisher : John Wiley & Sons
Page : 352 pages
File Size : 47,5 Mb
Release : 2020-08-27
Category : Mathematics
ISBN : 9781119671664

Get Book

An Introduction to the Finite Element Method for Differential Equations by Mohammad Asadzadeh Pdf

Master the finite element method with this masterful and practical volume An Introduction to the Finite Element Method (FEM) for Differential Equations provides readers with a practical and approachable examination of the use of the finite element method in mathematics. Author Mohammad Asadzadeh covers basic FEM theory, both in one-dimensional and higher dimensional cases. The book is filled with concrete strategies and useful methods to simplify its complex mathematical contents. Practically written and carefully detailed, An Introduction to the Finite Element Method covers topics including: An introduction to basic ordinary and partial differential equations The concept of fundamental solutions using Green's function approaches Polynomial approximations and interpolations, quadrature rules, and iterative numerical methods to solve linear systems of equations Higher-dimensional interpolation procedures Stability and convergence analysis of FEM for differential equations This book is ideal for upper-level undergraduate and graduate students in natural science and engineering. It belongs on the shelf of anyone seeking to improve their understanding of differential equations.

Author : A. J. Davies
Publisher : OUP Oxford
Page : 312 pages
File Size : 45,7 Mb
Release : 2011-09-08
Category : Mathematics
ISBN : 9780191630347

Get Book

The Finite Element Method by A. J. Davies Pdf

The finite element method is a technique for solving problems in applied science and engineering. The essence of this book is the application of the finite element method to the solution of boundary and initial-value problems posed in terms of partial differential equations. The method is developed for the solution of Poisson's equation, in a weighted-residual context, and then proceeds to time-dependent and nonlinear problems. The relationship with the variational approach is also explained. This book is written at an introductory level, developing all the necessary concepts where required. Consequently, it is well-placed to be used as a textbook for a course in finite elements for final year undergraduates, the usual place for studying finite elements. There are worked examples throughout and each chapter has a set of exercises with detailed solutions.

Author : Jonathan Whiteley
Publisher : Springer
Page : 232 pages
File Size : 50,7 Mb
Release : 2017-01-26
Category : Science
ISBN : 9783319499710

Get Book

Finite Element Methods by Jonathan Whiteley Pdf

This book presents practical applications of the finite element method to general differential equations. The underlying strategy of deriving the finite element solution is introduced using linear ordinary differential equations, thus allowing the basic concepts of the finite element solution to be introduced without being obscured by the additional mathematical detail required when applying this technique to partial differential equations. The author generalizes the presented approach to partial differential equations which include nonlinearities. The book also includes variations of the finite element method such as different classes of meshes and basic functions. Practical application of the theory is emphasised, with development of all concepts leading ultimately to a description of their computational implementation illustrated using Matlab functions. The target audience primarily comprises applied researchers and practitioners in engineering, but the book may also be beneficial for graduate students.

Author : G. Evans,J. Blackledge,P. Yardley
Publisher : Springer Science & Business Media
Page : 308 pages
File Size : 40,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781447103790

Get Book

Analytic Methods for Partial Differential Equations by G. Evans,J. Blackledge,P. Yardley Pdf

This is the practical introduction to the analytical approach taken in Volume 2. Based upon courses in partial differential equations over the last two decades, the text covers the classic canonical equations, with the method of separation of variables introduced at an early stage. The characteristic method for first order equations acts as an introduction to the classification of second order quasi-linear problems by characteristics. Attention then moves to different co-ordinate systems, primarily those with cylindrical or spherical symmetry. Hence a discussion of special functions arises quite naturally, and in each case the major properties are derived. The next section deals with the use of integral transforms and extensive methods for inverting them, and concludes with links to the use of Fourier series.

Author : Andrew R. Mitchell,R. Wait
Publisher : Unknown
Page : 198 pages
File Size : 41,8 Mb
Release : 1978
Category : Differential equations, Partial
ISBN : OCLC:1123444092

Get Book

The Finite Element Method in Partial Differential Equations by Andrew R. Mitchell,R. Wait Pdf

Author : Zhangxin Chen
Publisher : Springer Science & Business Media
Page : 415 pages
File Size : 54,8 Mb
Release : 2005-10-14
Category : Science
ISBN : 9783540280781

Get Book

Finite Element Methods and Their Applications by Zhangxin Chen Pdf

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.

Author : Wolfgang Bangerth,Rolf Rannacher
Publisher : Birkhäuser
Page : 216 pages
File Size : 55,8 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9783034876056

Get Book

Adaptive Finite Element Methods for Differential Equations by Wolfgang Bangerth,Rolf Rannacher Pdf

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.

Author : Andrew Ronald Mitchell,R. Wait
Publisher : Unknown
Page : 198 pages
File Size : 40,9 Mb
Release : 1977
Category : Electronic
ISBN : OCLC:718588836

Get Book

The Finite Element Method in Partial Differential Equations by Andrew Ronald Mitchell,R. Wait Pdf

Author : Carl de Boor
Publisher : Academic Press
Page : 430 pages
File Size : 55,9 Mb
Release : 2014-05-10
Category : Mathematics
ISBN : 9781483268071

Get Book

Mathematical Aspects of Finite Elements in Partial Differential Equations by Carl de Boor Pdf

Mathematical Aspects of Finite Elements in Partial Differential Equations addresses the mathematical questions raised by the use of finite elements in the numerical solution of partial differential equations. This book covers a variety of topics, including finite element method, hyperbolic partial differential equation, and problems with interfaces. Organized into 13 chapters, this book begins with an overview of the class of finite element subspaces with numerical examples. This text then presents as models the Dirichlet problem for the potential and bipotential operator and discusses the question of non-conforming elements using the classical Ritz- and least-squares-method. Other chapters consider some error estimates for the Galerkin problem by such energy considerations. This book discusses as well the spatial discretization of problem and presents the Galerkin method for ordinary differential equations using polynomials of degree k. The final chapter deals with the continuous-time Galerkin method for the heat equation. This book is a valuable resource for mathematicians.