hp-Finite Element Methods for Singular Perturbations
hp-Finite Element Methods for Singular Perturbations

Author: Jens M. Melenk

Publisher: Springer

Published: 2004-10-19

Total Pages: 331

ISBN-13: 354045781X

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Many partial differential equations arising in practice are parameter-dependent problems that are of singularly perturbed type. Prominent examples include plate and shell models for small thickness in solid mechanics, convection-diffusion problems in fluid mechanics, and equations arising in semi-conductor device modelling. Common features of these problems are layers and, in the case of non-smooth geometries, corner singularities. Mesh design principles for the efficient approximation of both features by the hp-version of the finite element method (hp-FEM) are proposed in this volume. For a class of singularly perturbed problems on polygonal domains, robust exponential convergence of the hp-FEM based on these mesh design principles is established rigorously.

Finite Element Methods and Their Applications
Finite Element Methods and Their Applications

Author: Zhangxin Chen

Publisher: Springer Science & Business Media

Published: 2005-06-23

Total Pages: 415

ISBN-13: 3540240780

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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.

Finite Element Methods
Finite Element Methods

Author: Jonathan Whiteley

Publisher: Springer

Published: 2017-01-26

Total Pages: 236

ISBN-13: 3319499718

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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.

Finite Element Methods
Finite Element Methods

Author: Michel Krizek

Publisher: CRC Press

Published: 1998-01-05

Total Pages: 370

ISBN-13: 9780824701482

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"Based on the proceedings of the first conference on superconvergence held recently at the University of Jyvaskyla, Finland. Presents reviewed papers focusing on superconvergence phenomena in the finite element method. Surveys for the first time all known superconvergence techniques, including their proofs."

The Finite Element Method: Theory, Implementation, and Applications
The Finite Element Method: Theory, Implementation, and Applications

Author: Mats G. Larson

Publisher: Springer Science & Business Media

Published: 2013-01-13

Total Pages: 403

ISBN-13: 3642332870

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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.​

p- and hp- Finite Element Methods
p- and hp- Finite Element Methods

Author: C. Schwab

Publisher: Clarendon Press

Published: 1998-10-15

Total Pages: 386

ISBN-13: 9780198503903

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The finite element method (FEM) is a numerical procedure for solving differential equations. Ever-increasing computing power means that engineers and applied mathematicians are seeking more complicated and sophisticated numerical methods to obtain progressively more accurate answers to problems in solid and fluid mechanics. The p- and hp- finite element methods are just such methods, and are therefore of great current interest. This book is the first to cover comprehensively the mathematical underpinnings of hp-FEM in one and two dimensions and pays particular attention to its applications in engineering.

The Finite Element Method
The Finite Element Method

Author: Patrick Ciarlet

Publisher: John Wiley & Sons

Published: 2023-08-29

Total Pages: 404

ISBN-13: 1786307685

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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.