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Author | : Peter Knabner |
Publisher | : Springer Science & Business Media |
Total Pages | : 437 |
Release | : 2006-05-26 |
Genre | : Mathematics |
ISBN | : 0387217622 |
Download Numerical Methods for Elliptic and Parabolic Partial Differential Equations Book in PDF, ePub and Kindle
This text provides an application oriented introduction to the numerical methods for partial differential equations. It covers finite difference, finite element, and finite volume methods, interweaving theory and applications throughout. The book examines modern topics such as adaptive methods, multilevel methods, and methods for convection-dominated problems and includes detailed illustrations and extensive exercises.
Author | : S. H, Lui |
Publisher | : John Wiley & Sons |
Total Pages | : 506 |
Release | : 2012-01-10 |
Genre | : Mathematics |
ISBN | : 1118111117 |
Download Numerical Analysis of Partial Differential Equations Book in PDF, ePub and Kindle
A balanced guide to the essential techniques for solving elliptic partial differential equations Numerical Analysis of Partial Differential Equations provides a comprehensive, self-contained treatment of the quantitative methods used to solve elliptic partial differential equations (PDEs), with a focus on the efficiency as well as the error of the presented methods. The author utilizes coverage of theoretical PDEs, along with the nu merical solution of linear systems and various examples and exercises, to supply readers with an introduction to the essential concepts in the numerical analysis of PDEs. The book presents the three main discretization methods of elliptic PDEs: finite difference, finite elements, and spectral methods. Each topic has its own devoted chapters and is discussed alongside additional key topics, including: The mathematical theory of elliptic PDEs Numerical linear algebra Time-dependent PDEs Multigrid and domain decomposition PDEs posed on infinite domains The book concludes with a discussion of the methods for nonlinear problems, such as Newton's method, and addresses the importance of hands-on work to facilitate learning. Each chapter concludes with a set of exercises, including theoretical and programming problems, that allows readers to test their understanding of the presented theories and techniques. In addition, the book discusses important nonlinear problems in many fields of science and engineering, providing information as to how they can serve as computing projects across various disciplines. Requiring only a preliminary understanding of analysis, Numerical Analysis of Partial Differential Equations is suitable for courses on numerical PDEs at the upper-undergraduate and graduate levels. The book is also appropriate for students majoring in the mathematical sciences and engineering.
Author | : Peter Knabner |
Publisher | : Springer Nature |
Total Pages | : 811 |
Release | : 2021-11-19 |
Genre | : Mathematics |
ISBN | : 3030793850 |
Download Numerical Methods for Elliptic and Parabolic Partial Differential Equations Book in PDF, ePub and Kindle
This text provides an application oriented introduction to the numerical methods for partial differential equations. It covers finite difference, finite element, and finite volume methods, interweaving theory and applications throughout. The book examines modern topics such as adaptive methods, multilevel methods, and methods for convection-dominated problems and includes detailed illustrations and extensive exercises.
Author | : Bilal Abbasi |
Publisher | : |
Total Pages | : |
Release | : 2018 |
Genre | : |
ISBN | : |
Download On the Applications of Numerical Methods for Elliptic Partial Differential Equations Book in PDF, ePub and Kindle
"The goal of this dissertation is to explore and demonstrate the applications of numerical methods for elliptic partial differential equations (PDEs). The numerical methods presented, as we will see, are applicable in a variety of contexts, ranging from computational geometry to machine learning. The general analytic framework of this dissertation is viscosity solutions for elliptic PDEs. The corresponding numerical framework belongs to Barles and Souganidis, with emphasis on its reinterpretation using elliptic finite difference schemes in lieu of monotone schemes. The first problem considered was building a multi-criteria anomaly detection algorithm that can be applied in a real-time setting. The algorithm was centered around a recently discovered PDE continuum limit for nondominated sorting. By exploiting the relatively low computational cost of numerically approximating the PDE we developed an efficient method to detect anomalies in two-dimensional data in real-time. We also derived a transport equation which characterizes sorting points within nondominated layers. This allowed us to add to our algorithm the ability of classifying anomalies. Our algorithm has an inherent ability to adapt to changes in the trend of data. In addition to demonstrating the effectiveness of our algorithm on synthetic and real data, we presented probabilistic arguments proving convergence rates for the PDE-based ranking.The second problem addressed the issue of computing the quasiconvex envelope of a given function. In a series of papers written by Barron, Goebel, and Jensen, first- and second-order differential operators characterizing quasiconvexity were rigourously developed. These characterizations, arising in the form of PDEs, unfortunately prove intractable in light of existing numerical methods. Hence, attempting to generate the quasiconvex envelope using these operators with an obstacle term, in a manner similar to Oberman, is not prudent. Our solution to this, and consequently our contribution, came two-fold (each of which is its own article, respectively): (i) a first-order nonlocal line solver which can compute the quasiconvex envelope in one dimension, and for which the extension to arbitrary dimensions follows naturally; (ii) a second-order operator which offers a more relaxed notion of quasiconvexity, and is more obliging to numerical approximation. Convergence of the algorithms presented in both solutions is proven, and numerical examples validating the arguments presented therein are demonstrated." --
Author | : William F. Ames |
Publisher | : |
Total Pages | : 386 |
Release | : 1977 |
Genre | : Mathematics |
ISBN | : |
Download Numerical Methods for Partial Differential Equations Book in PDF, ePub and Kindle
This volume is designed as an introduction to the concepts of modern numerical analysis as they apply to partial differential equations. The book contains many practical problems and their solutions, but at the same time, strives to expose the pitfalls--such as overstability, consistency requirements, and the danger of extrapolation to nonlinear problems methods used on linear problems. Numerical Methods for Partial Differential Equations, Third Edition reflects the great accomplishments that have taken place in scientific computation in the fifteen years since the Second Edition was published. This new edition is a drastic revision of the previous one, with new material on boundary elements, spectral methods, the methods of lines, and invariant methods. At the same time, the new edition retains the self-contained nature of the older version, and shares the clarity of its exposition and the integrity of its presentation. Key Features * Material on finite elements and finite differences have been merged, and now constitute equal partners * Additional material has been added on boundary elements, spectral methods, the method of lines, and invariant methods * References have been updated, and reflect the additional material * Self-contained nature of the Second Edition has been maintained * Very suitable for PDE courses
Author | : Joël Chaskalovic |
Publisher | : Springer |
Total Pages | : 362 |
Release | : 2014-05-16 |
Genre | : Mathematics |
ISBN | : 3319035630 |
Download Mathematical and Numerical Methods for Partial Differential Equations Book in PDF, ePub and Kindle
This self-tutorial offers a concise yet thorough introduction into the mathematical analysis of approximation methods for partial differential equation. A particular emphasis is put on finite element methods. 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. This English edition is based on the Finite Element Methods for Engineering Sciences by Joel Chaskalovic.
Author | : Boris N. Khoromskij |
Publisher | : Springer Science & Business Media |
Total Pages | : 304 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 3642187773 |
Download Numerical Solution of Elliptic Differential Equations by Reduction to the Interface Book in PDF, ePub and Kindle
During the last decade essential progress has been achieved in the analysis and implementation of multilevel/rnultigrid and domain decomposition methods to explore a variety of real world applications. An important trend in mod ern numerical simulations is the quick improvement of computer technology that leads to the well known paradigm (see, e. g. , [78,179]): high-performance computers make it indispensable to use numerical methods of almost linear complexity in the problem size N, to maintain an adequate scaling between the computing time and improved computer facilities as N increases. In the h-version of the finite element method (FEM), the multigrid iteration real izes an O(N) solver for elliptic differential equations in a domain n c IRd d with N = O(h- ) , where h is the mesh parameter. In the boundary ele ment method (BEM) , the traditional panel clustering, fast multi-pole and wavelet based methods as well as the modern hierarchical matrix techniques are known to provide the data-sparse approximations to the arising fully populated stiffness matrices with almost linear cost O(Nr log?Nr), where 1 d Nr = O(h - ) is the number of degrees of freedom associated with the boundary. The aim of this book is to introduce a wider audience to the use of a new class of efficient numerical methods of almost linear complexity for solving elliptic partial differential equations (PDEs) based on their reduction to the interface.
Author | : W. Hackbusch |
Publisher | : Springer Science & Business Media |
Total Pages | : 334 |
Release | : 1992 |
Genre | : Language Arts & Disciplines |
ISBN | : 9783540548225 |
Download Elliptic Differential Equations Book in PDF, ePub and Kindle
Derived from a lecture series for college mathematics students, introduces the methods of dealing with elliptical boundary-value problems--both the theory and the numerical analysis. Includes exercises. Translated and somewhat expanded from the 1987 German version. Annotation copyright by Book News, Inc., Portland, OR
Author | : Francisco J. Sayas |
Publisher | : CRC Press |
Total Pages | : 492 |
Release | : 2019-01-16 |
Genre | : Mathematics |
ISBN | : 0429016204 |
Download Variational Techniques for Elliptic Partial Differential Equations Book in PDF, ePub and Kindle
Variational Techniques for Elliptic Partial Differential Equations, intended for graduate students studying applied math, analysis, and/or numerical analysis, provides the necessary tools to understand the structure and solvability of elliptic partial differential equations. Beginning with the necessary definitions and theorems from distribution theory, the book gradually builds the functional analytic framework for studying elliptic PDE using variational formulations. Rather than introducing all of the prerequisites in the first chapters, it is the introduction of new problems which motivates the development of the associated analytical tools. In this way the student who is encountering this material for the first time will be aware of exactly what theory is needed, and for which problems. Features A detailed and rigorous development of the theory of Sobolev spaces on Lipschitz domains, including the trace operator and the normal component of vector fields An integration of functional analysis concepts involving Hilbert spaces and the problems which can be solved with these concepts, rather than separating the two Introduction to the analytical tools needed for physical problems of interest like time-harmonic waves, Stokes and Darcy flow, surface differential equations, Maxwell cavity problems, etc. A variety of problems which serve to reinforce and expand upon the material in each chapter, including applications in fluid and solid mechanics
Author | : Karsten Urban |
Publisher | : Numerical Mathematics and Scie |
Total Pages | : 509 |
Release | : 2009 |
Genre | : Mathematics |
ISBN | : 0198526059 |
Download Wavelet Methods for Elliptic Partial Differential Equations Book in PDF, ePub and Kindle
Wavelet methods are by now a well-known tool in image processing (jpeg2000). These functions have been used successfully in other areas, however. Elliptic Partial Differential Equations which model several processes in, for example, science and engineering, is one such field. This book, based on the author's course, gives an introduction to wavelet methods in general and then describes their application for the numerical solution of elliptic partial differential equations. Recently developed adaptive methods are also covered and each scheme is complemented with numerical results , exercises, and corresponding software.