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Title Details:
Numerical Analysis of Partial Differential Equations
Authors: Georgoulis, Emmanuil H.
Numerical Analysis
Numerical Solution of Partial Differential Equations
Finite Difference Methods
Finite Element Methods
This book aspires to provide an extensive introduction to the Numerical Analysis of Partial Differential Equation (PDE) problems and to serve as a textbook for both advanced undergraduate and graduate courses in this topic. The presentation of two popular categories of methods is chosen: Finite Difference Methods (FDM) and Finite Element Methods (FEM). The volume contains a brief, and inevitably incomplete, review of basic PDE theory (Chapters 1 and 7) covering key concepts, properties and results, as well as some advanced topics in modern PDE theory, such as the weak form, Lebesgue and Sobolev function spaces, energy method etc. Chapter 2 gives a brief introduction to divided differences and presents the basic finite difference method for the one-dimensional second-order boundary value problem. We proceed by presenting finite difference methods for parabolic (Chapter 3), linear hyperbolic (Chapter 4), nonlinear hyperbolic (Chapter 5), and elliptic PDE problems (Chapter 6), with special attention to the study of consistency, stability and convergence of the methods, for which we prove basic error estimates. Next, we proceed with the presentation and analysis of finite element methods for linear elliptic problems (Chapter 8). We present FEM for parabolic problems in Chapter 9. In Chapter 10, an introduction to the so-called, a posteriori error analysis is given, which is then used to define algorithms for automatically adjusting the local "resolution" of FEM for elliptic and parabolic problems. Chapter 11 gives the basic principles and error analysis of the Discontinuous Galerkin Method for these problems. Finally, in Chapter 12, we present some advanced topics in FEM theory. In particular, some basic concepts are given for the definition of high-order finite elements, as well as a space-time FEM with time stepping defined via a discontinuous Galerkin method.
Linguistic Editors: Chatzigeorgiou, Athina
Technical Editors: Georgoulis, Emmanuil H.
Graphic Editors: Georgoulis, Emmanuil H.
Other contributors: The cover image cluster is from calculations by Oliver J. Sutton for the paper: Cangiani, Α., Georgoulis, Ε.Η., Morozov, A. Yu., and Sutton, O. J.. Revealing new dynamical patterns in a reaction-diffusion model with cyclic competition via a novel computational framework. Proceedings of the Royal Society A 474(2213), 2018.
Type: Postgraduate textbook
Creation Date: 08-04-2024
Item Details:
ISBN 978-618-228-229-8
License: Attribution - NonCommercial - ShareAlike 4.0 International (CC BY-NC-SA 4.0)
Bibliographic Reference: Georgoulis, E. (2024). Numerical Analysis of Partial Differential Equations [Postgraduate textbook]. Kallipos, Open Academic Editions.
Language: Greek
Consists of:
1. Brief Introduction to Partial Differential Equations
2. Finite Differences and the Three-Point Method
3. Finite Difference Methods for Parabolic Problems
4. Finite Difference Methods for Linear Hyperbolic Problems
5. Finite Difference Methods for Nonlinear Hyperbolic Problems
6. Finite Difference Methods for Elliptic Problems
7. Function Spaces and Partial Differential Equations in Weak Form
8. The Finite Element Method for Elliptic Problems
9. Finite Element Methods for Parabolic Problems
10. A Posteriori Error Analysis
11. The Discontinuous Galerkin Method for Hyperbolic Problems
12. Advanced Topics in Finite Element Methods
Number of pages 265
Publication Origin: Kallipos, Open Academic Editions
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