| Title | Numerical Treatment of Partial Differential Equations |
| Author(s) | Christian Grossmann, Hans-Görg Roos, Martin Stynes |
| Publisher | Springer |
| Year | 2007 |
| Edition | 1st English Edition |
| Language | English |
| Pages | 596 |
| ISBN | 9783540715825 |
| Genre / Domain | Mathematics, Numerical Analysis, Engineering |
| Series | Universitext |
| Size | 3.67 MB |
| Extension |
Summary
Numerical Treatment of Partial Differential Equations is a comprehensive graduate-level textbook that provides a rigorous introduction to the numerical solution of partial differential equations (PDEs). Authored by Christian Grossmann, Hans-Görg Roos, and Martin Stynes, this English edition was published by Springer in 2007 as part of the Universitext series [citation:1]. The book represents a translation and revision of the third German edition, originally published in 2005, and incorporates corrections, additional material, and updated references. The authors, all distinguished numerical analysts at leading European institutions, wrote this text with the aim of providing students with the fundamental concepts and ideas behind the most important numerical methods for PDEs [citation:1].
The book covers a wide spectrum of numerical techniques for solving elliptic, parabolic, and hyperbolic partial differential equations. It addresses both the theoretical foundations, including convergence analyses, and the practical implementational aspects relevant to software packages. The authors deliberately avoid excessive abstraction, maintaining a balance that makes the material accessible to graduate students while still preparing them for advanced research. The text includes 86 black-and-white illustrations that aid in understanding the geometric and structural properties of numerical methods. An index and a summary of notations are provided to facilitate navigation through the material [citation:1].
One of the book's strengths is its systematic treatment of finite difference and finite element methods, which are the primary approaches for discretizing PDEs. The authors present the basic concepts behind these methods and then develop the convergence theory for each. The text also addresses singularly perturbed problems, a topic on which Hans-Görg Roos is a recognized expert. This inclusion makes the book particularly valuable for students interested in problems where standard numerical methods may fail due to boundary or interior layers. The comprehensive coverage ensures that readers develop both a broad understanding of the field and the ability to apply specific techniques to practical problems.
The target audience for this book is graduate students in mathematics, engineering, and related fields who have a solid foundation in calculus, linear algebra, and basic numerical analysis. The Universitext series is designed for master's level and beyond, and this book fits that description perfectly [citation:8]. It is also suitable for researchers and practicing engineers who need a rigorous yet accessible reference on numerical PDEs. The book's careful balance of theory and practice makes it an excellent resource for self-study as well as for classroom use.
Key Features
- The book provides a comprehensive introduction to numerical methods for elliptic, parabolic, and hyperbolic partial differential equations, covering both theory and implementation.
- It is part of the prestigious Universitext series from Springer, which presents well-class-tested textbooks at master's level and beyond.
- The text includes 86 black-and-white illustrations that aid in understanding the geometric and structural properties of numerical methods.
- It addresses both convergence analyses and implementational aspects of software packages, preparing readers for practical computational work.
- The authors are distinguished numerical analysts from the Technical University of Dresden and University College Cork.
- This English edition is a translation and revision of the third German edition, incorporating corrections, additional material, and updated references.
- The book covers singularly perturbed problems, a critical area for understanding numerical difficulties with boundary and interior layers.
- An index and a summary of notations are included to facilitate quick reference and efficient study.
- The text balances theoretical rigor with accessibility, avoiding excessive abstraction while maintaining mathematical precision.
- It serves as both a graduate textbook and a comprehensive reference for researchers and practicing engineers.
- The book has been highly recommended by reviewers for students, engineers, and numerical analysts alike [citation:1].
About the Author
Christian Grossmann is a professor of numerical analysis who was affiliated with the Technical University of Dresden, Germany, as of 2007 [citation:3]. Born in 1946, Grossmann has made significant contributions to the field of numerical analysis, particularly in the numerical treatment of differential equations. His research interests span the theoretical foundations of numerical methods and their practical applications in science and engineering. As a co-author of the original German editions of this textbook, he brings decades of teaching and research experience to the English edition.
Hans-Görg Roos, born in 1949, is a professor of mathematics at the Technical University of Dresden, Germany [citation:14]. He is internationally recognized for his work on singularly perturbed problems and numerical methods for differential equations. His expertise in layer-adapted meshes and uniformly convergent numerical methods is reflected in the coverage of these topics in the book. Roos has published extensively in leading numerical analysis journals and has supervised numerous doctoral students in the field.
Martin Stynes, born in 1951, is a professor of mathematics at University College Cork, Ireland [citation:18]. He is the translator and revisor of this English edition, and his contributions to the text go beyond mere translation. Stynes is known for his research on numerical methods for singularly perturbed differential equations and has authored several influential books in the field, including a work on numerical methods for singularly perturbed differential equations [citation:11]. His collaboration with Grossmann and Roos ensures that the English edition maintains the high standards of the original German text while being fully accessible to an English-speaking audience.
Related Books
- Numerical Methods for Singularly Perturbed Differential Equations — Hans-Görg Roos, Martin Stynes, and Lutz Tobiska
- Finite Element Methods for Navier-Stokes Equations — Vivette Girault and Pierre-Arnaud Raviart
- Numerical Solution of Partial Differential Equations — Gordon D. Smith
- Finite Difference Methods for Ordinary and Partial Differential Equations — Randall J. LeVeque
- The Mathematical Theory of Finite Element Methods — Susanne C. Brenner and L. Ridgway Scott
- Numerical Approximation of Partial Differential Equations — Alfio Quarteroni and Alberto Valli
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FAQ
Q : What is the relationship between this English edition and the original German text?
R : This English edition is a translation and revision of the third German edition, published in 2005 by Teubner [citation:1]. The authors have corrected minor errors, added additional material, and updated references to create a more polished text for English-speaking readers. Martin Stynes, a professor at University College Cork, translated and mathematically revised the work, ensuring accuracy and clarity. The result is an edition that retains the strengths of the original while being fully accessible to a new audience.
Q : What mathematical background is required to understand this book?
R : The book assumes a solid foundation in calculus, linear algebra, and basic numerical analysis, consistent with the Universitext series' target audience of master's level and beyond [citation:8]. Readers should be comfortable with partial differential equations at an introductory level and have some familiarity with functional analysis concepts such as Sobolev spaces. The text does not shy away from mathematical rigor, but it balances this with practical explanations and illustrations to aid comprehension.
Q : What types of partial differential equations does the book cover?
R : The book covers the three main classes of partial differential equations: elliptic, parabolic, and hyperbolic [citation:1]. For each class, the authors present the fundamental numerical methods, including finite difference and finite element approaches. The convergence theory for these methods is developed rigorously, and practical implementational aspects are discussed. This comprehensive treatment ensures readers understand both the theoretical underpinnings and the practical application of numerical methods across the spectrum of PDE problems.
Q : Does the book address singularly perturbed problems?
R : Yes, the book includes coverage of singularly perturbed problems, an area where Hans-Görg Roos has particular expertise. These problems are characterized by small parameters multiplying the highest derivatives, leading to solutions with boundary or interior layers. Standard numerical methods often fail or perform poorly on such problems, requiring specialized techniques such as layer-adapted meshes. The inclusion of this topic makes the book especially valuable for students and researchers working on challenging practical problems where conventional methods may be inadequate.
Q : How is the book structured for teaching purposes?
R : The book is designed as a textbook, with material organized to support a graduate course in numerical PDEs. The authors present the basic concepts behind each method before developing the theory, allowing students to build understanding progressively. The 86 illustrations help visualize key ideas, and the notation summary and index facilitate navigation. The Universitext series is known for well-class-tested texts, and this book reflects that tradition of pedagogical effectiveness [citation:8].
Q : What makes this book suitable for engineers as well as mathematicians?
R : The book addresses both theoretical foundations and implementational aspects relevant to software packages, making it valuable for engineers who need to apply numerical methods in practice. The authors avoid excessive abstraction and include discussion of practical computational considerations. Engineers working with simulation software will benefit from understanding the convergence properties and limitations of the methods they use. The book's comprehensive coverage and practical orientation have led reviewers to recommend it for students and engineers as well as numerical analysts [citation:1].
Q : Is this book still relevant given the rapid development of numerical methods?
R : The fundamental concepts and methods presented in this book remain foundational to the field. While new developments have occurred since its publication, the core ideas behind finite difference and finite element methods, convergence theory, and singular perturbation analysis remain unchanged. The book provides a rigorous grounding that enables readers to understand and evaluate newer developments. As a textbook in the Universitext series, it has been class-tested and refined over multiple editions, ensuring its enduring pedagogical value.
Q : What are the affiliations of the authors?
R : At the time of publication, Christian Grossmann and Hans-Görg Roos were affiliated with the Technical University of Dresden in Germany [citation:1][citation:3][citation:14]. Martin Stynes was affiliated with University College Cork in Ireland [citation:1][citation:18]. This collaboration between German and Irish numerical analysts brings together expertise from two leading European centers for research in numerical analysis and differential equations.
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