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Many well-known models in the natural sciences and engineering, and today even in economics, depend on partial di?erential equations. Thus the e?cient numerical solution of such equations plays an ever-increasing role in state-- the-art technology. This demand and the computational power available from current computer hardware have together stimulated the rapid development of numerical methods for partial di?erential equations-a development that encompasses convergence analyses and implementational aspects of software packages. In 1988 we started work on the ?rst German edition of our book, which appeared in 1992. Our aim was to give students a textbook that contained the basic concepts and ideas behind most numerical methods for partial di?er- tial equations. The success of this ?rst edition and the second edition in 1994 encouraged us, ten years later, to write an almost completely new version, taking into account comments from colleagues and students and drawing on the enormous progress made in the numerical analysis of partial di?erential equations in recent times. The present English version slightly improves the third German edition of 2005: we have corrected some minor errors and added additional material and references.
List of contents
Partial Differential Equations: Basics and Explicit Representation of Solutions.- Finite Difference Methods.- Weak Solutions, Elliptic Problems and Sobolev Spaces.- The Finite Element Method.- Finite Element Methods for Unsteady Problems.- Singular Perturbations.- Numerical Methods for Variational Inequalities and Optimal Control.- Numerical Methods for Discretized Problems.
Summary
This book deals with discretization techniques for elliptic, parabolic and hyperbolic partial differential equations. It provides an introduction to the main principles of discretizations and presents to the reader the ideas and analysis of advanced numerical methods in this area. The book also covers recent research developments - these topics of current interest are rarely considered in other textbooks. It is the authors' aim to give mathematically-inclined students, scientists and engineers a textbook that contains all the basic discretization techniques for the three fundamental types of partial differential equations and in which the reader can find analytical tools, properties of discretizations, and some advice on algorithmic aspects. While finite element methods are the main focus of the book, finite difference methods and finite volume techniques are also presented. Furthermore, the book provides the basic tools needed to solve the discrete problems generated.
Additional text
From the reviews:
"This textbook is the translation and revision of the third German edition of 2005. ... the book deals with different aspects of the numerical solution of elliptic, parabolic and hyperbolic partial differential equations. ... An index and two pages with a summary of the notations used complete the presentation. ... It can be highly recommended for students and engineers but also for numerical analysts." (Riidiger Weiner, Zentralblatt fiir Angewandte Mathematik and Mechanik, Vol. 88 (12), 2008)
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From the reviews:
"This textbook is the translation and revision of the third German edition of 2005. ... the book deals with different aspects of the numerical solution of elliptic, parabolic and hyperbolic partial differential equations. ... An index and two pages with a summary of the notations used complete the presentation. ... It can be highly recommended for students and engineers but also for numerical analysts." (Riidiger Weiner, Zentralblatt fiir Angewandte Mathematik and Mechanik, Vol. 88 (12), 2008)
