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Numerical mathematics 3. Adaptive solutions of partial differential equations. (Numerische Mathematik 3. Adaptive Lösung partieller Differentialgleichungen.) (German) Zbl 1230.65093

de Gruyter Lehrbuch. Berlin: de Gruyter (ISBN 978-3-11-021802-2/pbk; 978-3-11-021803-9/ebook). x, 432 p. (2011).
The volume 3 of the textbook Numerical Mathematics by Peter Deuflhard and coauthors deals with numerical methods of partial differential equations. [See: P. Deuflhard and A. Hohmann, Numerical mathematics. 1: An algorithmically oriented introduction. 4th rev. ed. Berlin: de Gruyter (2008; Zbl 1179.65001) and P. Deuflhard and F. Bornemann, Numerical mathematics 2. Ordinary differential equations. 3rd rev. ed. Berlin: de Gruyter (2008; Zbl 1167.65303)]. The book – written in German – is addressed to students of mathematics and computer science as well as graduate specialists in physics, chemistry and engineering who have to solve complex applications efficiently. The knowledge of Volume 1 is assumed. Elliptic and parabolic systems are the main focus of the textbook.
The basic orientations of the authors are detailed derivations of efficient adaptive algorithms, a clear orientation to scientific computing, an elementary (but sufficiently exact) presentation of the corresponding mathematical theory, and the application of the proposed methods in solving problems of natural sciences, technology and medicine. The focus is on the efficiency (speed, reliability and robustness) of the described algorithms, which leads to adaptive algorithms with respect to space- and time-discretization.
The first chapter gives some mathematical examples and a classification of elementary partial differential equations. Chapter 2 demonstrates that these equations indeed appear to describe applications in electrodynamics, fluid dynamics and elastomechanics. Difference methods on equidistant grids are derived.
Strategies for non-equidistant grids are discussed for Poisson problems. In Chapter 4, the Galerkin method – spectral methods as well as finite-element-methods – are presented. An adaptive version of the Fourier-Galerkin method is described. Numerical methods for finding the solution of linear elliptic problems and the resulting linear equations are given, the results of Volume 1 are essential. Chapter 6 deals with the construction of adaptive hierarchical grids, proving the convergence of the approximations on adaptive grids. In the next chapter, adaptive multigrid-methods are used to solve boundary value problems. A useful tool are subspace correction methods. Adaptive numerical solutions of nonlinear elliptic boundary value problems are the object of Chapter 8, which studies the adaptive integration of parabolic problems with initial and boundary values. The time-discretization is described for stiff differential equations. Space-time-discretizations are considered for parabolic equations.
The authors present very interesting applications in medicine (facial orthodontics and the electrical excitement of the heart muscle) and in physics (plasmon-polariton waves). Mathematical theorems necessary for understanding the book are added in an appendix (20 pages). Every chapter is completed by exercises; 105 figures illustrate the text. The book is well readable and correctly written. There are some (13) hints to software packages. The textbook is completed by an extensive (227 entries) biography. The book is excellent!

MSC:

65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65Nxx Numerical methods for partial differential equations, boundary value problems
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
92C50 Medical applications (general)
68W30 Symbolic computation and algebraic computation
35Jxx Elliptic equations and elliptic systems
35Kxx Parabolic equations and parabolic systems
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