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Adaptive numerical solution of PDEs. (English) Zbl 1268.65146
de Gruyter Textbook. Berlin: de Gruyter (ISBN 978-3-11-028310-5/pbk; 978-3-11-028311-2/ebook). xi, 421 p. (2012).
The present book gives a thorough overview of some of the most important methods in use nowadays to adaptively solve partial differential equations (PDEs). One of the authors, Peter Deuflhard, founded the Zuse Institute in Berlin where from the beginning in the 1980s adaptive algorithms for PDEs were in the focus. Hence, the book under review draws on first hand expert knowledge.
The first chapter gives an introduction to some elementary types of PDEs. The Laplace, the Poisson, and the Helmholtz equation are treated in more detail than the wave equation or the Schrödinger equation. Chapter 2 gives an overview of the modeling with PDEs, namely the role of Maxwell’s equations in electrodynamics, Euler’s and Navier-Stokes’s equations in Fluid Dynamics, and the equations of linear and nonlinear elastomechanics. Chapter 3 discusses finite difference discretisations of the Poisson equation while in the fourth chapter Galerkin methods for this type of PDE are introduced. Here, also spectral methods are discussed.
In Chapter 5, some numerical algorithms for the solution of linear elliptic grid equations are given, including direct elimination and iterative solvers like multigrid or hierarchical-basic methods. The first real occurrence of adaptive methods is in Chapter 6 where adaptive hierarchical methods are in the focus. The approach is fairly general and includes a thorough discussion of errors and error indicators. A convergence result is given for a model refinement strategy. Adaptive multigrid methods for linear elliptic problems constitute the content of Chapter 7. A combination of finite element analysis, multigrid methods, and adaptive hierarchical grids allows the treatment of the boundary problems in a modern functional analytic way. The cascadic multigrid method is also presented and explained.
In Chapter 8, the view is enlarged to allow for nonlinear elliptic problems and it is here where different Newton methods are discussed. In the last Chapter 9, the focus is on parabolic problems, leading to stiff differential equations. An Appendix gives information on Fourier Analysis, Integral Theorems, Sobolev spaces and the delta distribution. Also given are the locations of software packages in the public domain. The book closes with a bibliography containing 235 items and an index.
Due to the focus of the authors own work on elliptic and parabolic equations the book does not contain any further material on adaptive methods for hyperbolic problems although much was done in this area in the past. However, the book is a highly valuable addition to the literature. Readers should have more than only a basic knowledge in numerical analysis to appreciate the contents of this book valuable to students, researchers, and practitioners alike.

##### MSC:
 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs 65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65F05 Direct numerical methods for linear systems and matrix inversion 65F08 Preconditioners for iterative methods 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 65N06 Finite difference methods for boundary value problems involving PDEs 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 65Y15 Packaged methods for numerical algorithms 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35J60 Nonlinear elliptic equations 35L05 Wave equation 35Q41 Time-dependent Schrödinger equations and Dirac equations 35Q30 Navier-Stokes equations 35Q61 Maxwell equations
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