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The numerical analysis of partial differential equations. 3rd revised and expanded ed. (Numerische Behandlung partieller Differentialgleichungen.) (German) Zbl 1086.65109

Teubner Studienbücher Mathematik. Wiesbaden: Teubner (ISBN 3-519-22089-X/pbk). 571 p. (2005).
This textbook is a complete revision of the earlier editions [Numerik partieller Differentialgleichungen. (The numerical analysis of partial differential equations). 2., durchges. Aufl. (German) Teubner Studienbücher: Mathematik. (Stuttgart: B. G. Teubner). (1994; Zbl 0822.65055), (Numerik partieller Differentialgleichungen. (Numerics of partial differential equations). (German) Teubner Studienbücher: Mathematik. (Stuttgart: B. G. Teubner). (1992; Zbl 0755.65087)].
The principal emphasis is still on finite-element methods for elliptic partial differential equations. There is also briefer discussion of finite difference and finite volume methods and a presentation of methods for the solution of hyperbolic and parabolic equations. Unusual for a book of this sort are the chapters on variational inequalities and singularly perturbed equations. One of the modernizations is the inclusion of local mesh refinement and error estimation. The final chapter gives an overview of solution methods for large sparse matrix problems.
Because of its emphasis on the practical details of the numerical methods, as well as the ample illustrations by simple examples, the book is an excellent introduction to the field.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65K10 Numerical optimization and variational techniques
65F50 Computational methods for sparse matrices
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35K15 Initial value problems for second-order parabolic equations
35L15 Initial value problems for second-order hyperbolic equations
49J40 Variational inequalities
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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