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Theorie und Numerik elliptischer Differentialgleichungen. (German) Zbl 0609.65065
Teubner Studienb├╝cher: Mathematik. Stuttgart: B. G. Teubner. 276 S. DM 38.00 (1986).
This textbook presents the fundamental concepts in the theory of elliptic partial differential equations, together with the construction and analysis of numerical methods for solving boundary-value problems for elliptic equations. The exposition in both subject areas is clear and concise. A summary of the topics covered in the book follows.
Basic properties of the Laplace and Poisson equations are discussed in the first few chapters. This is followed by the construction and analysis of the standard finite-difference methods for Dirichlet and Neumann boundary-value problems for these equations. The reviewer was pleased to find discussed practical details regarding the treatment of curved boundaries and of the arbitrary additive constant for solutions of the Neumann problem.
Roughly, half of the book is devoted to the finite-element method, including the necessary background material in functional analysis and regularity theory. Practical questions are also discussed, including the handling of singularities at reentrant corners and the adaptation of finite elements to domains with curved boundaries.
The book closes with a chapter on the theory and numerical methods for the Stokes equation for slow flow of an incompressible fluid.
Two topics which might have been expected in such a book, but not found here, are numerical linear algebra and nonlinear elliptic equations. Still, the reviewer regards the book as an excellent introduction to the field.
Reviewer: G.Hedstrom

MSC:
65Nxx Numerical methods for partial differential equations, boundary value problems
35Jxx Elliptic equations and elliptic systems
65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
76D07 Stokes and related (Oseen, etc.) flows