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Anisotropic finite elements: Local estimates and applications. (English) Zbl 0934.65121

Advances in Numerical Mathematics. Leipzig: Teubner. Chemnitz: Technische Univ. 261 p. (1999).
Problems with boundary layers or edge singularities require finite element grids where the ratio of the size in the \(x\)-direction and the \(y\)-direction may differ by several orders of magnitude. In these cases the standard estimates hold only with constants that heavily depend on the ratio. The necessity of adjusted meshes and of appropriate tools of analysis is illustrated in Chapter 1.
Chapter 2 deals with interpolation on slim elements. An essential requirement is that the error of a derivative \(D^\alpha (u-I_h u)\) is estimated in terms of \(D^\beta u\) with \(\beta > \alpha\) and that not all derivatives of the total order \(|\beta |\) enter into the inequality. As a consequence, one cannot apply the Bramble-Hilbert lemma in the standard way.
Interpolation of Clément type as the Scott-Zhang procedure is even more involved since the mapping is only nearly local. Therefore, modifications of the operators and estimates with weighted norms are investigated in Chapter 3.
The Chapters 4 and 5 are concerned with the application to problems with edge singularities and with boundary layers, respectively. There are not only theoretical results, but numerical results are also presented.
Elliptic differential equations with strong anisotropics have attracted much interest in the last years. Therefore a systematic treatment as it is done in this book is very useful. The estimates on the various interpolation procedures contain some technical parts, but this seems to be unavoidable. The author has compensated this in the chapters on the applications by presenting a large number of figures with typical grids or numerical results.
It is a useful book for those who have to deal with anisotropic problems.
Reviewer: D.Braess (Bochum)

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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