##
**Multigrid methods for finite elements. Updated and revised transl. by N. B. Urusova.**
*(English)*
Zbl 0837.65118

Mathematics and its Applications (Dordrecht). 318. Dordrecht: Kluwer Academic Publishers. xiv, 331 p. (1995).

The book is a translation of a somewhat revised version of the book published before in Russian (1989; Zbl 0682.65065). The most noticeable alternations are in Chapter 4 (“General description of multigrid algorithms”). Actually only a rather traditional approach to multigrid methods is considered (it is assumed that the a priori estimate \(|u|_{H^2(\Omega)}\leq C|f|_{L_2(\Omega)}\) holds for the solution of the second-order elliptic equation and that \(f\in L_2(\Omega)\)). But special attention is paid to the use of special Chebyshev iterative parameters in the smoothing procedure; many important results for multigrid methods that apply for more general cases are not mentioned at all [see, e.g., the reviewer, “Optimization in solving elliptic problems”, CRC Press, Boca Raton (1995), and references to papers and books of O. Axelsson, R. Bank, J. Bramble, the reviewer, P. Vassilevskij, O. Widlund, H. Yserentant and others cited therein].

It should be noted that some of the errors and misprints in the Russian text are corrected but a lot of them are preserved. For example, on p. 34, Theorem 1.14 (see also (1.146)) states convergence of approximate eigenfunctions \(u^h_i\) to an eigenfunction \(w\), but actually, \(\{u^h_i\}\), \(h\to 0\), may have no limit unless these functions are specially chosen. On p. 2, bounded domain is defined as “an open connected set”. In many estimates like (1.130), it is not specified that the constant \(c\) is independent of \(h\). In Theorem 4f3, the author writes that a set is the solution of problem (4.22) but (4.22) does not contain minimization with respect to these parameters. On p. 180, the author writes about convergence of eigensubspaces but actually the approximating subspace may be a direct sum of eigensubspaces. It is strange but even such a basic notion as the Hilbert space is not defined; the author writes on p. 29 about a complete Hilbert space (he follows here the presentation in the book of Mikhlin, where uncomplete Hilbert spaces were considered as well) but Theorem 1.11 (p. 5) is true only when the Hilbert space is complete. Moreover, on p. 28 there is a fantastic case of this notion: “\(L_p(\Omega)\), \(p\in [1, \infty]\) is the Hilbert space”.

It should be noted that some of the errors and misprints in the Russian text are corrected but a lot of them are preserved. For example, on p. 34, Theorem 1.14 (see also (1.146)) states convergence of approximate eigenfunctions \(u^h_i\) to an eigenfunction \(w\), but actually, \(\{u^h_i\}\), \(h\to 0\), may have no limit unless these functions are specially chosen. On p. 2, bounded domain is defined as “an open connected set”. In many estimates like (1.130), it is not specified that the constant \(c\) is independent of \(h\). In Theorem 4f3, the author writes that a set is the solution of problem (4.22) but (4.22) does not contain minimization with respect to these parameters. On p. 180, the author writes about convergence of eigensubspaces but actually the approximating subspace may be a direct sum of eigensubspaces. It is strange but even such a basic notion as the Hilbert space is not defined; the author writes on p. 29 about a complete Hilbert space (he follows here the presentation in the book of Mikhlin, where uncomplete Hilbert spaces were considered as well) but Theorem 1.11 (p. 5) is true only when the Hilbert space is complete. Moreover, on p. 28 there is a fantastic case of this notion: “\(L_p(\Omega)\), \(p\in [1, \infty]\) is the Hilbert space”.

Reviewer: E.D’yakonov (Moskva)

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |

65H10 | Numerical computation of solutions to systems of equations |

35Jxx | Elliptic equations and elliptic systems |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

35P15 | Estimates of eigenvalues in context of PDEs |