Finite element approximation of variational problems and applications.

*(English)*Zbl 0708.65106
Pitman Monographs and Surveys in Pure and Applied Mathematics, 50. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc. 239 p. £36.00 (1990).

Concerning the purpose of this book, the authors write on page 1: “Our aim was to compress the standard theory of the finite element method (FEM) for elliptic problems (chapters 2-7) to a form that would be understandable to any reader who is familiar with basic linear algebra and functional analysis. The second aim was to extend the standard material of books on the FEM to cover also some recent special results obtained in this field.”

In fact, they achieve their first aim in a sensible way, starting from the variational formulation of an elliptic boundary value problem and ending up with the numerical solution of the finite linear systems, without major omission. Restricted to 90 pages for this part, they often have to refer to the literature for proofs.

In the second part, they cover (on 130 pages) an enormous amount of topics: superconvergence, extrapolation, 4th order equations, parabolic and hyperbolic problems, systems with prescribed divergence and rotation, Stokes’ problem, the time-harmonic Maxwell equation, the Helmholtz equation, contact problems, eigenvalue problems, bifurcation, monotone operators and axisymmetric elliptic problems.

The style of exposition varies between that of a research monograph, a textbook and a journal paper, and while a lot of facts and references are presented, a reader who wants to gain real insight into these topics will need additional sources of information.

Most of the discussion is illustrated with results from actual computations. This book is useful since for many topics one gets impressions and hints on how one could deal with them.

In fact, they achieve their first aim in a sensible way, starting from the variational formulation of an elliptic boundary value problem and ending up with the numerical solution of the finite linear systems, without major omission. Restricted to 90 pages for this part, they often have to refer to the literature for proofs.

In the second part, they cover (on 130 pages) an enormous amount of topics: superconvergence, extrapolation, 4th order equations, parabolic and hyperbolic problems, systems with prescribed divergence and rotation, Stokes’ problem, the time-harmonic Maxwell equation, the Helmholtz equation, contact problems, eigenvalue problems, bifurcation, monotone operators and axisymmetric elliptic problems.

The style of exposition varies between that of a research monograph, a textbook and a journal paper, and while a lot of facts and references are presented, a reader who wants to gain real insight into these topics will need additional sources of information.

Most of the discussion is illustrated with results from actual computations. This book is useful since for many topics one gets impressions and hints on how one could deal with them.

Reviewer: M.Brokate

##### MSC:

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 |

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

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

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

35Jxx | Elliptic equations and elliptic systems |

35Q60 | PDEs in connection with optics and electromagnetic theory |