Analyse mathématique et calcul numérique pour les sciences et les techniques. Volume 6: Méthodes intégrales et numériques. (Mathematical analysis and numerical calculus. Vol. 6: Integral and numerical methods.) Avec la collaboration de Michel Artola, Philippe Benilan, Michel Bernadou, Michel Cessenat, Jean-Claude Nedelec, Jacques Planchard. (Reproduction intégrale en 9 vol. brochés de l’ouvrage paru en trois tomes reliés, chez Masson en 1984 et 1985). Reproduction intégrale en 9 vol. brochés de l’ouvrage paru en trois tomes reliés, chez Masson en 1984 et 1985. (French) Zbl 0652.45001

Dautray, Robert; Lions, Jacques-Louis

Commissariat à l’Énergie Atomique, Gif-sur-Yvette (France). Inst. National des Sciences et Techniques Nucléaires. Collection Enseignement. Paris etc.: Masson. xi, p. 553-1061 FF 200.00 (1988).

This book contains chapters XI, XII and XIII of the authors’ multi-volume treatise “Analyse mathématique et calcul numérique”. The authors have well succeeded here in presenting an appealing and detailed text on integral equations (chapter XI), numerical methods for stationary problems (chapter XII), approximation of integral equations by finite elements (chapter XIII), singular integrals (annex). Some keywords from the paragraph headings may give more information. Chapter XI: Wiener-Hopf method, sectionally analytic functions, Hilbert problem, application to some physical problems, Sobolev spaces with weight, integral equations for boundary problems of electrostatics, for the Helmholtz equation, for problems of linear elasticity and for the system of Stokes. Chapter XII: finite elements for the problem of linear elasticity, non-conforming elements, problems of plates and shells, approximation of eigenvalues and eigenvectors. Chapter XIII: finite elements for Dirichlet and Neumann problems in \({\mathbb{R}}^ 3\) (simple and double layers), approximation of surfaces. Annex: convolution operators, Hilbert transformation, singular integral operators, Calderon-Zygmund theorem, Marcinkiewicz spaces. Although there are several references to previous volumes the book can independently be used with profit by a reader who has some general knowledge of partial differential equations, theory of functions and functional analysis.

Reviewer: R.Gorenflo

MSC:

45Exx	Singular integral equations
65R20	Numerical methods for integral equations
65-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
45-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral equations
35C15	Integral representations of solutions to PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
74B10	Linear elasticity with initial stresses
74S05	Finite element methods applied to problems in solid mechanics

Keywords:

integral equations; stationary problems; finite elements; singular integrals; Wiener-Hopf method; sectionally analytic functions; Hilbert problem; Sobolev spaces; electrostatics; Helmholtz equation; linear elasticity; system of Stokes; non-conforming elements; plates; shells; eigenvalues; eigenvectors; convolution operators; Hilbert transformation; singular integral operators; Calderon-Zygmund theorem