zbMATH — the first resource for mathematics

Stochastic finite elements: a spectral approach. (English) Zbl 0722.73080
New York etc.: Springer-Verlag. x, 214 p. DM 98.00/hbk (1991).
In this monograph the authors develop their ideas on how PDE’s with stochastic coefficients may be treated numerically with the finite element method. The main focus is here on the treatment of the stochastic side, since the finite element discretization may be found in many textbooks. By developing the covariance kernel of the stochastic fields in its eigenfunctions, a natural discretization for the stochastic part is initiated, followed by a representation in the finite element subspace. Several methods of computing the response are discussed, generally yielding the response as an explicit function of the stochastic variables - a so-called response surface. This may then in turn be used to estimate probability distributions or compute reliabilities. The book is concluded by some numerical examples and contains an extensive bibliography.
The manuscript was supplied camera-ready by the authors, and this unfortunately shows in quite a few typing errors. Also some paragraphs are very scatchy, and the book partly resembles a research report which was edited a bit into a monograph. However, even though there are a number of mathematical inaccuracies, the basic ideas presented are very interesting, and the book should be read by anyone interested in combining stochastic or reliability analysis with finite elements.

74S30 Other numerical methods in solid mechanics (MSC2010)
74S05 Finite element methods applied to problems in solid mechanics
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
60H15 Stochastic partial differential equations (aspects of stochastic analysis)