Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation
2005 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 194, no 16-dec, 1251-1294 p.Article in journal (Refereed) Published
This work studies a linear elliptic problem with uncertainty. The introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations. The major emphasis of this work is the probabilistic treatment of uncertainty, addressing the problem of solving linear elliptic boundary value problems with stochastic coefficients. If the stochastic coefficients are known functions of a random vector, then the stochastic elliptic boundary value problem is turned into a parametric deterministic one with solution u(y, x), y is an element of Gamma, x is an element of D, where D subset of R-d, d = 1, 2, 3, and Gamma is a high-dimensional cube. In addition, the function u is specified as the solution of a deterministic variational problem over Gamma x D. A tensor product finite element method, of h-version in D and k-, or, p-version in Gamma, is proposed for the approximation of it. A priori error estimates are given and an adaptive algorithm is also proposed. Due to the high dimension of Gamma, the Monte Carlo finite element method is also studied here. This work compares the asymptotic complexity of the numerical methods, and shows results from numerical experiments. Comments on the uncertainty in the probabilistic characterization of the coefficients in the stochastic formulation are included.
Place, publisher, year, edition, pages
2005. Vol. 194, no 16-dec, 1251-1294 p.
stochastic elliptic equation, perturbation estimates, Karhunen-Loeve expansion, finite elements, Monte Carlo method, k x h-version, p x h-version, expected value, error estimates, adaptive methods, error control, partial-differential-equations, groundwater models cannot, sensitivity-analysis, validation, verification, science, domain
IdentifiersURN: urn:nbn:se:kth:diva-14584DOI: 10.1016/j.cma.2004.02.026ISI: 000227483200002OAI: oai:DiVA.org:kth-14584DiVA: diva2:332625
QC 201005252010-08-052010-08-05Bibliographically approved