Sparse finite elements for elliptic problems with stochastic loading
type: Item
instance of: scholarly article
title: Sparse finite elements for elliptic problems with stochastic loading
zbMATH Open document ID: 1044.65085
author: Christoph Schwab
author: Radu Todor
published in: Numerische Mathematik
publication date: 2003-11-09T00:00:00Z
full work available at URL: http://hdl.handle.net/20.500.11850/53177
review text: This paper develops the entire numerical analysis of a class of elliptic boundary value problems with stochastic loading in a bounded domain. Such a kind of problems is of course very important in practice due to the uncetainties on the data in current physical problems. The objective of this work is a rigorous study of this class of problems together with the development of efficient numerical methods. In this way, the authors first introduce appropriate functional spaces for data and solutions in order to obtain a well-posed problem. Next, with the stochastic problem, they associate deterministic partial differential equations which involve the moments of first and second order of the random solution and they consider the regularity of corresponding equation. Next, they propose an efficient finite element method combined with preconditioning techniques in order to solve efficiently the associate deterministic equations. Finally, they analyze the complexity of the algorithm and they present some convincing numerical experiments. This a very strong, complete and nicely written paper on this very important class of problems. Such an approach constitutes a very attractive and efficient alternative to the more classical Monte Carlo method.
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 65N55
zbMATH DE Number: 2001640
zbMATH Keywords: stochastic Sobolev spaces
zbMATH Keywords: stochastic loading
zbMATH Keywords: hierarchic finite element spaces
zbMATH Keywords: asymptotical rates of convergence
zbMATH Keywords: multilevel preconditioning
zbMATH Keywords: comparison of methods
zbMATH Keywords: algorithm
zbMATH Keywords: numerical experiments
zbMATH Keywords: Monte Carlo method
reviewed by: Michel Bernadou
MaRDI profile type: Publication
OpenAlex ID: W2135329423
DOI: 10.1007/S00211-003-0455-Z
description: scientific article; zbMATH DE number 2001640