Sparse finite elements for elliptic problems with stochastic loading
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.
scientific article; zbMATH DE number 2001640