## Abstract

In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuška, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800-825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the "probability error" with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

Original language | English (US) |
---|---|

Pages (from-to) | 1005-1034 |

Number of pages | 30 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 45 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 2007 |

## Keywords

- Collocation method
- Exponential convergence
- Finite elements
- Stochastic partial differential equations
- Uncertainty quantification

## ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics