A stochastic collocation method for elliptic partial differential equations with random input data.

*(English)*Zbl 1151.65008The authors 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 by I. Babuška, R. Tempone, and G. E. Zouraris [SIAM J. Numer. Anal. 42, No. 2, 800–825 (2004; Zbl 1080.65003)].

It allows to treat easily a wider range of situations, such as input data that depend non linearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. A rigorous convergence analysis is provided and exponential convergence of the probability error with respect to the number of Gauss points in each direction in the probability space is proved under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

It allows to treat easily a wider range of situations, such as input data that depend non linearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. A rigorous convergence analysis is provided and exponential convergence of the probability error with respect to the number of Gauss points in each direction in the probability space is proved under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

Reviewer: Dominique Lepingle (Orléans)

##### MSC:

65C30 | Numerical solutions to stochastic differential and integral equations |

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

65N15 | Error bounds for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

35R60 | PDEs with randomness, stochastic partial differential equations |

60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |