*(English)*Zbl 1088.65002

Authors’ abstract: Stationary systems modelled by elliptic partial differential equations – linear as well as nonlinear – with stochastic coefficients (random fields) are considered. The mathematical setting as a variational problem, existence theorems, and possible discretisations – in particular with respect to the stochastic part – are given and investigated with regard to stability.

Different and increasingly sophisticated computational approaches involving both Wiener’s polynomial chaos as well as the Karhunen-Loève expansion are addressed in conjunction with stochastic Galerkin procedures, and stability within the Galerkin framework is established. New and effective algorithms to compute the mean and covariance of the solution are proposed.

The similarities and differences with better known Monte Carlo methods are exhibited, as well as alternatives to integration in high-dimensional spaces. Hints are given regarding the numerical implementation and parallelisation. Numerical examples serve as illustration.

##### MSC:

65C30 | Stochastic differential and integral equations |

60H15 | Stochastic partial differential equations |

35R60 | PDEs with randomness, stochastic PDE |

60H35 | Computational methods for stochastic equations |

35J25 | Second order elliptic equations, boundary value problems |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

65N12 | Stability and convergence of numerical methods (BVP of PDE) |

65C05 | Monte Carlo methods |