*(English)*Zbl 1128.65009

A new Galerkin-type method for the numerical solution of ordinary and partial differential equations with stochastic initial or boundary conditions is considered based on the multi-element generalized polynomial chaos (ME-gPC) expansion of random variables (probability measures). In ME-gPC the space on which the random variable is defined is decomposed into a set of rectangular elements. The probability measure is conditioned on each element and a sequence of the polynomials orthogonal w.r.t. this conditional probability is constructed on each element separately. Then the random element is expanded over all elements and all polynomials.

Numerical problems of the approximation and a scheme of adaptive decomposition on the elements are discussed. Applications to the analysis of heat transfer in a grooved channel and to stochastic elliptic problems are considered. Numerical results are presented for the solution of the Kraichnan-Orszag problem which demonstrate that this technique can be applied to problems with a stochastic discontinuity.

##### MSC:

65C30 | Stochastic differential and integral equations |

60H15 | Stochastic partial differential equations |

35R60 | PDEs with randomness, stochastic PDE |

65N30 | Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) |

65L60 | Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE |

65M60 | Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) |

60H10 | Stochastic ordinary differential equations |

60H35 | Computational methods for stochastic equations |