Multi-element generalized polynomial chaos for arbitrary probability measures. (English) Zbl 1128.65009

Wan, Xiaoliang; Karniadakis, George Em

SIAM J. Sci. Comput. 28, No. 3, 901-928 (2006).

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.

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

65C30	Numerical solutions to stochastic differential and integral equations
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35R60	PDEs with randomness, stochastic partial differential equations
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65L60	Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35	Computational methods for stochastic equations (aspects of stochastic analysis)

Keywords:

differential equation with random input; Wiener chaos; Galerkin projection; Kraichnan-Orszag problem; heat transfer in grooved channel; stochastic elliptic problems; numerical results; stochastic discontinuity