Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. (English) Zbl 1016.65001

Summary: We present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs. In particular, we focus on the solution of the Poisson equation with random diffusivity, forcing and boundary conditions. The stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the Askey scheme, as a generalization of the original polynomial chaos idea of N. Wiener [Am. J. Math. 60, 897-936 (1938)]. A Galerkin projection in random space is applied to derive the equations in the weak form. The resulting set of deterministic equations for each random mode is solved iteratively by a block Gauss-Seidel iteration technique. Both discrete and continuous random distributions are considered, and convergence is verified in model problems and against Monte Carlo simulations.


65C30 Numerical solutions to stochastic differential and integral equations
35R60 PDEs with randomness, stochastic partial differential equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65C05 Monte Carlo methods
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)


Zbl 0019.35406
Full Text: DOI


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