Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos

• Published in: Computer methods in applied mechanics and engineering Vol. 191; no. 43; pp. 4927 - 4948
• Main Authors: Xiu, Dongbin, Em Karniadakis, George
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 27.09.2002; Elsevier
• Subjects: Analytical and numerical techniques; Computational techniques; Exact sciences and technology; Finite-element and galerkin methods; Fundamental areas of phenomenology (including applications); Heat transfer; Mathematical methods in physics; Physics; Polynomial chaos; Random diffusion; Uncertainty; 79; Uncertainty; Polynomial chaos; 36; Random diffusion; Chaos; Thermal diffusion; Uncertain systems; Probabilistic approach; Poisson equation; Orthogonal polynomial; Galerkin-Petrov method; Weak solution; Elliptic equations; Modelling; Deterministic approach; Gauss Seidel method; Heat transfer; Stochastic equation
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/S0045-7825(02)00421-8

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 Wiener [Amer. J. Math. 60 (1938) 897]. 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.