Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos
BROWN UNIV PROVIDENCE RI DIV OF APPLIED MATHEMATICS
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We present a generalized polynomial chaos algorithms for the solution of stochastic elliptic partial differential equations subject to uncertain inputs. In particular, were 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 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.
- Numerical Mathematics
- Theoretical Mathematics
- Statistics and Probability