POD-Galerkin Modeling and Sparse-Grid Collocation for a Natural Convection Problem with Stochastic Boundary Conditions

Abstract

The computationally most expensive part of the stochastic collocation method are usually the numerical solutions of a deterministic equation at the collocation points. We propose a way to reduce the total computation time by replacing the deterministic model with its Galerkin projection on the space spanned by a small number of basis functions. The proper orthogonal decomposition (POD) is used to compute the basis functions from the solutions of the deterministic model at a few collocation points. We consider the computation of the statistics of the Nusselt number for a two-dimensional stationary natural convection problem with a stochastic temperature boundary condition. It turns out that for the estimation of the mean and the probability density, the number of finite element simulations can be significantly reduced by the help of the POD-Galerkin reduced-order model.