Intrusive and non-intrusive chaos approximation for a two-dimensional steady state Navier-Stokes system with random forcing

While convergence of a chaos approximation for linear equations is relatively well understood, a lot less is known for non-linear equations. The paper investigates this convergence, by establishing the corresponding a priori error bounds, for a particular equation with quadratic nonlinearity and for two different approximations: stochastic Galerkin and discrete projection. Stochastic Galerkin approximation reduces the stochastic equation to a system of deterministic equation to compute the coefficients in the chaos expansion. The approximation is called intrusive because the resulting system of equations is highly coupled and is harder to solve than the original system; there is also a special condition for uniqueness of solution. An alternative approximation of the chaos coefficients, using the discrete projection version of the stochastic collocation method, is non-intrusive and requires the solution of the original equation for specially chosen realizations of the random input. Compared to the Galerkin approximation, this non-intrusive procedure is easier to analyze and implement, but the resulting approximation error and computational costs can be higher.