Variational multiscale stabilized FEM formulations for transport equations: stochastic advection–diffusion and incompressible stochastic Navier–Stokes equations

An extension of the deterministic variational multiscale (VMS) approach with algebraic subgrid scale (SGS) modeling is considered for developing stabilized finite element formulations for the stochastic advection and the incompressible stochastic Navier–Stokes equations. The stabilized formulations are numerically implemented using the spectral stochastic formulation of the finite element method (SSFEM). Generalized polynomial chaos and Karhunen–Loève expansion techniques are used for representation of uncertain quantities. The proposed stabilized method is then applied to various standard advection–diffusion and fluid-flow examples with uncertainty in essential boundary conditions. Comparisons are drawn between the numerical solutions and Monte-Carlo/analytical solutions wherever possible.