A stochastic variational multiscale method for diffusion in heterogeneous random media

A stochastic variational multiscale method with explicit subgrid modelling is provided for numerical solution of stochastic elliptic equations that arise while modelling diffusion in heterogeneous random media. The exact solution of the governing equations is split into two components: a coarse-scale solution that can be captured on a coarse mesh and a subgrid solution. A localized computational model for the subgrid solution is derived for a generalized trapezoidal time integration rule for the coarse-scale solution. The coarse-scale solution is then obtained by solving a modified coarse formulation that takes into account the subgrid model. The generalized polynomial chaos method combined with the finite element technique is used for the solution of equations resulting from the coarse formulation and subgrid models. Finally, various numerical examples are considered for evaluating the method.