A multiscale stochastic finite element method on elliptic problems involving uncertainties

Abstract

In this study a multiscale stochastic finite element method (MsSFEM) is developed to resolve scale-coupling stochastic elliptic problems, based on formulations of a stochastic variational approach and scale-bridging multiscale shape functions. By employing polynomial chaos expansions and Lagrange polynomials, stochastic Galerkin method and stochastic collocation method are introduced and combined with deterministic multiscale methods, i.e. Variational Multiscale Method and Multiscale Finite Element Method. The resulting spectral and pseudo-spectral stochastic finite element methods are incorporated into the fine- and coarse-mesh computation of the MsSFEM. A benchmark multiscale model and the numerical experiment based on the formulated MsSFEM are provided for illustration. It is expected that the proposed MsSFEM can act as a paradigm for solving of general stochastic partial differential equations involving multiscale stochastic data.