  In this paper we use a splitting technique to develop new multiscale basis
functions for the multiscale finite element method (MsFEM). The multiscale
basis functions are iteratively generated using a Green's kernel. The Green's
kernel is based on the first differential operator of the splitting. The
proposed MsFEM is applied to deterministic elliptic equations and stochastic
elliptic equations, and we show that the proposed MsFEM can considerably reduce
the dimension of the random parameter space for stochastic problems. By
combining the method with sparse grid collocation methods, the need for a
prohibitive number of deterministic solves is alleviated. We rigorously analyze
the convergence of the proposed method for both the deterministic and
stochastic elliptic equations. Computational complexity discussions are also
offered to supplement the convergence analysis. A number of numerical results
are presented to confirm the theoretical findings.

Numerical Analysis