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In this paper we use a splitting technique to develop new multiscale basisfunctions for the multiscale finite element method (MsFEM). The multiscalebasis functions are iteratively generated using a Green's kernel. The Green'skernel is based on the first differential operator of the splitting. Theproposed MsFEM is applied to deterministic elliptic equations and stochasticelliptic equations, and we show that the proposed MsFEM can considerably reducethe dimension of the random parameter space for stochastic problems. Bycombining the method with sparse grid collocation methods, the need for aprohibitive number of deterministic solves is alleviated. We rigorously analyzethe convergence of the proposed method for both the deterministic andstochastic elliptic equations. Computational complexity discussions are alsooffered to supplement the convergence analysis. A number of numerical resultsare presented to confirm the theoretical findings.

A resourceful splitting technique with applications to deterministic and stochastic multiscale finite element methods