Convergence Of A Multiscale Finite Element Method For Elliptic Problems With Rapidly Oscillating Coefficients

. We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are adapted to the local property of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such simplifying assumption is not required by our method, it allows us to use the homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discuss..