  This paper addresses a multi-scale finite element method for second order
linear elliptic equations with arbitrarily rough coefficient. We propose a
local oversampling method to construct basis functions that have optimal local
approximation property. Our methodology is based on the compactness of the
solution operator restricted on local regions of the spatial domain, and does
not depend on any scale-separation or periodicity assumption of the
coefficient. We focus on a special type of basis functions that are harmonic on
each element and have optimal approximation property. We first reduce our
problem to approximating the trace of the solution space on each edge of the
underlying mesh, and then achieve this goal through the singular value
decomposition of an oversampling operator. Rigorous error estimates can be
obtained through thresholding in constructing the basis functions. Numerical
results for several problems with multiple spatial scales and high contrast
inclusions are presented to demonstrate the compactness of the local solution
space and the capacity of our method in identifying and exploiting this compact
structure to achieve computational savings.
