  A two-level overlapping Schwarz method is developed for second order elliptic
problems with highly oscillatory and high contrast coefficients, for which it
is known that the standard coarse problem fails to give a robust
preconditioner. In this paper, we develop energy minimizing multiscale finite
element functions to form a more robust coarse problem. First, a local spectral
problem is solved in each non-overlapping coarse subdomain, and dominant
eigenfunctions are selected as auxiliary functions, which are crucial for the
high contrast case. The required multiscale basis functions are then obtained
by minimizing an energy subject to some orthogonality conditions with respect
to the auxiliary functions. Due to an exponential decay property, the
minimization problem is solved locally on oversampling subdomains, that are
unions of a few coarse subdomains. The coarse basis functions are therefore
local and can be computed efficiently. The resulting preconditioner is shown to
be robust with respect to the contrast in the coefficients as well as the
overlapping width in the subdomain partition. Numerical results are presented
to validate the theory and show the performance.
