  In this paper we propose simple multiscale basis functions with constraint
energy minimization to solve elliptic problems with high contrast medium. Our
methodology is based on the recently developed non-local multicontinuum method
(NLMC). The main ingredient of the method is the construction of suitable local
basis functions with the capability of capturing multiscale features and
non-local effects. In our method, each coarse block is decomposed into various
regions according to the contrast ratio, and we require that the contrast ratio
should be relatively small within each region. The basis functions are
constructed by solving a local problem defined on the oversampling domains and
they have mean value one on the chosen region and zero mean otherwise.
Numerical analysis shows that the resulting basis functions can be localizable
and have a decay property. The convergence of the multiscale solution is also
proved. Finally, some numerical experiments are carried out to illustrate the
performances of the proposed method. They show that the proposed method can
solve problem with high contrast medium efficiently. In particular, if the
oversampling size is large enough, then we can achieve the desired error.
