  This note constructs a local generalized finite element basis for elliptic
problems with heterogeneous and highly varying coefficients. The basis
functions are solutions of local problems on vertex patches. The error of the
corresponding generalized finite element method decays exponentially with
respect to the number of element layers in the patches. Hence, on a uniform
mesh of size H, patches of diameter H\log(1/H) are sufficient to preserve a
linear rate of convergence in H without any pre-asymptotic or resonance
effects. The analysis does not rely on regularity of the solution or scale
separation in the coefficient. This result motivates new and justifies old
classes of variational multiscale methods.
