  A convergence result for a discontinuous Galerkin multiscale method for a
second order elliptic problem is presented. We consider a heterogeneous and
highly varying diffusion coefficient in $L^\infty(\Omega,\mathbb{R}^{d\times
d}_{sym})$ with uniform spectral bounds and without any assumption on scale
separation or periodicity. The multiscale method uses a corrected basis that is
computed on patches/subdomains. The error, due to truncation of corrected
basis, decreases exponentially with the size of the patches. Hence, to achieve
an algebraic convergence rate of the multiscale solution on a uniform mesh with
mesh size $H$ to a reference solution, it is sufficient to choose the patch
sizes as $\mathcal{O}(H|\log(H^{-1})|)$. We also discuss a way to further
localize the corrected basis to element-wise support leading to a slight
increase of the dimension of the space. Improved convergence rate can be
achieved depending on the piecewise regularity of the forcing function. Linear
convergence in energy norm and quadratic convergence in $L^2$-norm is obtained
independently of the forcing function. A series of numerical experiments
confirms the theoretical rates of convergence.
