  In this paper, we introduce a multiscale framework based on adaptive edge
basis functions to solve second-order linear elliptic PDEs with rough
coefficients. One of the main results is that we prove the proposed multiscale
method achieves nearly exponential convergence in the approximation error with
respect to the computational degrees of freedom. Our strategy is to perform an
energy orthogonal decomposition of the solution space into a coarse scale
component comprising $a$-harmonic functions in each element of the mesh, and a
fine scale component named the bubble part that can be computed locally and
efficiently. The coarse scale component depends entirely on function values on
edges. Our approximation on each edge is made in the Lions-Magenes space
$H_{00}^{1/2}(e)$, which we will demonstrate to be a natural and powerful
choice. We construct edge basis functions using local oversampling and singular
value decomposition. When local information of the right-hand side is
adaptively incorporated into the edge basis functions, we prove a nearly
exponential convergence rate of the approximation error. Numerical experiments
validate and extend our theoretical analysis; in particular, we observe no
obvious degradation in accuracy for high-contrast media problems.
