It was an open problem that how to make multigrid/DD methods convergence (nearly) uniformly for the elliptic equations with strongly discontinuous coefficients. Recently, we proved that the multilevel and DD preconditioners lead to a nearly uniform convergent preconditioned conjugate gradient methods. In this talk, I will present the theoretical and numerical justification of these results. As an application of these elliptic solvers, I will also present the auxiliary space preconditioners (Hiptmair and Xu 2007) for H(curl) and H(div) systems, which convert solving H(curl) or H(div) systems into solving several Poisson equations. Another way to interpret these preconditioners is to cast the H(curl) and H(div) systems into a compatible discretization framework. Using this framework, I will derive the algorithm for solving H(div) systems, and use it to solve the mixed formulation of Poisson equation by augmented Lagrange method.