making multigrid/dd methods converge (nearly) uniformly for elliptic equations with strongly discontinuous coefficients was an open problem. 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 the augmented lagrange method.