In this paper, we analyze two-level preconditioners for second and fourth order elliptic boundary value problems. These preconditioners involve smoothing on the original problem and the solution (or preconditioning) of an auxiliary problem on a related mesh. Two abstract theorems are provided as a basis for this analysis. Properties needed to apply these theorems are developed for general finite element approximation spaces. These results are then applied to the second order and Biharmonic Dirichlet problems. Uniform preconditioning estimates are proved in the general case when the triangulations are only assumed to be shape regular but not necessarily quasiuniform. In the case when the meshes are of locally comparable size, this analysis applies to both conforming and nonconforming finite element approximations. When the preconditioning mesh is genuinely coarser than the original, an analysis is given in the case that the auxiliary problem is conforming. For this application, it is shown that appropriate smoothers can be defined based on overlapping Schwarz methods.