  The goal of this paper is to design optimal multilevel solvers for the finite
element approximation of second order linear elliptic problems with piecewise
constant coefficients on bisection grids. Local multigrid and BPX
preconditioners are constructed based on local smoothing only at the newest
vertices and their immediate neighbors. The analysis of eigenvalue
distributions for these local multilevel preconditioned systems shows that
there are only a fixed number of eigenvalues which are deteriorated by the
large jump. The remaining eigenvalues are bounded uniformly with respect to the
coefficients and the meshsize. Therefore, the resulting preconditioned
conjugate gradient algorithm will converge with an asymptotic rate independent
of the coefficients and logarithmically with respect to the meshsize. As a
result, the overall computational complexity is nearly optimal.
