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The goal of this paper is to design optimal multilevel solvers for the finiteelement approximation of second order linear elliptic problems with piecewiseconstant coefficients on bisection grids. Local multigrid and BPXpreconditioners are constructed based on local smoothing only at the newestvertices and their immediate neighbors. The analysis of eigenvaluedistributions for these local multilevel preconditioned systems shows thatthere are only a fixed number of eigenvalues which are deteriorated by thelarge jump. The remaining eigenvalues are bounded uniformly with respect to thecoefficients and the meshsize. Therefore, the resulting preconditionedconjugate gradient algorithm will converge with an asymptotic rate independentof the coefficients and logarithmically with respect to the meshsize. As aresult, the overall computational complexity is nearly optimal.

Local multilevel preconditioners for elliptic equations with jump coefficients on bisection grids