Uniform Convergent Multigrid Methods for Elliptic Problems With Strongly Discontinuous Coefficients

This paper gives a solution to an open problem concerning the performance of various multilevel preconditioners for the linear finite element approximation of second-order elliptic boundary value problems with strongly discontinuous coefficients. By analyzing the eigenvalue distribution of the BPX preconditioner and multigrid V-cycle preconditioner, we prove that only a small number of eigenvalues may deteriorate with respect to the discontinuous jump or meshsize, and we prove that all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and meshsize. As a result, we prove that the convergence rate of the preconditioned conjugate gradient methods is uniform with respect to the large jump and meshsize. We also present some numerical experiments to demonstrate the theoretical results.