In this paper we generalize the analysis of classical multigrid and two-level overlapping Schwarz methods for 2nd order elliptic boundary value problems to problems with large discontinuities in the coefficients that are not resolved by the coarse grids or the subdomain partition. The theoretical results provide a recipe for designing hierarchies of standard piecewise linear coarse spaces such that the multigrid convergence rate and the condition number of the Schwarz preconditioned system do not depend on the coefficient variation or on any mesh parameters. An assumption we have to make is that the coarse grids are sufficiently fine in the vicinity of cross points or where regions with large diffusion coefficients are separated by a narrow region where the coefficient is small. We do not need to align them with possible discontinuities in the coefficients. The proofs make use of novel stable splittings based on weighted quasi-interpolants and weighted Poincaré-type inequalities. Numerical experiments are included that illustrate the sharpness of the theoretical bounds and the necessity of the technical assumptions.
Scheichl, R., Vassilevski, P. S., & Zikatanov, L. T. (2012). Multilevel methods for elliptic problems with highly varying coefficients on nonaligned coarse grids. SIAM Journal on Numerical Analysis (SINUM), 50(3), 1675-1694. https://doi.org/10.1137/100805248