Robust Aggregation-based Coarsening for Additive Schwarz in the Case of Highly Variable Coefficients

We study two–level overlapping domain decomposition preconditioners with coarse spaces obtained by smoothed aggregation in iterative solvers for finite element discretisations of second-order elliptic problems. We are particularly interested in the situation where the diffusion coefficient (or the permeability) α is highly variable throughout the domain. Our motivating example is Monte-Carlo simulation for flow in rock with permeability modelled by log-normal random fields. By using the concept of strongly-connected graph r-neighbourhoods (suitably adapted from the algebraic multigrid context) we design a two–level additive Schwarz preconditioner that is robust to strong variations in α as well as to mesh refinement. We give upper bounds on the condition number of the preconditioned system which do not depend on the size of the subdomains (not available previously in the literature) and make explicit the interplay between the coefficient function and the coarse space basis functions in this bound. In particular, we are able to show that the condition number can be bounded independent of the ratio of the two values of α in a binary medium even when the discontinuities in the coefficient function are not resolved by the coarse mesh. Our numerical results show that the bounds with respect to the mesh parameters are sharp and that the method is indeed robust to strong variations in α. We compare the method to other preconditioners (aggregation-type AMG and classical additive Schwarz) as well as to a sparse direct solver, and show its superiority over those methods for highly variable coefficient functions α.