We consider two-level overlapping domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients, as they arise in practice, for example, in the computation of flows in heterogeneous porous media, in both the deterministic and (Monte-Carlo simulated) stochastic cases. We propose and analyse new robust coarsening strategies for these problems, e.g. based on smoothed aggregation and multiscale finite elements, and study different ways of combining the coarse solve with the local solves (i.e. additive, multiplicative and deflation). We will use and extend a recently developed general theoretical framework for analysing the resulting preconditioners to show their robustness for a variety of model situations. In particular, our estimates prove very precisely the previously made empirical observation that the use of low-energy coarse spaces can lead to robust preconditioners. Numerical experiments will be used to illustrate the robustness and efficiency of the proposed methods and the sharpness of our theoretical results.

This is joint work with I.G. Graham and J. Van lent (University of Bath) and E. Vainikko (University of Tartu, Estonia).