The simulation of processes in highly heterogeneous media comes with many challenges. In particular many domain decomposition methods do not perform well in this case, specially if the decomposition into subdomains does not accommodate the coefficient variations. For three popular domain decomposition methods (two level additive Schwarz, BDD and FETI) we have proposed a remedy to this problem in previous work with coauthors. Here we present the strategy which was used by applying it to the Hybrid Schwarz preconditioner. It is based on identifying a bottleneck estimate in the proof of convergence which cannot be satisfied for the entire solution space. Then the part of the solution which is problematic is isolated via a generalized eigenvalue problem and solved separately.
