The solution of large sparse linear systems is a critical operationfor many numerical simulations. To cope with the hierarchical designof modern supercomputers, hybrid solvers based on Domain DecompositionMethods (DDM) have been been proposed. Among them, approachesconsisting of solving the problem on the interior of the domains witha sparse direct method and the problem on their interface with apreconditioned iterative method applied to the related SchurComplement have shown an attractive potential as they can combine therobustness of direct methods and the low memory footprint of iterativemethods. In this report, we consider an additive Schwarz preconditionerfor the Schur Complement, which represents a scalable candidate butwhose numerical robustness may decrease when the number of domainsbecomes too large. We thus propose a two-level MPI/thread parallelapproach to control the number of domains and hence the numericalbehaviour. We illustrate our discussion with large-scale matricesarising from real-life applications and processed on both a moderncluster and a supercomputer. We show that the resulting method canprocess matrices such as tdr455k for which we previously either ranout of memory on few nodes or failed to converge on a larger number ofnodes. Matrices such as Nachos_4M that could not be correctly processedin the past can now be efficiently processed up to a very large numberof CPU cores (24,576 cores). The corresponding code has beenincorporated into the MaPHyS package
