Domain decomposition methods. Application to high-performance computing

Abstract : This thesis introduces a unified framework for various domain decomposition methods:those with overlap, so-called Schwarz methods, and those based on Schur complements,so-called substructuring methods. It is then possible to switch with a high-level of abstractionbetween methods and to build different preconditioners to accelerate the iterativesolution of large sparse linear systems. Such systems are frequently encountered in industrialor scientific problems after discretization of continuous models. Even though thesepreconditioners naturally exhibit good parallelism properties on distributed architectures,they can prove inadequate numerical performance for complex decompositions or multiscalephysics. This lack of robustness may be alleviated by concurrently solving sparse ordense local generalized eigenvalue problems, thus identifying modes that hinder the convergenceof the underlying iterative methods a priori. Using these modes, it is then possibleto define projection operators based on what is usually referred to as a coarse solver. Theseauxiliary tools tend to solve the aforementioned issues, but typically decrease the parallelefficiency of the preconditioners. In this dissertation, it is shown in three points thatthe newly developed construction is efficient: 1) by performing large-scale numerical experimentson Curie—a European supercomputer, and by comparing it with state of the art2) multigrid and 3) direct solvers.
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Pierre Jolivet. Domain decomposition methods. Application to high-performance computing. General Mathematics [math.GM]. Université Grenoble Alpes, 2014. English. ⟨NNT : 2014GRENM040⟩. ⟨tel-01155718⟩

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