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Substructured domain decomposition (DD) methods have been extensivelystudied, and they are usually associated with nonoverlapping decompositions. Weintroduce here a substructured version of Restricted Additive Schwarz (RAS)which we call SRAS, and we discuss its advantages compared to the standardvolume formulation of the Schwarz method when they are used both as iterativesolvers and preconditioners for a Krylov method. To extend SRAS to nonlinearproblems, we introduce SRASPEN (Substructured Restricted Additive SchwarzPreconditioned Exact Newton), where SRAS is used as a preconditioner forNewton's method. We study carefully the impact of substructuring on theconvergence and performance of these methods as well as their implementations.We finally introduce two-level versions of nonlinear SRAS and SRASPEN.Numerical experiments confirm the advantages of formulating a Schwarz method atthe substructured level.

Linear and nonlinear substructured Restricted Additive Schwarz iterations and preconditioning