Linear and nonlinear substructured Restricted Additive Schwarz iterations and preconditioning

Substructured domain decomposition (DD) methods have been extensively studied, and they are usually associated with nonoverlapping decompositions. We introduce here a substructured version of Restricted Additive Schwarz (RAS) which we call SRAS, and we discuss its advantages compared to the standard volume formulation of the Schwarz method when they are used both as iterative solvers and preconditioners for a Krylov method. To extend SRAS to nonlinear problems, we introduce SRASPEN (Substructured Restricted Additive Schwarz Preconditioned Exact Newton), where SRAS is used as a preconditioner for Newton's method. We study carefully the impact of substructuring on the convergence 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 at the substructured level.