  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.
