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.
翻译：亚结构化域分解(DD)方法已经进行了广泛研究,通常与非重叠分解(DD)有关,我们在此介绍一个亚结构化版本,称为SRAS(RAS),我们讨论其优缺点,与Schwarz方法的标准体积配方相比,当它们同时作为迭代解答器和Krylov方法的先决条件使用时,我们讨论了Schwarz方法的优点。为了将SRASS扩大到非线性问题,我们采用了SRASPON(结构化受限制的Additive Schwarz先设附加的Exact Newton),其中将SRAS用作Newton方法的先决条件。我们仔细研究了次结构化对这些方法的趋同和性能及其实施的影响。我们最后引进了非线性SRAS和SRASPEN的两级版本。数字实验证实了在亚结构化一级制定Swarz方法的优点。
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