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For linear problems, domain decomposition methods can be used directly asiterative solvers, but also as preconditioners for Krylov methods. In practice,Krylov acceleration is almost always used, since the Krylov method finds a muchbetter residual polynomial than the stationary iteration, and thus convergesmuch faster. We show in this paper that also for non-linear problems, domaindecomposition methods can either be used directly as iterative solvers, or onecan use them as preconditioners for Newton's method. For the concrete case ofthe parallel Schwarz method, we show that we obtain a preconditioner we callRASPEN (Restricted Additive Schwarz Preconditioned Exact Newton) which issimilar to ASPIN (Additive Schwarz Preconditioned Inexact Newton), but with allcomponents directly defined by the iterative method. This has the advantagethat RASPEN already converges when used as an iterative solver, in contrast toASPIN, and we thus get a substantially better preconditioner for Newton'smethod. The iterative construction also allows us to naturally define a coarsecorrection using the multigrid full approximation scheme, which leads to aconvergent two level non-linear iterative domain decomposition method and a twolevel RASPEN non-linear preconditioner. We illustrate our findings withnumerical results on the Forchheimer equation and a non-linear diffusionproblem.

Nonlinear Preconditioning: How to use a Nonlinear Schwarz Method to Precondition Newton's Method