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

For linear problems, domain decomposition methods can be used directly as iterative solvers but also as preconditioners for Krylov methods. In practice, Krylov acceleration is almost always used, since the Krylov method finds a much better residual polynomial than the stationary iteration and thus converges much faster. We show in this paper that also for nonlinear problems, domain decomposition methods can be used either directly as iterative solvers or as preconditioners for Newton's method. For the concrete case of the parallel Schwarz method, we show that we obtain a preconditioner we call RASPEN (restricted additive Schwarz preconditioned exact Newton), which is similar to ASPIN (additive Schwarz preconditioned inexact Newton) but with all components directly defined by the iterative method. This has the advantage that RASPEN already converges when used as an iterative solver, in contrast to ASPIN, and we thus get a substantially better preconditioner for Newton's method. The iterative construction also allows us to naturally define a coarse correction using the multigrid full approximation scheme, which leads to a convergent two-level nonlinear iterative domain decomposition method and a two level RASPEN nonlinear preconditioner. We illustrate our findings with numerical results on the Forchheimer equation and a nonlinear diffusion problem.