NONLINEAR PRECONDITIONING: HOW TO USE A NONLINEAR SCHWARZ METHOD TO PRECONDITION NEWTON'S METHOD

Victorita Dolean (1, 2) , Martin J. Gander (3) , Walid Kheriji (2) , F Kwok (4) , Roland Masson (2)

1 University of Strathclyde [Glasgow]
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 Section de mathématiques [Genève]
4 UNIGE - Université de Genève = University of Geneva

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 non-linear problems, domain decomposition methods can either be used directly as iterative solvers, or one can use them 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 non-linear iterative domain decomposition method and a two level RASPEN non-linear preconditioner. We illustrate our findings with numerical results on the Forchheimer equation and a non-linear diffusion problem. 1. Introduction. Non-linear partial differential equations are usually solved after discretization by Newton's method or variants thereof. While Newton's method converges well from an initial guess close to the solution, its convergence behaviour can be erratic and the method can lose all its effectiveness if the initial guess is too far from the solution. Instead of using Newton, one can use a domain decomposition iteration , applied directly to the non-linear partial differential equations, and one obtains then much smaller subdomain problems, which are often easier to solve by Newton's method than the global problem. The first analysis of an extension of the classical alternating Schwarz method to non-linear monotone problems can be found in [24], where a convergence proof is given at the continuous level for a minimization formulation of the problem. A two-level parallel additive Schwarz method for non-linear problems was proposed and analyzed in [12], where the authors prove that the non-linear iteration converges locally at the same rate as the linear iteration applied to the linearized equations about the fixed point, and also a global convergence result is given in the case of a minimization formulation under certain conditions. In [25], the classical alternating Schwarz method is studied at the continuous level, when applied

Mots clés: Non-Linear Preconditioning; Two-Level Non-Linear Schwarz Methods; Precondi-tioning Newton's Method AMS subject classifications 65M55; 65F10; 65N22

Domaines: Analyse numérique [math.NA]

Victorita Dolean, Martin J. Gander, Walid Kheriji, F Kwok, Roland Masson. NONLINEAR PRECONDITIONING: HOW TO USE A NONLINEAR SCHWARZ METHOD TO PRECONDITION NEWTON'S METHOD. SIAM Journal on Scientific Computing, 2016. ⟨hal-01171167⟩