Abstract. The multiplicative Schwarz preconditioned inexact Newton (MSPIN) algorithm is presented as a complement to additive Schwarz preconditioned inexact Newton (ASPIN). At an algebraic level, ASPIN and MSPIN are variants of the same strategy to improve the convergence of systems with unbalanced nonlinearities; however, they have natural complementarity in practice. MSPIN is naturally based on partitioning of degrees of freedom in a nonlinear PDE system by field type rather than by subdomain, where a modest factor of concurrency can be sacrificed for physically motivated convergence robustness. ASPIN, originally introduced for decompositions into subdomains, is natural for high concurrency and reduction of global synchronization. We consider both types of inexact Newton algorithms in the field-split context, and we augment the classical convergence theory of ASPIN for the multiplicative case. Numerical experiments show that MSPIN can be significantly more robust than Newton methods based on global linearizations, and that MSPIN can be more robust than ASPIN and maintain fast convergence even for challenging problems, such as high Reynolds number Navier-Stokes equations.Key words. nonlinear equations, nonlinear preconditioning, field splitting, Newton method, Navier-Stokes equations AMS subject classifications. 65H10, 65H20, 65N22, 65N55 DOI. 10.1137/140970379 1. Introduction. Newton-like methods in their many variants are often favored for the solution of nonlinear systems, especially large algebraic systems arising from discretized differential equations. However, the problem of "nonlinear stiffness" frequently arises, in which progress in updating the state variables with a global Newton step is retarded by an often small subset of the variable-equation pairs. Global Newton-like methods may waste considerable computational resources while the majority of state variables barely evolve for many iterations until some critical feature (e.g., boundary or interior layer, aerodynamic shock, reaction zone, contact discontinuity, phase transition) of the solution falls into place, following which the desired superlinear asymptotic convergence of Newton polishes the root. These unproductive iterations typically require the solution of large linear systems with ill-conditioned Jacobians. Worse, Newton may stagnate indefinitely.To conquer unbalanced nonlinearities and improve global convergence properties, the additive Schwarz preconditioned inexact Newton (ASPIN) was devised in [5] as an inner-outer Newton solver, with most of the work performed on independent subproblems in inner iterations, plus relatively few outer iterations on a transformed system, on which Newton is supposed to converge quickly. The key idea of ASPIN is to transform the original system into a modified system with the same root, and to solve it using a Jacobian-free [19] inexact Newton method.
