The paper is concerned with numerical solving of nonlinear systems by an inexact Newton method of a special type. The application of the inexact Newton method can fail when it is used to solve some difficult nonlinear equations.
To exceed this, recently was proposed a nonlinearly preconditioned inexact Newton algorithm was recently proposed by X. C. Cai, M. Druyja and M. Sarkis [SIAM J. Numer. Anal. 41, 1209–1231 (2003; Zbl 1052.65036)] which consists in converting the given nonlinear system into another nonlinear system having the same solution, but the new system has more uniform nonlinearities, so it is relatively easy to solve. The corresponding method is the ASPIN method (additive Schwarz preconditioned inexact Newton method). This method is very effective for solving some nonlinear problems with a strong nonbalanced nonlinearity.
In this paper the author discuss the local convergence and the convergence rate of the ASPIN method. It is proved that the ASPIN method is quadratically convergent under suitable conditions.