A class of parallel two-level nonlinear Schwarz preconditioned inexact Newton algorithms

We propose and test a new class of two-level nonlinear additive Schwarz preconditioned inexact Newton algorithms (ASPIN). The two-level ASPIN combines a local nonlinear additive Schwarz preconditioner and a global linear coarse preconditioner. This approach is more attractive than the two-level method introduced in [X.-C. Cai, D.E. Keyes, L. Marcinkowski, Nonlinear additive Schwarz preconditioners and applications in computational fluid dynamics, Int. J. Numer. Methods Fluids, 40 (2002), 1463-1470], which is nonlinear on both levels. Since the coarse part of the global function evaluation requires only the solution of a linear coarse system rather than a nonlinear coarse system derived from the discretization of original partial differential equations, the overall computational cost is reduced considerably. Our parallel numerical results based on an incompressible lid-driven flow problem show that the new two-level ASPIN is quite scalable with respect to the number of processors and the fine mesh size when the coarse mesh size is fine enough, and in addition the convergence is not sensitive to the Reynolds numbers.