Non-linear additive Schwarz preconditioners and application in computational fluid dynamics

The focus of this paper is on the numerical solution of large sparse non-linear systems of algebraic equations on parallel computers. Such non-linear systems often arise from the discretization of non-linear partial differential equations, such as the Navier-Stokes equations for fluid flows, using finite element or finite difference methods. A traditional inexact Newton method, applied directly to the discretized system, does not work well when the non-linearities in the algebraic system become unbalanced. In this paper, we study some preconditioned inexact Newton algorithms, including the single-level and multilevel non-linear additive Schwarz preconditioners. Some results for solving the high Reynolds number incompressible Navier-Stokes equations are reported.