### A Parallel Nonlinear Additive Schwarz Preconditioned Inexact
Newton Algorithm for Incompressible Navier-Stokes Equations

Feng-Nan Hwang

Department of Applied Mathematics,
University of Colorado,
Boulder, CO 80309

Xiao-Chuan Cai

Department of Computer Science,
University of Colorado,
Boulder, CO 80309

Abstract
A nonlinear additive Schwarz preconditioned inexact
Newton method (ASPIN)
was introduced recently for solving large
sparse nonlinear systems of equations obtained
from the discretization of nonlinear partial differential equations,
and the method has proved numerically to be more robust than the
traditional
inexact Newton methods, especially for problems with unbalanced
nonlinearities.
In this talk, we discuss some recent development of ASPIN for
solving the steady state
incompressible Navier-Stokes equations in the velocity-pressure
formulation.
The sparse nonlinear system is obtained
by using a Galerkin least squares finite element discretization on
two dimensional unstructured meshes.
The key idea of ASPIN
is that we find the solution of the original
system F(u)=0 by solving a nonlinearly preconditioned system G(u)=0
that has the same solution as the original system, but with more
balanced nonlinearities. Our numerical results show that
ASPIN is more robust than the traditional inexact Newton method
when the Reynolds number is high and when the number of processors is
large. In this talk we present some results obtained on
parallel computers for high Reynolds number flows and compare our
approach with some inexact Newton method with different forcing terms.