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


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.