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

A nonlinear additive Schwarz preconditioned inexact Newton method (ASPIN) was introduced recently for solving large sparse highly nonlinear system of equations obtained from the discretization of nonlinear partial differential equations. In this paper, we discuss some extensions of ASPIN for solving the steady-state incompressible Navier-Stokes equations with high Reynolds numbers in the velocity-pressure formulation. The key idea of ASPIN is to find the solution of the original system by solving a nonlinearly preconditioned system that has the same solution as the original system, but with more balanced nonlinearities. Our parallel nonlinear preconditioner is constructed using the nonlinear overlapping additive Schwarz method. To show the robustness and scalability of the algorithm, we present some numerical results obtained on a parallel computer for two benchmark problems: A driven cavity flow problem and a backward-facing step problem with high Reynolds numbers. The sparse nonlinear system is obtained by applying a Q1−Q1 Galerkin least squares finite element discretization on two-dimensional unstructured meshes. We compare our approach with an inexact Newton method with backtracking and with different choices of forcing terms. Our numerical results show that ASPIN has good convergence and is more robust than the traditional inexact Newton method with respect to certain parameters such as the Reynolds number, the mesh size, and the number of processors.