GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Stabilized finite element methods. I: Application to the advective-diffusive model
Computer Methods in Applied Mechanics and Engineering
Stabilized finite element methods. II: The incompressible Navier-Stokes equations
Computer Methods in Applied Mechanics and Engineering
Choosing the forcing terms in an inexact Newton method
SIAM Journal on Scientific Computing - Special issue on iterative methods in numerical linear algebra; selected papers from the Colorado conference
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Journal of Computational Physics
Convergence Analysis of Pseudo-Transient Continuation
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
A Multigrid-Preconditioned Newton--Krylov Method for the Incompressible Navier--Stokes Equations
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Nonlinearly Preconditioned Inexact Newton Algorithms
SIAM Journal on Scientific Computing
On backtracking failure in newton-GMRES methods with a demonstration for the navier-stokes equations
Journal of Computational Physics
A parallel block multi-level preconditioner for the 3D incompressible Navier--Stokes equations
Journal of Computational Physics
Pseudotransient Continuation and Differential-Algebraic Equations
SIAM Journal on Scientific Computing
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
Journal of Computational Physics
ICCS '09 Proceedings of the 9th International Conference on Computational Science: Part I
Journal of Computational Physics
Journal of Computational Physics
A new parallel finite element algorithm for the stationary Navier-Stokes equations
Finite Elements in Analysis and Design
SIAM Journal on Scientific Computing
A combined linear and nonlinear preconditioning technique for incompressible navier-stokes equations
PARA'04 Proceedings of the 7th international conference on Applied Parallel Computing: state of the Art in Scientific Computing
A Parallel Domain Decomposition Method for 3D Unsteady Incompressible Flows at High Reynolds Number
Journal of Scientific Computing
Hi-index | 31.46 |
A nonlinear additive Schwarz preconditioned inexact Newton method (ASPIN) was introduced recently for solving large sparse highly nonlinear systems of equations obtained from the discretization of nonlinear partial differential equations. In this paper, we discuss some extensions of ASPIN for solving 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 a 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 Q"1-Q"1 Galerkin least squares finite element discretization on two-dimensional unstructured meshes. We compare our approach with an inexact Newton method using 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.