Preconditioning Techniques for the Incompressible Navier-Stokes Equations Open Access

Wang, Zhen (2011)

Permanent URL: https://etd.library.emory.edu/concern/etds/05741r787?locale=en
Published

Abstract

We study different preconditioning techniques for the incompressible Navier-Stokes equations in two and three space dimensions. Both steady and unsteady problems are considered.

First we analyze different variants of the augmented Lagrangian-based block triangular preconditioner. The preconditioners are used to accelerate the convergence of the Generalized Minimal Residual (GMRES) method applied to both stable and stabilized finite element and MAC discretizations of the Oseen problem. We study the eigenvalues of the preconditioned matrices obtained from Picard linearization, and we devise a simple and effective method for the choice of the augmentation parameter based on Fourier analysis. Numerical experiments on a wide range of model problems demonstrate the robustness of these preconditioners, yielding fast convergence independent of mesh size and only mildly dependent on viscosity on both uniform and stretched grids. Good results are also obtained on linear systems arising from Newton linearization. We also show that performing inexact preconditioner solves with an algebraic multigrid algorithm results in excellent scalability. Comparisons of the modified augmented Lagrangian preconditioners with other state-of-the-art techniques show the competitiveness of our approach. Implementation on parallel architectures is also considered.

Moreover, we study a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting preconditioner. Numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the RDF preconditioner with respect to both mesh size and viscosity.

Table of Contents

1 Introduction 1
2 Augmented Lagrangian-based preconditioners 14
2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Ideal AL preconditioner for stable finite elements . . . . . . . 15
2.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Modified AL preconditioner . . . . . . . . . . . . . . . . . . . 26
2.4.1 Eigenvalue analysis . . . . . . . . . . . . . . . . . . . . 28
2.4.2 Estimation of the optimal augmentation parameter γ
using Fourier analysis . . . . . . . . . . . . . . . . . . . 35
2.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 39
2.5.1 Empirical rule for choosing γ . . . . . . . . . . . . . . 39
2.5.2 Fourier analysis-based approach for choosing γ . . . . . 41
2.5.3 Comparison with other preconditioners . . . . . . . . . 46
2.5.4 Inexact solves . . . . . . . . . . . . . . . . . . . . . . . 49
2.5.5 Parallel results . . . . . . . . . . . . . . . . . . . . . . 51
2.5.6 Unsteady problems . . . . . . . . . . . . . . . . . . . . 52
2.5.7 Extension to 3D problems . . . . . . . . . . . . . . . . 55
2.5.8 Numerical experiments: 3D examples . . . . . . . . . . 57
2.5.9 Parallel results: 3D examples . . . . . . . . . . . . . . 60

3 The stabilized case 63
3.1 Ideal AL preconditioners . . . . . . . . . . . . . . . . . . . . . 63
3.1.1 Problem formulation . . . . . . . . . . . . . . . . . . . 64
3.1.2 Augmented linear systems and ideal AL preconditioners 65
3.1.3 Eigenvalue analysis . . . . . . . . . . . . . . . . . . . . 68
3.2 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 76
3.3 Modified AL preconditioner . . . . . . . . . . . . . . . . . . . 78
3.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 80
3.4.1 Empirical rule for choosing γ . . . . . . . . . . . . . . 80
3.4.2 Fourier analysis-based approach for choosing γ . . . . . 80
3.4.3 Comparison with other preconditioners . . . . . . . . . 85
3.4.4 Inexact solves . . . . . . . . . . . . . . . . . . . . . . . 86
4 A relaxed dimensional factorization preconditioner 88
4.1 Dimensional splitting preconditioner . . . . . . . . . . . . . . 88
4.2 Relaxed dimensional factorization preconditioner . . . . . . . . 91
4.2.1 Practical implementation of the RDF preconditioner . 95
4.2.2 Estimation of the optimal relaxation parameter α using
Fourier analysis . . . . . . . . . . . . . . . . . . . . . . 97
4.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 98
4.3.1 Comparison with other preconditioners . . . . . . . . . 105
4.4 The stabilized case . . . . . . . . . . . . . . . . . . . . . . . . 106
4.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 112
5 Conclusions 114
Bibliography 117

About this Dissertation

Rights statement
  • Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.
School
Department
Degree
Submission
Language
  • English
Research field
Keyword
Committee Chair / Thesis Advisor
Committee Members
Last modified

Primary PDF

Supplemental Files