Preconditioning Techniques for the Incompressible Navier-Stokes Equations Público

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

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Palabra Clave	saddle point problems Fourier analysis dimensional factorization augmented Lagrangian method parallel computing Krylov subspace methods inexact solves eigenvalue analysis preconditioning AMG Oseen problem
Committee Chair / Thesis Advisor	Benzi, Michele, Emory University
Committee Members	Veneziani, Alessandro, Emory University Nagy, James, Emory University

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