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4 Acknowledgements First I owe a great debt of gratitude to Walter Zulehner, not only for advising my thesis but for arousing my interest and deepening my understanding in the scientific field I am currently working in. I would also like to thank Ulrich Langer who, as leader of the project and head of the institute I am employed in, provided me with scientific and financial background, and who made constructive suggestions on many details of this thesis, and I want to thank Arnold Reusken for co-refereeing this dissertation. Special thanks go to Ferdinand Kickinger who supplied me with industrial test cases and with whom I had some very fruitful discussions, and to all other people who are responsible for some aspects of this thesis by “asking the right questions”. This work has been supported by the Austrian Science Foundation “Fonds zur Förderung der wissenschaftlichen Forschung (FWF)” under the grant P14953 “Robust Algebraic Multigrid Methods and their Parallelization”.
Contents Notation 7 1 Introduction 9 2 Preliminaries 12 2.1 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Analysis of the Associated Stokes Problem — the Inf-Sup Condition 14 2.2 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Examples of Mixed Elements . . . . . . . . . . . . . . . . . . . . . 18 2.2.2 Multi-Element Meshes . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 The Non-Stationary Problem . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 The Convective Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.1 Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.2 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Iterative Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.1 Krylov Space Methods . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.2 SIMPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5.3 Inexact Uzawa Methods . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Multigrid Methods 35 3.1 A General Algebraic Multigrid Method . . . . . . . . . . . . . . . . . . . . 35 3.1.1 Basic Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Examples in the Scalar Elliptic Case . . . . . . . . . . . . . . . . . . . . . 39 3.2.1 Geometric Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.2 Algebraic Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4 AMG Methods for the Mixed Problem 48 4.1 Construction of the Coarse Level Systems . . . . . . . . . . . . . . . . . . 49 4.1.1 The P 1 isoP 2 -P 1 Hierarchy . . . . . . . . . . . . . . . . . . . . . . . 51 4.1.2 The P 1 -P 1 -stab Hierarchy . . . . . . . . . . . . . . . . . . . . . . . 53 4.1.3 The P1 nc-P 0 Hierarchy — Applying AMGe . . . . . . . . . . . . . . 58 4.2 Smoothers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2.1 Standard Smoothers for the Squared System . . . . . . . . . . . . . 64 5
Page 1 and 2: J O H A N N E S K E P L E R U N I V
Page 3: 3 Abstract If the Navier-Stokes equ
Page 7 and 8: Notation We generally use standard
Page 9 and 10: Chapter 1 Introduction A very impor
Page 11 and 12: CHAPTER 1. INTRODUCTION 11 overview
Page 13 and 14: CHAPTER 2. PRELIMINARIES 13 Physica
Page 15 and 16: CHAPTER 2. PRELIMINARIES 15 Define
Page 17 and 18: CHAPTER 2. PRELIMINARIES 17 Figure
Page 21 and 22: CHAPTER 2. PRELIMINARIES 21 2.2.1.3
Page 23 and 24: CHAPTER 2. PRELIMINARIES 23 The fir
Page 27 and 28: CHAPTER 2. PRELIMINARIES 27 2.5 Ite
Page 29 and 30: CHAPTER 2. PRELIMINARIES 29 y = Kp
Page 31 and 32: CHAPTER 2. PRELIMINARIES 31 — for
Page 33 and 34: CHAPTER 2. PRELIMINARIES 33 2.5.3.1
Page 35 and 36: Chapter 3 Multigrid Methods In the
Page 37 and 38: CHAPTER 3. MULTIGRID METHODS 37 Alg
Page 39 and 40: CHAPTER 3. MULTIGRID METHODS 39 The
Page 41 and 42: CHAPTER 3. MULTIGRID METHODS 41 Rem
Page 43 and 44: CHAPTER 3. MULTIGRID METHODS 43 wit
Page 45 and 46: CHAPTER 3. MULTIGRID METHODS 45 3.2
Page 47 and 48: CHAPTER 3. MULTIGRID METHODS 47 Alg
Page 49 and 50: CHAPTER 4. AMG METHODS FOR THE MIXE
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CHAPTER 4. AMG METHODS FOR THE MIXE
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Chapter 5 Software and Numerical St
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CHAPTER 5. SOFTWARE AND NUMERICAL S
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BIBLIOGRAPHY 93 [Ber02] [BF91] [BMR
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BIBLIOGRAPHY 95 [Hac76] W. Hackbusc
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BIBLIOGRAPHY 97 [Pat80] [Pir89] S.
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BIBLIOGRAPHY 99 [VdV92] [Ver84a] [V
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INDEX 101 restriction matrix, 36 se
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