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CONTENTS 6 4.2.2 Transforming Smoothers . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2.3 Braess-Sarazin Smoother . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.4 Vanka Smoothers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5 Software and Numerical Studies 71 5.1 The Software Package AMuSE . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.1.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2.1 2D Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2.2 3D Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Notation We generally use standard characters for scalar values and scalar functions (p, q,. . . ) and boldface characters for vectors and vector valued functions (u, v,. . . ). We will use the underline notation (which will be introduced in detail in section 2.2) for finite-element vectors associated to scalar or vector valued functions (p, q,. . . , resp. u, v, . . . ). The components of vectors are denoted by (u 1 , . . . , u n ) T = u. For matrices we use capital letters (A,. . . ), or component-notation A = (a ij ) i,j . G is an open, connected subset of R d with space dimension d (generally d = 2 or 3), ∂G its boundary. Operators u · v = ∑ d i=1 u iv i (scalar product). u ⊗ v = (u i v j ) i,j=1,...,d (tensor product). ∂ j p = ∂p ∂x j (partial derivative of p with respect to x j ). ∂ j u = (∂ j u i ) i=1,...,d . ∂ t p = ∂p (partial derivative of p with respect to t) . ∂t ∂ t u = (∂ t u i ) i=1,...,d . ∇p = (∂ i p) i=1,...,d (gradient of p). ∇u = (∂ i u j ) i,j=1,...,d . div u = ∑ d i=1 ∂ iu i (divergence of u). (u · ∇)ϕ = ∑ d j=1 u j∂ j ϕ. (u · ∇)v = ( ∑ d j=1 u j∂ j v i ) i=1,...,d . Function spaces C(G) space of continuous functions on G. C k (G) space of functions with continuous k-th derivative on G . C0 ∞ (G) space of infinitely smooth functions with compact support in G. ∞ C (Ḡ) space of infinitely smooth functions on Ḡ. L p (G) Lebesgue space of measurable functions q with finite norm ‖q‖ 0,p = (∫ |q|p) 1 p . G 7
Page 1 and 2: J O H A N N E S K E P L E R U N I V
Page 3 and 4: 3 Abstract If the Navier-Stokes equ
Page 5: Contents Notation 7 1 Introduction
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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