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CHAPTER 4. THEORY 109 Table 4.1 shows the performance of GMRES when applied to the finite-dimensional problem where the preconditioner ˜ M =(K − 1 τ I)−1 . Here, we show the average number of Newton iterations per continuation step and the average number of GMRES iterations per Newton iteration. As the table shows, these values remain relatively flat as the grid is refined. Table 4.1: Krylov and Newton Iterations as Mesh Is Refined Nx Nk Avg. Newton Its. Per Cont. Step Avg. Krylov Its. Per Newton 86 72 2.6 176 172 144 2.5 174 344 288 2.5 173 688 576 2.5 186 This is a reflection of the rapid convergence we expect in the infinite-dimensional setting. If we were able to prove that Z : L 2 → X was a compact operator, we could then show Z ′ : L 2 → X is a compact linear operator, and we would be able to apply the theory given in [21]. In [21], the authors look at a separable real Hilbert space ˜ H with a subspace X that has a compact linear operator Z ′ that maps ˜ H to X where they want to solve the equation (I − Z ′ )u = f. In this paper, by using a modified version of GMRES, the authors prove the error (not the residual) of the GMRES iterates converge in the X to 0.
CHAPTER 4. THEORY 110 4.3 Application of Schauder’s Fixed Point Theo- rem The Schauder’s Fixed Point Theorem [34] says that if an operator R : X → X • is compact and continuous • maps a convex subset of X back to itself, then there exist a fixed point of R, that is there is a u ∈ X such that u = R(u). Define the operator R : X → X by R(u)(x, k) =Z(u)(x, k)+ā(x, k) for u ∈ X. Since we have shown that Z is a compact and continuous operator on X, then it follows that R is a compact and continuous operator on X. Let¯r>0. Let B¯r = {u ∈ X|uX ≤ ¯r} denote the closed ball of radius ¯r (measured in the X-norm) about the zero function. Now for any ¯r >0, B¯r is a convex set. Let u ∈ B¯r. Thenwehave R(u)X = Z(u)+āX ≤ Z(u)X + āX ≤ G1(u)X + G2(u)X + āX ≤ 2Kmaxh∞uX + 4τ 2KmaxT ∞uX + =āX h ≤ 2Kmax h∞ + 4τ T ∞ uX + āX. h
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Numerical Methods for the Wigner-Po
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NUMERICAL METHODS FOR THE WIGNER-PO
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Acknowledgments First and foremost,
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Table of Contents List of Tables ix
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4.3 ApplicationofSchauder’sFixedP
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List of Tables 1.1 PhysicalConstant
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3.11 I-V Curve with 30 Percent Redu
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CHAPTER 1. OVERVIEW 2 being heavily
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CHAPTER 1. OVERVIEW 4 Doping Spacer
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CHAPTER 1. OVERVIEW 6 and identfyin
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CHAPTER 1. OVERVIEW 8 for predictin
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CHAPTER 1. OVERVIEW 10 by Schrödin
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CHAPTER 1. OVERVIEW 12 ranging term
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CHAPTER 1. OVERVIEW 14 But because
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CHAPTER 1. OVERVIEW 16 = h 2π ,we
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CHAPTER 1. OVERVIEW 18 Table 1.1: P
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CHAPTER 1. OVERVIEW 20 where q is t
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CHAPTER 1. OVERVIEW 22 get K(f)
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CHAPTER 1. OVERVIEW 24 rule. The pr
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CHAPTER 1. OVERVIEW 26 The weights
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Chapter 2 Temporal Integration 2.1
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CHAPTER 2. TEMPORAL INTEGRATION 30
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Chapter 3 Bifurcation Analysis 3.1
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CHAPTER 3. BIFURCATION ANALYSIS 46
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Page 71 and 72: CHAPTER 3. BIFURCATION ANALYSIS 58
Page 87 and 88: Chapter 4 Theory 4.1 Steady-State T
Page 89 and 90: CHAPTER 4. THEORY 76 If we write ou
Page 91 and 92: CHAPTER 4. THEORY 78 So if we write
Page 93 and 94: CHAPTER 4. THEORY 80 Combining Equa
Page 95 and 96: CHAPTER 4. THEORY 82 We now estimat
Page 97 and 98: CHAPTER 4. THEORY 84 pendent of x,
Page 99 and 100: CHAPTER 4. THEORY 86 interval we ha
Page 101 and 102: CHAPTER 4. THEORY 88 So an estimate
Page 103 and 104: CHAPTER 4. THEORY 90 We want to cho
Page 105 and 106: CHAPTER 4. THEORY 92 The exact same
Page 107 and 108: CHAPTER 4. THEORY 94 Since un → u
Page 109 and 110: CHAPTER 4. THEORY 96 thereexistsaco
Page 111 and 112: CHAPTER 4. THEORY 98 For 0
Page 113 and 114: CHAPTER 4. THEORY 100 denote this d
Page 115 and 116: CHAPTER 4. THEORY 102 Now, we can t
Page 117 and 118: CHAPTER 4. THEORY 104 operator. 4.2
Page 119 and 120: CHAPTER 4. THEORY 106 we have at th
Page 121: CHAPTER 4. THEORY 108 we have at th
Page 125 and 126: CHAPTER 4. THEORY 112 We need to fi
Page 127 and 128: Chapter 5 Conclusion In this work,
Page 129 and 130: CHAPTER 5. CONCLUSION 116 • Findi
Page 131 and 132: REFERENCES 118 ODE solver. Technica
Page 133 and 134: REFERENCES 120 [21] C. T. Kelley an
Page 135 and 136: REFERENCES 122 [37] Homer F. Walker
Page 137 and 138: APPENDIX A. NOTATION 124 ˜B Genera
Page 139 and 140: APPENDIX A. NOTATION 126 k Wave num
Page 141 and 142: APPENDIX A. NOTATION 128 ∆t Incre
Page 143 and 144: APPENDIX A. NOTATION 130 ɛc ɛk ɛ
Page 145 and 146: APPENDIX B. TRILINOS 132 user-given
Page 147 and 148: APPENDIX B. TRILINOS 134 Trilinos,
Page 149 and 150: APPENDIX B. TRILINOS 136 Minimum Re
Page 151 and 152: APPENDIX B. TRILINOS 138 A very use
Page 153 and 154: Appendix C Guide to RTD Simulation
Page 155 and 156: APPENDIX C. GUIDE TO RTD SIMULATION
Page 163 and 164: Appendix D GMRES and Arnoldi Iterat
Page 165 and 166: APPENDIX D. GMRES AND ARNOLDI ITERA
Page 167 and 168: APPENDIX D. GMRES AND ARNOLDI ITERA
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