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36 Yu.A. Kuznetsov At the second step, we eliminate the vector ¯p in (24). Then, we get the system ̂Mū H + C T ¯λ =ḡ (27) complemented by the interface and boundary conditions for the components of ū H . Here, ̂M H = ˜M H + ˜B HS T −1 ˜B h H (28) and ḡ = ˜B HS T −1 h F. (29) To analyze the matrix ̂M H in (28), we consider the eigenvalue problem S h ¯w = µΣ ¯w. (30) Let ν be the dimension of S h . Then problem (30) has ν positive eigenvalues and ν corresponding Σ-orthonormal eigenvectors 1=µ 1
Mixed FE Methods on Polyhedral Meshes 37 To derive the required final discretization for the problem (4) we introduce the new variable ˆp by the formula ˆp = 1 ] [ē T σ 2 ˜BH ū H − ē T F ≡ 1 [ ] σ 2 BHū 0 H + |E|f , (38) where f = − |E|ēT 1 F. (39) Then, we get the system in terms of ū (k) H and ̂p k (we return the index k): M 0,(k) H ū(k) H + [ B 0,(k) ] T H ˆpk + Ck T ¯λ =ĝ k , B 0,(k) H ū(k) H − c k|E k |ˆp k = −|E k |f k , (40) k = 1,n, complemented by the equations of continuity of normal fluxes on the interfaces between neighboring polyhedral cells and by the equations for the normal fluxes on ∂Ω. Here, ĝ k =ḡ k − 1 σ 2 k [ ] ˜B(k) T H ēk ē T k F k (41) and the values of c k and f k are defined in (8). Recall that σ 2 k = c k|E k |. Now, we return to the system (23) and consider the case when the coefficient c ≡ 0inE k , i.e. Σ k is the zero matrix. In this case, the matrix S h = B h M −1 h BT h (42) in (26) is singular. Let us consider the eigenvalue problem S h ¯w = µD ¯w, (43) where the subindex k staying for the number of the cell E = E k is again omitted. This eigenvalue problem has one zero eigenvalue µ 1 =0andν − 1 positive eigenvalues µ 2 ≤ µ 3 ≤ ··· ≤ µ ν where ν is the dimension of S h .We denote the system of D-orthonormal eigenvectors of problem (43) by ¯w 1 , ¯w 2 ,..., ¯w ν , (44) where ¯w 1 = 1 ē. (45) |E| 1/2 The spectral decomposition of the matrix S h with respect to eigenvalue problem (43) is defined by the following formula: S h = DWΛW T D, (46)
Page 1 and 2: Partial Differential Equations
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Page 5 and 6: Dedicated to Olivier Pironneau
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Page 11 and 12: List of Contributors Yves Achdou UF
Page 13 and 14: List of Contributors XV Claude Le B
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102 I. Sazonov et al. H z 1 exact F
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106 I. Sazonov et al. Scattering Wi
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108 I. Sazonov et al. 6.4 Scatterin
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110 I. Sazonov et al. (a) (b) Fig.
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112 I. Sazonov et al. [MHP96] K. Mo
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126 R. Sanders and A.M. Tesdall 0.3
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128 R. Sanders and A.M. Tesdall [TR
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132 S. Lapin et al. Ω R γ Ω 2 Γ
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134 S. Lapin et al. ∫ ∂ 2 ∫
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136 S. Lapin et al. Ω R γ Fig. 3.
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138 S. Lapin et al. 4 Energy Inequa
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140 S. Lapin et al. 5 Numerical Exp
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142 S. Lapin et al. Fig. 6. Contour
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144 S. Lapin et al. Fig. 9. Obstacl
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Domain Decomposition and Electronic
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Domain Decomposition Approach for C
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Numerical Analysis of a Finite Elem
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so that |u| 2 1,ω h ≤ 2 Numerica
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188 A. Bonito et al. of the model a
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190 A. Bonito et al. Fig. 1. The sp
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192 A. Bonito et al. 3 1 16 4 1 1 1
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194 A. Bonito et al. ∫ v n+1 h
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196 A. Bonito et al. −pn +2µD(v)
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204 A. Bonito et al. Fig. 8. Jet bu
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206 A. Bonito et al. References [AM
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208 A. Bonito et al. [Set96] J. A.
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210 J. Hao et al. due to shear flow
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212 J. Hao et al. The backward reac
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214 J. Hao et al. where u and p den
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218 J. Hao et al. and solve for V n
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220 J. Hao et al. Table 2. The calc
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222 J. Hao et al. References [ASS80
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Computing the Eigenvalues of the La
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Eigenvalues of the Laplace-Beltrami
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A Fixed Domain Approach in Shape Op
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Shape Optimization Problems with Ne
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Reduced-Order Modelling of Dispersi
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Calibration of Lévy Processes with
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We have proved Calibration of Lévy
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