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Fluid Flows with Complex Free Surfaces 205 Fig. 9. Fingering instabilities. Shape of the liquid region at times t = 0 s (left) and t =0.745 s (right). Fig. 10. Fingering instabilities. Horizontal cuts through the middle of the liquid region at times t =0.119 s, t =0.245 s, t =0.364 s, t =0.49 s (first row) and times t =0.609 s, t =0.735 s, t =0.854 s, t =0.98 s (second row). 6 Conclusions An efficient computational model for the simulation of two-phases flows has been presented. It allows to consider both Newtonian and non-Newtonian flows. It relies on an Eulerian framework and couples finite element techniques with a forward characteristics method. Numerical results illustrate the large range of applications covered by the model. Extensions are being investigated (1) to couple viscoelastic and surface tension effects, (2) to reduce the CPU time required to solve Stokes problems, and (3) to improve the reconstruction of the interface and the computation of surface tension effects. Acknowledgement. The authors wish to thank Vincent Maronnier for his contribution to this project and his implementation support.
206 A. Bonito et al. References [AMS04] [BCP06a] [BCP06b] [BCP07] [BDJ80] [BKZ92] [BPL06] [BPS01] E. Aulisa, S. Manservisi, and R. Scardovelli. A surface marker algorithm coupled to an area-preserving marker redistribution method for threedimensional interface tracking. J. Comput. Phys., 197(2):555–584, 2004. A. Bonito, Ph. Clément, and M. Picasso. Finite element analysis of a simplified stochastic Hookean dumbbells model arising from viscoelastic flows. M2AN Math. Model. Numer. Anal., 40(4):785–814, 2006. A. Bonito, Ph. Clément, and M. Picasso. Mathematical analysis of a simplified Hookean dumbbells model arising from viscoelastic flows. J. Evol. Equ., 6(3):381–398, 2006. A. Bonito, Ph. Clément, and M. Picasso. Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow. Numer. Math., 107(2):213–255, 2007. R. B. Bird, N. L. Dotson, and N. L. Johnson. Polymer solution rheology based on a finitely extensible bead-spring chain model. J. Non- Newtonian Fluid Mech., 7:213–235, 1980. J. U. Brackbill, D. B. Kothe, and C. Zemach. A continuum method for modeling surface tension. J. Comput. Phys., 100:335–354, 1992. A. Bonito, M. Picasso, and M. Laso. Numerical simulation of 3D viscoelastic flows with free surfaces. J. Comput. Phys., 215(2):691–716, 2006. J. Bonvin, M. Picasso, and R. Stenberg. GLS and EVSS methods for a three-field Stokes problem arising from viscoelastic flows. Comput. Methods Appl. Mech. Engrg., 190(29–30):3893–3914, 2001. [BRLH02] A. Bach, H. K. Rasmussen, P.-Y. Longin, and O. Hassager. Growth of non-axisymmetric disturbances of the free surface in the filament stretching rheometer: experiments and simulation. J. Non-Newtonian Fluid Mech., 108:163–186, 2002. [Cab05] [Cab06] A. Caboussat. Numerical simulation of two-phase free surface flows. Arch. Comput. Methods Engrg., 12(2):165–210, 2005. A. Caboussat. A numerical method for the simulation of free surface flows with surface tension. Comput. & Fluids, 35(10):1205–1216, 2006. [CPR05] A. Caboussat, M. Picasso, and J. Rappaz. Numerical simulation of free surface incompressible liquid flows surrounded by compressible gas. J. Comput. Phys., 203(2):626–649, 2005. [CR05] A. Caboussat and J. Rappaz. Analysis of a one-dimensional free surface flow problem. Numer. Math., 101(1):67–86, 2005. [DLCB03] D. Derks, A. Lindner, C. Creton, and D. Bonn. Cohesive failure of [FCD + 06] [FF92] thin layers of soft model adhesives under tension. J. Appl. Phys., 93(3):1557–1566, 2003. M. M. Francois, S. J. Cummins, E. D. Dendy, D. B. Kothe, J. M. Sicilian, and M. W. Williams. A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework. J. Comput. Phys., 213(1):141–173, 2006. L. P. Franca and S. L. Frey. Stabilized finite element method: II. The incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Engrg., 99:209–233, 1992.
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Partial Differential Equations
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Partial Differential Equations Mode
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Dedicated to Olivier Pironneau
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VIII Preface computers has been at
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Contents List of Contributors .....
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List of Contributors Yves Achdou UF
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List of Contributors XV Claude Le B
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Discontinuous Galerkin Methods Vive
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Discontinuous Galerkin Methods 5 2
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Discontinuous Galerkin Methods 7 g
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Discontinuous Galerkin Methods 9 (
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Discontinuous Galerkin Methods 15 W
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Let a h and b h denote the bilinear
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Discontinuous Galerkin Methods 19 t
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Discontinuous Galerkin Methods 21 l
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Table 1. Primal DG for transport Di
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Discontinuous Galerkin Methods 25 [
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Mixed Finite Element Methods on Pol
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Mixed FE Methods on Polyhedral Mesh
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4 Hybridization and Condensation Mi
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is symmetric and positive definite,
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with some coefficient α ∈ R wher
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44 E.J. Dean and R. Glowinski so fa
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46 E.J. Dean and R. Glowinski 2 A L
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48 E.J. Dean and R. Glowinski S:T=
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50 E.J. Dean and R. Glowinski minim
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52 E.J. Dean and R. Glowinski 6 On
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54 E.J. Dean and R. Glowinski Fig.
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56 E.J. Dean and R. Glowinski and
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58 E.J. Dean and R. Glowinski 7 Num
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60 E.J. Dean and R. Glowinski Fig.
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62 E.J. Dean and R. Glowinski Assum
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Higher Order Time Stepping for Seco
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u n+1 h Optimal Higher Order Time D
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Optimal Higher Order Time Discretiz
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96 I. Sazonov et al. To provide a p
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98 I. Sazonov et al. In the first s
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100 I. Sazonov et al. Fig. 1. An ex
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102 I. Sazonov et al. H z 1 exact F
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104 I. Sazonov et al. (a) (b) Fig.
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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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114 R. Sanders and A.M. Tesdall I R
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116 R. Sanders and A.M. Tesdall imp
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118 R. Sanders and A.M. Tesdall alo
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120 R. Sanders and A.M. Tesdall (a)
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122 R. Sanders and A.M. Tesdall D C
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124 R. Sanders and A.M. Tesdall 8.6
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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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Calibration of Lévy Processes with
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We have proved Calibration of Lévy
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280 S. Ikonen and J. Toivanen the p
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