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184 B. Maury The source term g is, therefore, a single layer distribution supported by ∂O, with weight −λ · n = −∂u/∂n (where n is the outward normal to ∂O). Note that it is in H 1/2 (∂O). Remark 7. Note that letting ε go to 0 for any h>0 leads to an estimate for a fictitious domain method (à laGlowinski, i.e. based on the use of Lagrange multiplier). In [GG95], an error estimate is obtained for such a method; it relies on two independent meshes for the primal and dual components of the solution (conditionally to some compatibility conditions between the sizes of the two meshes). We recover this estimate in the situation where the local mesh is simply the restriction of the covering mesh to the obstacle (to the reduced obstacle O h , to be more precise). Remark 8. The approach we presented can be extended to other situations, like the one we already considered in Example 2, as soon as a H 1 -penalty is used. The functional to minimize is then J ε (v) = 1 ∫ ∫ |∇v| 2 − fv + 1 ∫ ( u 2 + |∇u| 2) , 2 2ε Ω Ω so that B identifies to the restriction operator from H 1 0 (Ω) toH 1 (O). The discrete inf-sup condition, as well as the approximation properties, are essentially the same as in the case we considered here. Concerning the original problem of simulating fluid-particle flows, an extra difficulty lies in the fact that two constraints of different types must be dealt with (global incompressibility and local rigid motion). It raises additional issues which shall be addressed in the future. Ω References [ABF99] Ph. Angot, Ch.-H. Bruneau, and P. Fabrie. A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math., 81(4):497–520, 1999. [AR] Ph. Angot and I. Ramière. Convergence analysis of the Q1-finite element method for elliptic problems with non boundary-fitted meshes. Submitted. [Bab73] I. Babuška. The finite element method with penalty. Math. Comp., 27:221–228, 1973. [BF91] F. Brezzi and M. Fortin. Mixed and hybrid finite element methods, volume 15 of Springer Series in Computational Mathematics. Springer- Verlag, New York, 1991. [DP02] S. Del Pino. Une méthode d’éléments finis pour la résolution d’EDP dans des domaines décrits par géométrie constructive. PhD thesis, Université Pierre et Marie Curie, Paris, 2002. [DPP03] S. Del Pino and O. Pironneau. A fictitious domain based general PDE solver. In E. Heikkola, editor, Numerical Methods for Scientific Computing, Barcelona, 2003.
Numerical Analysis of a Finite Element/Volume Penalty Method 185 [ff3] [FFp] [GG95] [JLM05] [JT96] [lif] [Mau99] [Mau01] [PG02] [RAB06] [RPVC05] [SMSTT05] [VCLR04] freeFEM3D (http://www.freefem.org/ff3d/). freeFEM++ (http://www.freefem.org/). V. Girault and R. Glowinski. Error analysis of a fictitious domain method applied to a Dirichlet problem. Japan J. Indust. Appl. Math., 12(3):487–514, 1995. J. Janela, A. Lefebvre, and B. Maury. A penalty method for the simulation of fluid-rigid body interaction. In CEMRACS 2004—mathematics and applications to biology and medicine, volume 14 of ESAIM Proc., pages 115–123 (electronic). EDP Sci., Les Ulis, 2005. A. A. Johnson and T. E. Tezduyar. Simulation of multiple spheres falling in a liquid-filled tube. Comput. Methods Appl. Mech. Engrg., 134(3-4): 351–373, 1996. LifeV (http://www.lifev.org/). B. Maury. Direct simulations of 2D fluid-particle flows in biperiodic domains. J. Comput. Phys., 156(2):325–351, 1999. B. Maury. A fat boundary method for the Poisson problem in a domain with holes. J. Sci. Comput., 16(3):319–339, 2001. T.-W. Pan and R. Glowinski. Direct simulation of the motion of neutrally buoyant circular cylinders in plane Poiseuille flow. J. Comput. Phys., 181(1):260–279, 2002. I. Ramière, Ph. Angot, and M. Belliard. A fictitious domain approach with spread interface for elliptic problems with general boundary conditions. Comput. Methods App. Mech. Engrg., 196(4–6):766–781, 2007. T. N. Randrianarivelo, G. Pianet, S. Vincent, and J. P. Caltagirone. Numerical modelling of solid particle motion using a new penalty method. Internat. J. Numer. Methods Fluids, 47:1245–1251, 2005. J. San Martín, J.-F. Scheid, T. Takahashi, and M. Tucsnak. Convergence of the Lagrange-Galerkin method for the equations modelling the motion of a fluid-rigid system. SIAM J. Numer. Anal., 43(4):1536–1571 (electronic), 2005. S. Vincent, J.-P. Caltagirone, P. Lubin, and T. N. Randrianarivelo. An adaptative augmented Lagrangian method for three-dimensional multimaterial flows. Comput. & Fluids, 33(10):1273–1289, 2004.
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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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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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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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Shape Optimization Problems with Ne
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Calibration of Lévy Processes with
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