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22 V. Girault and M.F. Wheeler Y, meters 600 500 400 300 200 100 Cone 1.0E+02 4.2E+03 2.2E+04 1.0E+03 4.0E+04 2.2E+04 1.0E+04 4.0E+05 2.2E+05 1.0E+05 4.0E+05 2.2E+05 1.0E+05 4.2E+07 2.2E+07 1.0E+07 4.2E+06 2.2E+06 1.0E+06 Y, meters 600 500 400 300 200 100 0 0 5000 10000 15000 X, meters 20000 25000 0 0 5000 10000 15000 20000 25000 X, meters Fig. 4. Simulation of nuclear reactive transport using DG - 1 600 500 Y, meters 400 300 200 100 0 0 5000 10000 15000 20000 25000 X, meters Fig. 5. Simulation of nuclear reactive transport using DG - 2 advection-dominated and diffusion-dominated problems. Figure 4 shows Iodine concentration at 200K years and Figure 5 at 2 million years. The low numerical diffusion of the DG method was also found to be important in this benchmark problem because of the long simulation time, cf. [WESR03]. Details regarding this simulation and several mesh adaptation strategies are discussed in [SW06a, SW06b]. The latter demonstrated that by employing dynamic adaptivity, time-dependent transport could be resolved without slope limiting for both long-term and short-term simulations. Moreover, mass conservation was retained locally during dynamic mesh modification. The theoretical and computational results obtained for primal DG methods for transport and flow are summarized in Table 1. Two rows provide a comparison of the methods for treating flow problems with highly varying
Table 1. Primal DG for transport Discontinuous Galerkin Methods 23 OBB-DG NIPG SIPG IIPG Penalty Term 0 ≥ 0 >σ 0 > 0 > 0 and ≪ σ 0 Optimality in L 2 (H 1 )orH 1 Yes Yes Yes Yes Optimality in L 2 (L 2 )orL 2 No No Yes No Robust probs. with highly var. coeffs. Yes Yes No Yes Scalar primary interest(transp.) No No Yes No Compatibility Flow Condition No No No Yes coefficients and for transport problems in which the scalar variable is of primary interest. These results were obtained from an extensive set of numerical experiments. The studies indicate that the non-symmetric DG formulations are more robust in handling rough coefficients. The symmetric form performs better for treating diffusion/advection/reaction problems since the SIPG form yield optimal L 2 and non-negative norm estimates. The last row summarizes a compatibility condition formulated in [DSW04] in which the objective is to choose a flow field that preserves positive concentrations in reactive transport. The IIPG method is the only primal DG for which this holds. DG methods are currently being investigated for modeling multiphase flow in porous media, e.g., see [BR04, KR06] for two-phase incompressible and for two and three phases compressible systems see [HF06, Esl05, SW]. While much progress has been made in modeling transport a major disadvantage for DG has been the development of efficient parallel solvers for large linear and nonlinear systems, the pressure equation or a fully implicit formulation for multiphase flow respectively. The development of DG solvers is an active area of research and new domain decomposition approaches are currently being developed, e.g., see [Kan05, Joh05, AA07, Esl05, BR00]. References [AA07] [ABCM02] [Arn79] [Arn82] [Bab73] P. F. Antonietti and B. Ayuso. Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case. M2AN Math. Model. Numer. Anal., 41(1):21–54, 2007. D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5):1749–1779, 2002. D. N. Arnold. An interior penalty finite element method with discontinuous elements. PhD thesis, University of Chicago, Chicago, IL, 1979. D. N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19(4):742–760, 1982. I. Babu˘ska. The finite element method with Lagrangian multipliers. Numer. Math., 20:179–192, 1973.
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
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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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110 I. Sazonov et al. (a) (b) Fig.
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112 I. Sazonov et al. [MHP96] K. Mo
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116 R. Sanders and A.M. Tesdall imp
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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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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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200 A. Bonito et al. The normal vec
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204 A. Bonito et al. Fig. 8. Jet bu
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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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216 J. Hao et al. * * * * * * * * *
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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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280 S. Ikonen and J. Toivanen the p
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288 S. Ikonen and J. Toivanen 1.6 1
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292 S. Ikonen and J. Toivanen [Mer7
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