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254 J.-M. Brun and B. Mohammadi 0.03 0.025 0.02 0.015 0.01 0.005 0 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 Fig. 3. Generalized similitude solution (right) for a 2-point based wind (similar to Fig. 1) compared to a direct simulation with a PDE based transport-diffusion model for the same wind. The similitude solution has been evaluated on all the nodes of the finite element mesh for comparison. Fig. 4. Regions affected from the treatment of two sources. The flow field has been built from three points of measurement indicated on the picture. 220 200 180 160 140 120 N 100 80 2 km N Fig. 5. Left: a typical digital terrain model (x and y coordinates range over 2 km). Dispersion in a uniform north wind with (middle) and without (right) the ground model (Section 2.2).
Reduced-Order Modelling of Dispersion 255 Fig. 6. Left: constructed flow field. Middle: dispersion from a vineyard. Right: sensitivity analysis for a dispersion detected on the lower left corner. One can, therefore, give possible origins of a pollution. together with this low-complexity modelling to introduce robustness issues in the prediction. In addition to the data assimilation inverse problem, inverse source reconstruction has been considered as a natural demand in environmental survilance. The current work concerns the introduction of stochastic analysis in the present model to produce regional parametric risk maps using Monte Carlo simulations which become achievable thanks to the low calculation cost of the approach. Acknowledgement. This contribution is dedicated to Professor O. Pironneau for his 60 th birthday. It has been realized for Cemagref at Montpellier, France. The authors would like to thank V. Bellon-Maurel, B. Bonicelli, B. Ruelle and C. Sinfort for their kindness and valuable comments. Thanks also to S. Labbe from Cemagref/Teledetection for having made available to us DTM models. References [AGFM02] [Arc06] [BGM97] [Bru06] [CFG + 01] F. Alauzet, P.-L. George, P. Frey, and B. Mohammadi. Transient fixed point based unstructured mesh adaptation. Internat. J. Numer. Methods Fluids, 43(6):729–745, 2002. ArcGIS. Geographic information system, 2006. http://www.esri.com/ software/arcgis. H. Borouchaki, P.-L. George, and B. Mohammadi. Delaunay mesh generation governed by metric specifications. Finite Element in Analysis and Design, 2:85–109, 1997. J. M. Brun. Modélisation à complexité réduite de la dérive. PhD thesis, University of Montpellier, 2006. G. Corliss, C. Faure, A. Griewank, L. Hascoet, and U. Naumann, editors. Automatic differentiation of algorithms: From simulation to optimization. Number 50 in Lect. Notes Comput. Sci. Eng. Springer, Berlin, 2001. Selected papers from the AD2000 Conference, Nice, France, June 2000.
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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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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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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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