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92 J.Ch. Gilbert and P. Joly • The effective efficiency of the new schemes should be tested on realistic wave propagation problems. • The impact of the modification of the initial schemes (the ones which are based on the modified equation technique) on the effective accuracy (we are only guaranteed that the order of approximation is preserved) should be analyzed thorough numerical dispersion studies. • Our various theoretical conjectures should be addressed in a rigorous way. These will be the subjects of forthcoming works. References [AJT00] [AKM74] [BGLS06] [BM05] L. Anné, P. Joly, and Q. H. Tran. Construction and analysis of higher order finite difference schemes for the 1D wave equation. Comput. Geosci., 4(3):207–249, 2000. R. M. Alford, K. R. Kelly, and Boore D. M. Accuracy of finite difference modeling of the acoustic wave equation. Geophysics, 39:834– 842, 1974. J. F. Bonnans, J. Ch. Gilbert, C. Lemaréchal, and C. Sagastizábal. Numerical Optimization – Theoretical and Practical Aspects. Universitext. Springer Verlag, Berlin, 2nd edition, 2006. S. Bellavia and B. Morini. An interior global method for nonlinear systems with simple bounds. Optim. Methods Softw., 20(4–5):453– 474, 2005. [CdLBL97] R. Carpentier, A. de La Bourdonnaye, and B. Larrouturou. On the derivation of the modified equation for the analysis of linear numerical methods. RAIRO Modél. Math. Anal. Numér., 31(4):459– 470, 1997. [CF05] [Che66] [CJ96] [CJKMVV99] G. Cohen and S. Fauqueux. Mixed spectral finite elements for the linear elasticity system in unbounded domains. SIAM J. Sci. Comput., 26(3):864–884 (electronic), 2005. E. W. Cheney. Introduction to Approximation Theory. McGraw-Hill, 1966. G. Cohen and P. Joly. Construction analysis of fourth-order finite difference schemes for the acoustic wave equation in nonhomogeneous media. SIAM J. Numer. Anal., 33(4):1266–1302, 1996. M. J. S. Chin-Joe-Kong, W. A. Mulder, and M. Van Veldhuizen. Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation. J. Engrg. Math., 35(4):405– 426, 1999. [CJRT01] G. Cohen, P. Joly, J. E. Roberts, and N. Tordjman. Higher order triangular finite elements with mass lumping for the wave equation. SIAM J. Numer. Anal., 38(6):2047–2078 (electronic), 2001. [Coh02] G. C. Cohen. Higher-order numerical methods for transient wave [Dab86] equations. Scientific Computation. Springer-Verlag, Berlin, 2002. M. A. Dablain. The application of high order differencing for the scalar wave equation. Geophysics, 51:54–56, 1986.
Optimal Higher Order Time Discretizations 93 [Deu04] P. Deuflhard. Newton Methods for Nonlinear Problems – Affine Invariance and Adaptative Algorithms. Number 35 in Computational Mathematics. Springer, Berlin, 2004. [DPJ06] S. Del Pino and H. Jourdren. Arbitrary high-order schemes for the linear advection and wave equations: application to hydrodynamics and aeroacoustics. C. R. Math. Acad. Sci. Paris, 342(6):441–446, 2006. [FLLP05] L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno. Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes. M2AN Math. Model. Numer. Anal., 39(6):1149–1176, 2005. [HW96] E. Hairer and G. Wanner. Solving ordinary differential equations. II, volume 14 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2nd edition, 1996. Stiff and differentialalgebraic problems. [HW02] J. S. Hesthaven and T. Warburton. Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell’s equations. J. Comput. Phys., 181(1):186–221, 2002. [Jol03] P. Joly. Variational methods for time-dependent wave propagation problems. In Topics in computational wave propagation, volume 31 of Lect. Notes Comput. Sci. Eng., pages 201–264. Springer, Berlin, 2003. [Kan01] Ch. Kanzow. An active set-type Newton method for constrained nonlinear systems. In M.C. Ferris, O.L. Mangasarian, and J.S. Pang, editors, Complementarity: applications, algorithms and extensions, pages 179–200, Dordrecht, 2001. Kluwer Acad. Publ. [LT86] P. Lascaux and R. Théodor. Analyse Numérique Matricielle Appliquée à l’Art de l’Ingénieur. Masson, Paris, 1986. [PFC05] S. Pernet, X. Ferrieres, and G. Cohen. High spatial order finite element method to solve Maxwell’s equations in time domain. IEEE Trans. Antennas and Propagation, 53(9):2889–2899, 2005. [RM67] R. D. Richtmyer and K. W. Morton. Difference methods for initialvalue problems, volume 4 of Interscience Tracts in Pure and Applied Mathematics. John Wiley & Sons, Inc., New York, 2nd edition, 1967. [RS78] M. Reed and B. Simon. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978. [SB87] G. R. Shubin and J. B. Bell. A modified equation approach to constructing fourth-order methods for acoustic wave propagation. SIAM J. Sci. Statist. Comput., 8(2):135–151, 1987. [Sch91] L. Schwartz. Analyse I – Théorie des Ensembles et Topologie. Hermann, Paris, 1991. [TT05] E. F. Toro and V. A. Titarev. ADER schemes for scalar non-linear hyperbolic conservation laws with source terms in three-space dimensions. J. Comput. Phys., 202(1):196–215, 2005. [Wei06] E. W. Weisstein. Chebyshev polynomial of the first kind. MathWorld. http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirst Kind.html, 2006.
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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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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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198 A. Bonito et al. each of its pa
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200 A. Bonito et al. The normal vec
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202 A. Bonito et al. with initial c
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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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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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Note that p ∗ satisfies Calibrati
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280 S. Ikonen and J. Toivanen the p
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282 S. Ikonen and J. Toivanen Merto
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284 S. Ikonen and J. Toivanen For H
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286 S. Ikonen and J. Toivanen and {
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288 S. Ikonen and J. Toivanen 1.6 1
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290 S. Ikonen and J. Toivanen 8 Con
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292 S. Ikonen and J. Toivanen [Mer7
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