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Electromagnetic Scattering 109 Scattering Width, dB Co-volume /120 Co-volume /15 uniform FETD /120 FETD /15 uniform FETD /15 refined Viewing Angle, degrees Fig. 12. Simulation of scattering of a plane TE wave by a PEC NACA0012 aerofoil of length λ showing a comparison between the computed and benchmark scattering width distributions. Table 4. Simulation of scattering of a plane TE wave by a PEC NACA0012 aerofoil of length λ. Mesh FETD Co-volume Speed up resolution spc time, s E SW spc time, s E SW ratio Uniform 59 12. 6.00 46 0.4 0.9 30 Refined 97 20. 2.14 99 0.6 0.5 33 shown in Fig. 13(a). The simulations are advanced for 150 cycles and the typical distribution of the contours of the computed total magnetic field in the domain, excluding the PML, is shown in Figure 13(b). A comparison of the computed scattering width distributions is given in Figure 14. Also shown on this figure is the scattering width distribution computed using a high order finite element frequency domain (FEFD) simulation [LMHW02]. The number ofstepspercycleis57fortheco-volumeschemeand59fortheFETDmethod and, for this example, the co-volume scheme requires 31 seconds of cpu time, while the FETD method requires 1980 seconds. This represents a speed-up of a factor of 65.
110 I. Sazonov et al. (a) (b) Fig. 13. Simulation of scattering of a plane TE wave by a PEC cavity showing (a) the unstructured mesh employed, (b) the computed total magnetic field after 150 cycles. Scattering Width, dB Viewing Angle, degrees Fig. 14. Simulation of scattering of a plane TE wave by a PEC cavity showing a comparison of the scattering width distributions computed, after 150 cycles, by FETD, the co-volume scheme and a FEFD method. 7 Conclusions The numerical performance of an explicit unstructured mesh co-volume time domain scheme and a standard finite element time domain method has been compared for a number of electromagnetic wave propagation and scattering examples. To ensure the efficiency of the co-volume approach, the smooth Delaunay–Voronoï dual meshes that are used are generated using a stitching method. The numerical examples that have been considered show that the co-volume method is 30–60 times faster than the finite element method for two-dimensional scattering problems. In addition, the co-volume method
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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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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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286 S. Ikonen and J. Toivanen and {
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292 S. Ikonen and J. Toivanen [Mer7
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