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Electromagnetic Scattering 105 100 E SW,dB 10 -1 10 -2 10-3 8 16 32 64 128 Fig. 7. Scattering of a plane TE wave by a circular PEC cylinder of diameter λ showing the variation of the computed error, with the number of elements, λ/δ, per wavelength, for the co-volume scheme and the FETD scheme on the unstructured meshes. Table 2. Scattering of a plane TE wave by a circular PEC cylinder of diameter λ. Co-volume FETD Speed up ratio λ/δ spc time E SW spc time E SW FETD/Co-volume 8 21 0.15 0.744 31 1.2 0.750 8 16 42 0.5 0.275 61 15. 0.102 30 32 83 4.0 0.060 122 117 0.026 30 64 165 37 0.019 242 922 0.007 25 128 239 250 0.006 485 7295 0.002 30 one location. The values of spc, time and E SW are shown in Table 2 for the co-volume scheme and the FETD scheme on these unstructured meshes. It can be observed that, for these simulations, the co-volume scheme is nearly 30 times faster than the FETD scheme. As a more challenging variation of this example, we also consider scattering of a plane single frequency wave by a perfectly conducting circular cylinder of diameter 15λ. The mesh employed is generated to meet a mesh spacing requirement of δ = λ/15. Again, the minimum distance from the PML region to the cylinder is λ. The solution is advanced for 50 cycles of the incident wave and the computed and exact scattering width distributions are compared in Figure 8(a). Excellent agreement with the exact solution is observed using both schemes. The distribution of the computed total magnetic field in the complete domain, including the PML, is shown in Figure 8(b). For this example, the co-volume scheme is nearly 34 times faster than the FETD scheme.
106 I. Sazonov et al. Scattering Width, dB (a) Viewing Angle, degrees (b) Fig. 8. Scattering of a plane TE wave by a circular PEC cylinder of diameter 15λ showing (a) a comparison between the exact and computed scattering width distributions, (b) computed contour distribution of the total magnetic field in the complete computational domain. 6.3 Scattering by a Square PEC Cylinder The next example involves scattering of a plane single frequency electromagnetic wave by a perfectly conducting cylinder of square cross section. The sides of the square are of length λ. The objective is to use this example to illustrate the accuracy of the FETD and co-volume schemes in the presence of singularities. This simple geometry means that the computational domain may be discretised using a structured mesh of square elements and, in this case, the co-volume scheme of the equations (11) and (12) reduces to the classical Yee scheme. The distribution of the scattering width obtained using the Yee scheme on a fine Cartesian grid, with 512 elements per wavelength, is taken as the benchmark solution. An unstructured mesh, termed mesh a, is generated with mesh spacing λ/16. The solution is advanced on this mesh for 8 cycles using both the co-volume and the FETD schemes. In this case, the error E SW is determined as the maximum difference, in absolute value, between the computed and the benchmark scattering width distributions. Table 3 shows the values of spc, time and E SW for this grid. It is apparent that the error in the FETD scheme is an order of magnitude greater than the error in the co-volume method. This is believed to be due to the singularity in the geometry, where higher mesh resolution will be required in a scheme such as FETD. Two further unstructured meshes are generated, by reducing the spacing by a factor of 2 (termed mesh b) and 4 (termed mesh c), in the vicinity of the corners. Details of the three meshes in the region of one of the corners are shown in Figure 9. Figure 10 shows the variation in the computed error with the near corner resolution that is employed. It can be seen that the error in the FETD results decreases as the mesh is refined. It is also clear that the
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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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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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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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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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