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A Lagrange Multiplier Based Domain Decomposition Method 141 Fig. 4. Example of a finite element mesh. Fig. 5. Contour plot of the real part of the solution for L =0.5 (left) and L =0.25 (right) meters. Incident wave coming from the left. we chose the time step to be ∆t = T/50, where T =1/f =1.66 × 10 −9 sec is a time period corresponding to L =0.5 meters and T =1/f =0.83 × 10 −9 sec for L =0.25 meters. The next set of numerical experiments contains the simulations of wave propagation through a domain with an obstacle completely consisting of a
142 S. Lapin et al. Fig. 6. Contour plot of the real part of the solution for L =0.5 (left) and L =0.25 (right) meters. Incident wave coming from the lower left corner with an angle of 45 degrees. coating material. We have taken the coating material coefficients to be ε 2 =1 and µ 2 = 9, implying that the speed of propagation in the coating material is three times slower than in air. As before Ω is a 2 meter × 2 meter rectangle and Ω 2 has the shape of an airfoil. For the solution of this problem for an incident frequency f =0.6 GHzwe have used a mesh with a total of 8435 nodes and 16228 elements. The time step was taken to be ∆t = T/50, where T =1/f =1.66 × 10 −9 sec is a time period. We used a mesh consisting of 20258 nodes (39514 elements) for solving the problem for an incident wave with the frequency f =1.2 GHz. The time step was equal to T/50, T =1/f =0.83 × 10 −9 sec. In Figures 7 and 8 we present the contour plot of the real part of the solution for the incident frequency L =0.5 andL =0.25. We also performed numerical computations for the case when the obstacle is an airfoil with a coating (Figure 9). The coating region is moon shaped and, as before, ε 2 =1 and µ 2 = 9. We show in Figure 10 the contour plot of the real part of the solution for the incident frequency L =0.5 meters and L =0.25 meters for the case when the incident wave is coming from the left. Figure 11 presents the contour plot of the real part of the solution for incident frequency, L =0.5 meters and L =0.25 meters for the case when incident wave is coming from the lower left corner with angle equal to 45 ◦ . An important observation for all of the numerical experiments mentioned is that, despite the fact that a mesh discontinuity takes place over γ together with a weak forcing of the matching conditions, we do not observe a discontinuity of the computed fields.
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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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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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280 S. Ikonen and J. Toivanen the p
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