The Boundary Element Method for the Helmholtz Equation ... - FEI VÅ B

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84 5 Numerical Experiments10.80.60.70.610.80.60.140.120.40.50.40.10.200.40.200.08−0.2−0.4−0.60.30.2−0.2−0.4−0.60.060.04−0.8−0.80.02−1−1−0.500.500.51 1(a) Real part.−0.5−10.10−1−1−0.500.500.51 1(b) Imaginary part.−0.5−10−0.02Figure 5.9: Solution to the exterior Neumann BVP on the sphere with E = 7432.0.310.510.250.80.60.40.80.60.20.40.20.30.40.20.150−0.20.20−0.20.1−0.4−0.60.1−0.4−0.60.05−0.8−1−0.500.50.511.5(a) Real part.0−0.5−10−0.1−0.8−1−0.500.500.511.5(b) Imaginary part.−0.5−10−0.05Figure 5.10: Solution to the exterior Neumann BVP on the elephant with E = 7510.In Tables 5.12, 5.13, 5.14 we provide the results, the error columns are given by (5.5)with the curve (5.9). Figures 5.9, 5.10 depict the solution to the exterior Neumann boundaryvalue problem on the sphere and the elephant, respectively.Let us also consider the scattering problem (2.4) with a sound-hard obstacle, namelythe cube. For the incoming plane wave we chooseu i (x) := e iκ⟨x,d⟩with the unit direction vector d := √ 13[1, 1, 1] T . The scattered wave u s is depicted inFigure 5.11, in Figure 5.12 you can see the total wave u = u s + u i .
8540.540.430.430.220.320100.20.110−0.2−10−0.1−1−0.4−2−0.2−2−0.6−3−0.3−3−0.8−4−4−2024024(a) Real part.−2−4−0.4−0.5−4−4−202024 4(b) Imaginary part.−2−4−1Figure 5.11: Scattered wave on the sound-hard cube with E = 1200.4413130.52210.51000−10−1−0.5−2−2−3−0.5−3−1−4−4−2024024(a) Real part.−2−4−1−4−4−202024 4(b) Imaginary part.−2−4−1.5Figure 5.12: Total wave on the sound-hard cube with E = 1200.5.6 Exterior Mixed Boundary Value ProblemFinally, we consider the exterior mixed boundary value problem (3.53) with κ = 2, thetesting solution (5.7) and Ω being the sphere and the cube. The testing domains aredecomposed in the same way as in the case of the interior mixed problem in Section 5.3.We solve the problem of finding the missing Cauchy data using the discretized Galerkinequations corresponding to the symmetric approach (3.56), i.e.,a(s h , t h , ψ k , ϕ l ) = F (ψ k , ϕ l )for all k ∈ E D , j ∈ N N
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I hereby declare that this thesis i
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AbstractIn this work we study the a
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1IntroductionThe boundary element m
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31 Function SpacesIn this section w
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5∂Ωy i 2ΩU + iU − iΓ iy i 1F
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7For a special choice of p = 2 we g
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9Note that the trace operator is no
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11Definition 1.8 (Weak Solution). C
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152 Helmholtz EquationLet us first
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20 2 Helmholtz Equation∂ΩΩ εε
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22 2 Helmholtz Equationthe normal v
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24 2 Helmholtz Equationand due to p
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26 2 Helmholtz Equationand thus l
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293 Boundary Integral EquationsAt t
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313.1 Single Layer Potential Operat
Page 45 and 46: 33For the jump of the Neumann trace
Page 47 and 48: 35with the surface curl operatorcur
Page 49 and 50: 37The following theorems are standa
Page 51 and 52: 39u ∈ H 1 (Ω, ∆+κ 2 ) if and
Page 53 and 54: 41Another approach to computing the
Page 55 and 56: 433.5.3 Interior Mixed Boundary Val
Page 57 and 58: 45respectively.The unique solvabili
Page 59: 47with an unbounded domain Ω ext :
Page 62 and 63: 50 4 Discretization and Numerical R
Page 87 and 88: 755 Numerical ExperimentsIn this pa
Page 89 and 90: 77E N Err N Err N,p Err ϑ3962 1983
Page 91 and 92: 792002000.81500.81500.60.41000.60.4
Page 93 and 94: 81The approximate solution u h to (
Page 95: 83with the unit direction vector d
Page 99: 87ConclusionIn the preceding sectio
Page 102: 90[15] SAUTER, Stefan; SCHWAB, Chri
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