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

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82 5 Numerical ExperimentsIn Tables 5.9, 5.10, 5.11 we provide the results with errors given by the formulae (5.3)and the testing curve ϑ: [0, 1] → R 3 defined as⎡ ⎤ ⎡ ⎤−1 2ϑ(t) := ⎣ 0 ⎦ + t ⎣0⎦ for t ∈ [0, 1]. (5.9)1.1 0For the solution on the elephant see Figure 5.6.0.61430.4430.80.620.220.41010.20−1−2−0.2−0.40−1−20−0.2−3−4−4−2024024(a) Real part.−2−4−0.6−0.8−1−3−4−4−202024 4(b) Imaginary part.−2−4−0.4−0.6−0.8−1Figure 5.7: Scattered wave on the sound-soft sphere with E = 1240.41413320.520.5110000−1−1−2−0.5−2−0.5−3−3−4−4−2024024(a) Real part.−2−4−1−4−4−202024 4(b) Imaginary part.−2−4−1Figure 5.8: Total wave on the sound soft-sphere with E = 1240.Let us also consider the scattering problem (2.4) with a sound-soft obstacle, namelythe sphere. For the incoming plane wave we chooseu i (x) := e iκ⟨x,d⟩
83with the unit direction vector d := √ 13[1, 1, 1] T . The scattered wave u s is depicted inFigure 5.7, in Figure 5.8 you can see the total wave u = u s + u i .5.5 Exterior Neumann Boundary Value ProblemNext we consider the exterior Neumann boundary value problem (3.48) with κ = 2 andthe testing solutions (5.7) for Ω being the sphere and the cube and (5.8) in the case of theelephant.To solve the exterior Neumann problem and to find the missing Dirichlet data weuse the discretized variational formulation related to the hypersingular boundary equation(3.51), i.e.,⟨D κ g D,h , ϕ i ⟩ ∂Ω =− 1 2 I − K∗ κ g N,h , ϕ i∂Ωfor all i ∈ {1, . . . , N}and the matrix formulationD κ,h g D =− 1 2 MT h − KT κ,h g N .E N Err D Err D,p Err ϑ320 162 1.92 · 10 −1 7.11 · 10 −2 1.92 · 10 −11240 622 6.36 · 10 −2 1.71 · 10 −2 5.34 · 10 −27432 3718 9.24 · 10 −3 4.39 · 10 −3 6.86 · 10 −3Table 5.12: Exterior Neumann BVP on the sphere.E N Err D Err D,p Err ϑ300 152 2.46 · 10 −1 2.32 · 10 −1 1.20 · 10 −11200 602 9.02 · 10 −2 4.18 · 10 −2 7.78 · 10 −27500 3752 1.86 · 10 −2 1.17 · 10 −2 1.11 · 10 −2Table 5.13: Exterior Neumann BVP on the cube.E N Err D Err D,p Err ϑ3962 1983 7.90 · 10 −2 5.59 · 10 −3 1.77 · 10 −27510 3757 7.14 · 10 −2 2.31 · 10 −3 1.78 · 10 −2Table 5.14: Exterior Neumann BVP on the elephant.
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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
Page 43 and 44: 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: 81The approximate solution u h to (
Page 97 and 98: 8540.540.430.430.220.320100.20.110
Page 99: 87ConclusionIn the preceding sectio
Page 102: 90[15] SAUTER, Stefan; SCHWAB, Chri
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