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

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80 5 Numerical ExperimentsE N Err D Err D,p Err N Err N,p Err ϑ320 162 3.09 · 10 −2 1.82 · 10 −2 1.78 · 10 −1 1.24 · 10 −1 7.94 · 10 −31240 622 7.96 · 10 −3 4.98 · 10 −3 9.60 · 10 −2 6.49 · 10 −2 1.68 · 10 −37432 3718 1.21 · 10 −3 7.46 · 10 −4 3.32 · 10 −2 2.62 · 10 −2 2.79 · 10 −4Table 5.7: Interior mixed BVP on the sphere.E N Err D Err D,p Err N Err N,p Err ϑ300 152 5.28 · 10 −2 4.54 · 10 −2 2.84 · 10 −1 1.88 · 10 −1 1.22 · 10 −21200 602 1.19 · 10 −2 1.03 · 10 −2 1.18 · 10 −1 9.47 · 10 −2 2.98 · 10 −37500 3752 1.80 · 10 −3 1.56 · 10 −3 4.15 · 10 −2 3.79 · 10 −2 5.37 · 10 −4Table 5.8: Interior mixed BVP on the cube.The results are summarized in Tables 5.7, 5.8 with errors given byErr D := ∥g D − g D,h ∥ L 2 (Γ N )∥g D ∥ L 2 (Γ N )Err N := ∥g N − g N,h ∥ L 2 (Γ D )∥g N ∥ L 2 (Γ D ), Err D,p := ∥g D − g D,p ∥ L 2 (Γ N ),∥g D ∥ L 2 (Γ N)and the curve ϑ: [0, 1] → R 3 defined by (5.4)., Err N,p := ∥g N − g N,p ∥ L 2 (Γ D ), Err ϑ := ∥u − u h∥ L 2 (ϑ)∥g N ∥ L 2 (Γ D)∥u∥ L 2 (ϑ)(5.6)5.4 Exterior Dirichlet Boundary Value ProblemNow we consider the exterior Dirichlet boundary value problem (3.43) with κ = 2. For thetesting solutions we chooseu(x) := v κ (x, y 1 ) (5.7)with y 1 := [0, 0, 0.9] T in the case of the sphere and the cube andu(x) := v κ (x, y 2 ) + v κ (x, y 3 ) (5.8)with y 2 := [0.9, 0.25, 0.7] T and y 3 := [0.9, −0.25, 0.7] T in the case of the elephant.For the computation of the missing Neumann data we use the discretized Galerkinequations⟨V κ g N,h , ψ k ⟩ ∂Ω = − 1 2 I + K κ g D,h , ψ k for all k ∈ {1, . . . , E}∂Ωleading to the system of linear equationsV κ,h g N =− 1 2 M h + K κ,h g D .
81The approximate solution u h to (3.43) is given by the discretized representation formula(4.19).E N Err N Err N,p Err ϑ320 162 7.99 · 10 −1 7.65 · 10 −1 7.18 · 10 −21240 622 4.97 · 10 −1 4.47 · 10 −1 1.20 · 10 −27432 3718 1.70 · 10 −1 1.51 · 10 −1 4.95 · 10 −4Table 5.9: Exterior Dirichlet BVP on the sphere.E N Err N Err N,p Err ϑ300 152 7.91 · 10 −1 6.82 · 10 −1 1.63 · 10 −11200 602 5.97 · 10 −1 5.43 · 10 −1 2.44 · 10 −27500 3752 2.38 · 10 −1 1.95 · 10 −1 1.77 · 10 −3Table 5.10: Exterior Dirichlet BVP on the cube.E N Err N Err N,p Err ϑ3962 1983 1.38 · 10 −1 1.08 · 10 −1 9.74 · 10 −57510 3757 9.76 · 10 −2 8.14 · 10 −2 2.64 · 10 −5Table 5.11: Exterior Dirichlet BVP on the elephant.0.50.151010.10.80.6−0.50.80.60.050.4−10.400.20.20−1.50−0.05−0.2−0.4−2−0.2−0.4−0.1−0.6−2.5−0.6−0.15−0.8−1−0.500.50.511.5(a) Real part.0−0.5−1−3−3.5−0.8−1−0.500.500.511.5(b) Imaginary part.−0.5−1−0.2−0.25Figure 5.6: Solution to the exterior Dirichlet BVP on the elephant with E = 7510.
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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
Page 41 and 42: 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: 792002000.81500.81500.60.41000.60.4
Page 95 and 96: 83with the unit direction vector d
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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