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

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86 5 Numerical Experimentswith the related system of linear equationsVκ,h −K κ,h s −¯V κ,h − 1 ¯M 2 h + ¯K κ,h gN=D κ,h t ¯M T h − ¯KT κ,h−¯D κ,h g D .K T κ,h− 1 2E N Err D Err D,p Err N Err N,p Err ϑ320 162 1.04 · 10 −2 6.32 · 10 −3 8.09 · 10 −1 7.75 · 10 −1 7.18 · 10 −21240 622 2.51 · 10 −3 1.74 · 10 −3 5.06 · 10 −1 4.54 · 10 −1 1.20 · 10 −27432 3718 4.16 · 10 −4 2.68 · 10 −4 1.73 · 10 −1 1.53 · 10 −1 4.95 · 10 −4Table 5.15: Exterior mixed BVP on the sphere.E N Err D Err D,p Err N Err N,p Err ϑ300 152 2.76 · 10 −1 2.59 · 10 −1 1.71 · 10 −1 1.13 · 10 −1 1.24 · 10 −11200 602 1.01 · 10 −1 4.68 · 10 −2 6.52 · 10 −2 5.71 · 10 −2 7.78 · 10 −27500 3752 2.08 · 10 −2 1.31 · 10 −2 2.41 · 10 −2 2.29 · 10 −2 1.11 · 10 −2Table 5.16: Exterior mixed BVP on the cube.The results are summarized in Tables 5.15, 5.16 with the errors given by (5.6) and thecurve ϑ: [0, 1] → R 3 defined by (5.9).
87ConclusionIn the preceding sections we described the application of the boundary element method forsolving the Helmholtz equation in 3D. We showed that the method is a good alternativeto the widely used finite element method and in the case of exterior problems especially,the boundary integral formulation seems more natural than the finite element approachwith an artificial boundary. In the last section we applied the Galerkin boundary elementmethod to various boundary value problems. In the tables provided we showed that theerror decreases with increasing number of elements not only on the boundary but also inthe domain itself.However, it should be noted that the method presented here is memory and computationallydemanding due to dense system matrices. This is a major drawback makingthe method inapplicable for large problems. To overcome this problem, fast boundary elementmethods have been developed, allowing to approximate some entries in the matricesgenerated by the discretized boundary integral operators. Due to a noticeable complexityreduction such methods offer very attractive alternatives to other standard approaches.To conclude, no elephant was hurt during the making of this thesis.
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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
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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 and 96: 83with the unit direction vector d
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Page 102: 90[15] SAUTER, Stefan; SCHWAB, Chri
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