The Boundary Element Method for the Helmholtz Equation ... - FEI VÅ B
The Boundary Element Method for the Helmholtz Equation ... - FEI VÅ B The Boundary Element Method for the Helmholtz Equation ... - FEI VÅ B
26 2 Helmholtz Equationand thus lim∂u 2ε→∞ ∂B ε(0) ∂n (y)Since+ κ 2 |u(y)| 2 ds + 2κ Im u(y) ∂ū∂B ε(0) ∂n (y) ds = 0. (2.10)Now we apply the first Green’s identity (1.7) in Ω ε for functions u and ū to obtain∂ū∆ū(y)u(y) dy + ∇ū(y)∇u(y) dy = (y)u(y) ds. (2.11)Ω ε Ω ε∂n∆ū + κ 2 ū = 0in Ω εand ∂Ω ε = ∂Ω ∪ ∂B ε (0), we can rewrite the formula (2.11) as|∇u(y)| 2 dy − κ 2 |u(y)| 2 ∂ūdy =Ω ε Ω ε ∂B ε(0) ∂n∂Ω(y)u(y) ds − ∂ū(y)u(y) ds, (2.12)∂nwhere we had to take into account the opposite direction of n for y ∈ ∂Ω (see Figure 2.3).Because the left-hand side of (2.12) is real, we obtain∂ūIm∂n∂Ω(y)u(y) ds = Im ∂ū(y)u(y) ds < ∞. (2.13)∂n∂B ε(0)Inserting (2.13) into (2.10) we getand thus lim∂u 2ε→∞ ∂B ε(0) ∂n (y)limε→∞∂B ε(0) ∂u 2∂n (y)∂Ω ε+ κ 2 |u(y)| 2 ds + 2κ Im u(y) ∂ū∂Ω ∂n (y) ds = 0+ κ 2 |u(y)| 2 ds = −2κ Im∂Ωu(y) ∂ū (y) ds < ∞.∂nBecause both terms on the left-hand side are non-negative, they have to be bounded forε → ∞. Therefore, we have proved that|u(y)| 2 ds = O(1) for ε → ∞. (2.14)∂B ε(0)From Theorem 2.3 we know that the radiation condition and thus also (2.14) is validfor the fundamental solution v κ . Using the Hölder inequality we thus obtain ∂vκI 1 := u(y) (x, y) − iκv κ (x, y) ds y∂B ε(0) ∂n y 1/2 ≤ |u(y)| 2 ds y ∂v κ2 1/2 (x, y) − iκv κ (x, y)∂B ε(0)∂B ε(0) ∂n y ds y → 0 for ε → ∞(2.15)
27andI 2 :=∂B ε(0) ∂uv κ (x, y)∂n (y) − iκu(y) ds y 1/2 ≤ |v κ (x, y)| 2 ds y ∂u2 1/2∂B ε(0)∂B ε(0) ∂n (y) − iκu(y) ds y → 0 for ε → ∞.(2.16)Finally, for the bounded domain Ω ε we can apply the representation formula (2.7) toobtain∂uu(x) =∂Ω ε∂n (y)v κ(x, y) ds y − u(y) ∂v κ(x, y) ds y∂Ω ε∂n y∂u= −∂n (y)v κ(x, y) ds y + u(y) ∂v κ(x, y) ds y + I 2 − I 1∂n y∂Ωfor all x ∈ Ω ε . Letting ε → ∞, using (2.15) and (2.16), we eventually getu(x) = u(y) ∂v κ∂u(x, y) ds y −∂n y ∂n (y)v κ(x, y) ds y∂Ωfor all x ∈ Ω ext , which was to be proved.Theorems 2.6 and 2.7 provide representation of the solution to the Helmholtz equationby means of boundary integrals. The functions( V ∂v κκ s)(x) := v κ (x, y)s(y) ds y and (W κ t)(x) := (x, y)t(y) ds y∂n y∂Ωare called potentials with density functions s, t. Although we have only discussed the caseof smooth data, in the following section we will show that the potentials can also be definedfor more general density functions and domains. The properties of the integral operatorsV κ and W κ will play a crucial role in obtaining the missing Cauchy data.∂Ω∂Ω∂Ω
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27andI 2 :=∂B ε(0) ∂uv κ (x, y)∂n (y) − iκu(y) ds y 1/2 ≤ |v κ (x, y)| 2 ds y ∂u2 1/2∂B ε(0)∂B ε(0) ∂n (y) − iκu(y) ds y → 0 <strong>for</strong> ε → ∞.(2.16)Finally, <strong>for</strong> <strong>the</strong> bounded domain Ω ε we can apply <strong>the</strong> representation <strong>for</strong>mula (2.7) toobtain∂uu(x) =∂Ω ε∂n (y)v κ(x, y) ds y − u(y) ∂v κ(x, y) ds y∂Ω ε∂n y∂u= −∂n (y)v κ(x, y) ds y + u(y) ∂v κ(x, y) ds y + I 2 − I 1∂n y∂Ω<strong>for</strong> all x ∈ Ω ε . Letting ε → ∞, using (2.15) and (2.16), we eventually getu(x) = u(y) ∂v κ∂u(x, y) ds y −∂n y ∂n (y)v κ(x, y) ds y∂Ω<strong>for</strong> all x ∈ Ω ext , which was to be proved.<strong>The</strong>orems 2.6 and 2.7 provide representation of <strong>the</strong> solution to <strong>the</strong> <strong>Helmholtz</strong> equationby means of boundary integrals. <strong>The</strong> functions( V ∂v κκ s)(x) := v κ (x, y)s(y) ds y and (W κ t)(x) := (x, y)t(y) ds y∂n y∂Ωare called potentials with density functions s, t. Although we have only discussed <strong>the</strong> caseof smooth data, in <strong>the</strong> following section we will show that <strong>the</strong> potentials can also be defined<strong>for</strong> more general density functions and domains. <strong>The</strong> properties of <strong>the</strong> integral operatorsV κ and W κ will play a crucial role in obtaining <strong>the</strong> missing Cauchy data.∂Ω∂Ω∂Ω
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