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)