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

25Ωn to ∂Ω ε0n to ∂Ω∂Ω∂B ε (o)n to ∂Ω εΩ εFigure 2.3: Illustration for the proof of Theorem 2.7.Proof. Let B ε (0) := {y ∈ R 3 : ∥y∥ < ε}, where ε is taken such that Ω ⊂ B ε (0). Furthermore,let Ω ε := B ε (0) \ Ω (see Figure 2.3). We start with showing that|u(y)| 2 ds = O(1) for ε → ∞.∂B ε(0)From the radiation condition we deduce lim∂u2ε→∞ ∂n (y) − iκu(y) ds = 0. (2.9)∂B ε(0)BecauseIm u ∂ū = Re u Im ∂ū∂ū∂u∂u+ Im u Re = − Re u Im + Im u Re∂n∂n ∂n ∂n ∂n ,we get for the integrand from (2.9) ∂u 2∂n − iκu =∂uRe ∂n∂u2 + i Im∂n − iκ Re u + κ Im u = Re ∂u∂n+ κ 2 (Im u) 2 + 2κ Im u Re ∂u ∂n + Im ∂u∂n− 2κ Im ∂u ∂n Re u = ∂u∂n2+ κ 2 |u| 2 + 2κ Im 2 2+ κ 2 (Re u) 2u ∂ū ∂n