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

17the boundary value problem (see, e.g., [6], [9])⎧∆u + κ 2 u = 0 in Ω ext ,⎪⎨⎪⎩x ∇u s (x),∥x∥u s + u i = u in Ω ext ,− iκu s (x)u = 0 on Γ D ,∂u∂n = 0 on Γ N, 1 = O ∥x∥ 2for ∥x∥ → ∞(2.4)with u := u s + u i denoting the total wave. The homogeneous Dirichlet and Neumannboundary conditions represent the so-called sound-soft and sound-hard scattering, respectively.Assuming that the source of the incident wave is remote enough, we can approximateu i by plane waves, i.e.,u i (x) := e iκ⟨x,d⟩with d denoting the normalized propagation direction. Because such u i satisfies the Helmhholtzequation, we can reduce the problem (2.4) to⎧⎪⎨⎪⎩ ∇u s (x),x∥x∥∆u s + κ 2 u s = 0 in Ω ext ,u s = −u i on Γ D ,∂u s∂n = −iκ⟨d, n⟩u i on Γ N , − iκu s (x)1 = O ∥x∥ 2 for ∥x∥ → ∞with n denoting the unit outward normal vector to ∂Ω. The total wave is then given bythe formula u = u s + u i .2.2 Fundamental SolutionThe knowledge of the fundamental solution is essential for the derivation of the representationformulae and the corresponding boundary integral equations. The fundamentalsolution for the Helmholtz equation in R 3 is introduced by the following definition.Definition 2.1 (Fundamental Solution). The function v : R 3 × R 3 → C defined asv κ (x, y) := 1 e iκ∥x−y∥4π ∥x − y∥is called the fundamental solution for the Helmholtz equation in R 3 .In the following theorems we provide some properties of the fundamental solution v κ ,which will be used for the derivation of the representation formulae.