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

33For the jump of the Neumann trace of the single layer potential V κ s on the boundarywe thus get[γ 1 Vκ s] := γ 1,ext Vκ s − γ 1,int Vκ s = −s for all s ∈ H −1/2 (∂Ω).3.3 Double Layer Potential OperatorLet us now consider the operator W κ defined by (3.4). We define the corresponding boundaryintegral operator as the composition of γ 0 and W κ to get the mappingγ 0 W κ : H 1/2 (∂Ω) → H 1/2 (∂Ω).From the properties of the Dirichlet trace operator and the integral operator W κ we getthat γ 0 W κ is linear and continuous.Theorem 3.12. For t ∈ H 1/2 (∂Ω) there holdsγ 0,int (W κ t)(x) = (σ(x) − 1)t(x) + (K κ t)(x)γ 0,ext (W κ t)(x) = σ(x)t(x) + (K κ t)(x)for x ∈ ∂Ω,for x ∈ ∂Ω,where K κ denotes the double layer potential operator∂v κ(K κ t)(x) := (x, y)t(y) ds y∂n y∂Ωfor x ∈ ∂Ωand σ is defined by (3.8).Proof. The proof can be found in [9], following Lemma 2.32.Remark 3.13. Similarly as in Remark 3.11, for Lipschitz domains we get simplified formulaeγ 0,int (W κ t)(x) = − 1 2 t(x) + (K κt)(x) for x ∈ ∂Ω, (3.11)γ 0,ext (W κ t)(x) = 1 2 t(x) + (K κt)(x) for x ∈ ∂Ω. (3.12)For the jump of the Dirichlet trace of the double layer potential W κ t on the boundarywe have[γ 0 W κ t] := γ 0,ext W κ t − γ 0,int W κ t = t for all t ∈ H 1/2 (∂Ω).3.4 Hypersingular Integral OperatorFinally, we consider the hypersingular integral operator defined as the negative Neumanntrace of the double layer potential operator, i.e.,D κ : H 1/2 (∂Ω) → H −1/2 (∂Ω), D κ := −γ 1 W κ .