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

35with the surface curl operatorcurl ∂Ω t(y) := n(y) × ∇˜t(y)for y ∈ ∂Ω,where ˜t is some locally defined extension of t into the neighbourhood of ∂Ω.Proof. For the proof and more details see [13], Theorem 3.4.2 and [15], Theorem 3.3.22and Corrolary 3.3.24.Theorem 3.15. The hypersingular operator D κ : H 1/2 (∂Ω) → H −1/2 (∂Ω) is coercive.Proof. The proof is taken from [15], Lemma 3.9.8. The regularised hypersingular operatorD 0 + I corresponding to the Laplace equation is H 1/2 (∂Ω)-elliptic. Due to the compactembedding H 1/2 (∂Ω) ↩→↩→ H −1/2 (∂Ω) and the compactness of D κ − D 0 , the operatorC := I + D 0 − D κ : H 1/2 (∂Ω) → H −1/2 (∂Ω) is compact. Thus, we have⟨(D κ + C)t, t⟩ = ⟨(D 0 + I)t, t⟩ ≥ c∥t∥ 2 H 1/2 (∂Ω)for all t ∈ H 1/2 (∂Ω),which completes the proof.To conclude, for the jump of the Neumann trace of the double layer potential W κ t onthe boundary we have[γ 1 W κ t] := γ 1,ext W κ t − γ 1,int W κ t = 0 for all t ∈ H 1/2 (∂Ω). (3.14)3.5 Boundary Integral EquationsRecall that the solution to an interior boundary value problem for the Helmholtz equationcan be represented asu = −W κ γ 0,int u + V κ γ 1,int u in Ω.Applying the interior Dirichlet trace operator and using the definition of V κ and the property(3.11) we get the boundary integral equation 1γ 0,int u =2 I − K κ γ 0,int u + V κ γ 1,int u on ∂Ω (3.15)with the identity operator I. Similarly, applying the Neumann trace operator and using(3.9) and the definition of the hypersingular integral operator we obtain 1γ 1,int u = D κ γ 0,int u +2 I + K∗ κ γ 1,int u on ∂Ω. (3.16)The boundary integral equations (3.15), (3.16) can be rewritten as γ 0,int uγ 1,int = C int γ 0,int uu γ 1,int u