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

46 3 Boundary Integral EquationsSimilarly, applying the Neumann trace operator we get the Fredholm integral equation ofthe first kind(D κ γ 0,ext u)(x) = − 1 2 g N(x) − (K ∗ κg N )(x) for x ∈ ∂Ω. (3.51)andInstead of equations (3.50) and (3.51) we may consider the variational problemsrespectively.− 1 2 I + K κ γ 0,ext u, s = ⟨V κ g N , s⟩ ∂Ω for all s ∈ H −1/2 (∂Ω),∂Ω⟨D κ γ 0,ext u, t⟩ ∂Ω =− 1 2 I − K∗ κ g N , t∂Ωfor all t ∈ H 1/2 (∂Ω), (3.52)The arguments for the unique solvability of the boundary integral equation (3.51) andthe corresponding variational problem (3.52) for κ 2 not coinciding with an eigenvalue of(3.36) are the same as in Section 3.5.2.Theorem 3.30. If u ∈ H 1 loc (Ωext , ∆ + κ 2 ) is a solution to the exterior Neumann problem(3.48), then the Dirichlet trace γ 0,ext u satisfies the boundary integral equation (3.51) andu has the representation (3.49).Conversely, if γ 0,ext u satisfies the boundary integral equation (3.51), then the representationformula (3.49) defines a solution u ∈ H 1 loc (Ωext , ∆ + κ 2 ) to the interior Neumannproblem (3.48).Note that Theorems 3.25 and 3.30 ensure that for all κ ∈ R + it holds− 1 2 I − K∗ κ g N ∈ Im D κand thus one of the methods mentioned at the end of the previous section can be used tocompute a solution to (3.51) with κ 2 coinciding with an eigenvalue of (3.36).3.5.6 Exterior Mixed Boundary Value ProblemFinally, let us consider the exterior mixed boundary value problem⎧⎪⎨⎪⎩ ∇u(x),x∥x∥∆u + κ 2 u = 0 in Ω ext ,γ 0,ext u = g D on Γ D ,γ 1,ext u = g N on Γ N , − iκu(x)1 = O ∥x∥ 2for ∥x∥ → ∞(3.53)