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

45respectively.The unique solvability of the boundary integral equation (3.45) and the related variationalproblem (3.47) for κ 2 not coinciding with an eigenvalue of (3.28) follows from thediscussion in Section 3.5.1.Theorem 3.29. If u ∈ H 1 loc (Ωext , ∆ + κ 2 ) is a solution to the exterior Dirichlet problem(3.43), then the Neumann trace γ 1,ext u satisfies the boundary integral equation (3.45) andu has the representation (3.44).Conversely, if γ 1,ext u satisfies the boundary integral equation (3.45), then the representationformula (3.44) defines a solution u ∈ H 1 loc (Ωext , ∆ + κ 2 ) to the exterior Dirichletproblem (3.43).Note that Theorems 3.25 and 3.29 ensure that for all κ ∈ R + it holds− 1 2 I + K κ g D ∈ Im V κeven though the equation (3.45) is not uniquely solvable for some values of κ. To overcomethis difficulty, several approaches have been proposed. The approach of Brakhageand Werner (see [5]) suggests seeking the solution in the form of some complex linearcombination of a single and double layer potentials, i.e.,u(x) = (W κ w)(x) − iη( V κ w)(x)for x ∈ Ω extwith an unknown density function w ∈ L 2 (∂Ω). In [4], Burton and Miller suggest analternative to direct methods, using a complex linear combination of the boundary integralequations (3.17) and (3.18). To conclude, we also mention the CHIEF method proposedby Schenck (see [2]).3.5.5 Exterior Neumann Boundary Value ProblemNow we consider the exterior Neumann boundary value problem⎧∆u + κ 2 u = 0 in Ω ext ,⎪⎨γ 1,ext u = g N on ∂Ω, ⎪⎩x ∇u(x), − iκu(x)1∥x∥ = O ∥x∥ 2 for ∥x∥ → ∞(3.48)with an unbounded domain Ω ext := R 3 \ Ω and the Neumann trace g N ∈ H −1/2 (∂Ω). Thesolution is given byu(x) = −( V κ g N )(x) + (W κ γ 0,ext u)(x) for x ∈ Ω ext (3.49)with unknown Dirichlet trace data γ 0,ext u. Applying the Dirichlet trace operator to (3.49)we obtain the Fredholm integral equation of the second kind− 1 2 γ0,ext u(x) + (K κ γ 0,ext u)(x) = (V κ g N )(x) for x ∈ ∂Ω. (3.50)