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

41Another approach to computing the missing Neumann data is based on Theorems 3.6and 3.7 suggesting that we may seek the solution in the form of a single layer potentialor a double layer potentialu(x) = ( V κ s)(x) for x ∈ Ω (3.29)u(x) = −(W κ t)(x) for x ∈ Ω (3.30)with unknown density functions s, t. The density functions usually have no physical meaningand these methods are thus called indirect. The representations (3.29) and (3.30) giverise to the boundary integral equationsand the corresponding variational problemsV κ s(x) = γ 0,int u(x) for x ∈ ∂Ω, (3.31)12 t(x) − (K κt)(x) = γ 0,int u(x) for x ∈ ∂Ω⟨V κ s, t⟩ ∂Ω = ⟨γ 0,int u, t⟩ ∂Ω for all t ∈ H −1/2 (∂Ω), 1κ2 I − K t, s = ⟨γ 0,int u, s⟩ ∂Ω for all s ∈ H −1/2 (∂Ω).∂ΩNote that the structure of (3.31) and (3.26) is similar, only the right-hand sides are different.Thus, for values of κ 2 not coinciding with an eigenvalue of (3.28) we have a uniquesolution to (3.31).Although the indirect approach can also be used for other types of boundary valueproblems, in the following sections we only consider the direct boundary element methods.3.5.2 Interior Neumann Boundary Value ProblemSolution to the interior Neumann boundary value problem∆u + κ 2 u = 0 in Ω,γ 1,int u = g Non ∂Ω(3.32)with a bounded Lipschitz domain Ω and the Neumann trace g N ∈ H −1/2 (∂Ω) is given bythe representation formulau(x) = ( V κ g N )(x) − (W κ γ 0,int u)(x) for x ∈ Ω (3.33)with unknown Dirichlet trace data γ 0,int u. Applying the Dirichlet trace operator to (3.33)we obtain the Fredholm integral equation of the second kind12 γ0,int u(x) + (K κ γ 0,int u)(x) = (V κ g N )(x) for x ∈ ∂Ω. (3.34)