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

44 3 Boundary Integral Equationsand the right-hand side 1F (q, r) :=2 I + K κ ˜g D , q − ⟨V κ˜g N , q⟩ ΓD +Γ D 12κI − K∗ ˜g N , r − ⟨D κ˜g D , r⟩ ΓN .Γ NTheorem 3.28. If u ∈ H 1 (Ω, ∆ + κ 2 ) is a solution to the mixed problem (3.37), thenthe functions s, t satisfy the system of boundary integral equations (3.40) and u has therepresentationu(x) = ( V κ (s + ˜g N ))(x) − (W κ (t + ˜g D ))(x) for x ∈ Ω. (3.42)Conversely, if s, t satisfy the system of boundary integral equations (3.40), then therepresentation formula (3.42) defines a solution u ∈ H 1 (Ω, ∆ + κ 2 ) to the interior mixedproblem (3.37).3.5.4 Exterior Dirichlet Boundary Value ProblemLet us denote Ω ext := R 3 \ Ω with a simply connected bounded Lipschitz domain Ω ⊂ R 3so that ∂Ω ext = ∂Ω. Now we consider the exterior Dirichlet boundary value problem⎧∆u + κ 2 u = 0 in Ω ext ,⎪⎨γ 0,ext u = g D on ∂Ω,⎪⎩∇u(x) x∥x∥ − iκu(x)1 = O ∥x∥ 2for ∥x∥ → ∞(3.43)with the Dirichlet boundary condition g D ∈ H 1/2 (∂Ω). The solution can be representedasu(x) = −( V κ γ 1,ext u)(x) + (W κ g D )(x) for x ∈ Ω ext (3.44)with unknown Neumann trace data γ 1,ext u. Applying the Dirichlet trace operator to (3.44)we obtain the Fredholm integral equation of the first kind(V κ γ 1,ext u)(x) = − 1 2 g D(x) + (K κ g D )(x) for x ∈ ∂Ω. (3.45)Applying the Neumann trace gives rise to the Fredholm integral equation of the secondkind12 γ1,ext u(x) + (K ∗ κγ 1,ext u)(x) = −(D κ g D )(x) for x ∈ ∂Ω. (3.46)The boundary integral equations (3.45) and (3.46) are equivalent to the variationalproblems⟨V κ γ 1,ext u, s⟩ ∂Ω = − 1 2 I + K κ g D , s for all s ∈ H −1/2 (∂Ω) (3.47)∂Ωand 12κI + K∗ γ 1,ext u, t = ⟨−D κ g D , t⟩ ∂Ω for all t ∈ H 1/2 (∂Ω),∂Ω