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

614.3.4 Exterior Dirichlet Boundary Value ProblemIn the following three sections we describe the solution to exterior boundary value problems.Since the discretization techniques are identical to those mentioned above, the discussionwill be more succinct.The approximate solution to the exterior Dirichlet boundary value problem (3.43) isgiven by the discretized representation formula (3.44), i.e.,u(x) ≈ u h (x) := −El=1g N lτ lv κ (x, y) ds y +Nj=1g D j∂Ωϕ j (y) ∂v κ∂n y(x, y) ds y (4.19)with unknown Neumann data g N,h . To obtain the missing data we use the variationalformulation corresponding to the Fredholm integral equation (3.45), i.e.,⟨V κ g N , s⟩ ∂Ω = − 1 2 I + K κ g D , s for all s ∈ H −1/2 (∂Ω). (4.20)Inserting the approximations of the Cauchy data (4.4), (4.5) into (4.20) we obtain thesystem of Galerkin equationsEgl N ⟨V κψ l , ψ k ⟩ ∂Ω =l=1Nj=1g D j∂Ω− 1 2 I + K κ ϕ j , ψ k∂Ωand the related system of linear equationsV κ,h g N =− 1 2 M h + K κ,h g Dwith the matrices given by (4.7), (4.8) and (4.9).for all k ∈ {1, . . . , E}4.3.5 Exterior Neumann Boundary Value ProblemThe solution to the exterior Neumann boundary value problem (3.48) is given by therepresentation formula (3.49) and its discretized variant (4.19). To compute the missingDirichlet data we use the hypersingular boundary integral equation (3.51) and the relatedvariational problem⟨D κ g D , t⟩ ∂Ω = − 1 2 I − K∗ κ g N , t for all t ∈ H 1/2 (∂Ω). (4.21)∂ΩSubstituting the approximate boundary data g D,h , g N,h into (4.21) we obtain the GalerkinequationsNgj D ⟨D κ ϕ j , ϕ i ⟩ ∂Ω =j=1El=1g N l− 1 2 I − K∗ κ ψ l , ϕ i∂Ωfor all i ∈ {1, . . . , N}