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

56 4 Discretization and Numerical RealizationThe Galerkin equations (4.6) can thus be rewritten into the system of linear equationsV κ,h g N = 12 M h + K κ,hg D .The approximate solution to (3.24) can be obtained using the discretized representationformulau(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 . (4.11)Another approach to the discretization of the boundary integral equations is the collocationmethod. Because the relation (4.3) is valid for all x ∈ ∂Ω, we can insert themidpoints x k∗ into (4.3) to obtain the system of linear equationswithV κ,h g N =C E×E ∋ V κ,h [k, l] := 1 e4πτ iκ∥xk∗ −y∥l∥x k∗ − y∥ ds y,R E×N ∋ M h [k, j] := ϕ j (x k∗ ),C E×N ∋ K κ,h [k, j] :=∂Ωϕ j (y) 14π 12 M h + K κ,hg De iκ∥xk∗ −y∥∥x k∗ − y∥ 3 (1 − iκ∥xk∗ − y∥)⟨x k∗ − y, n(y)⟩ ds y .Contrary to the Galerkin scheme, the system matrix V κ,h arising from the collocationmethod is in general non-symmetric and the stability of the method is still an open problem.Although the collocation scheme can be derived for other boundary integral equations inthe same way, in the following sections we prefer the Galerkin discretization.4.3.2 Interior Neumann Boundary Value ProblemThe solution to the interior Neumann boundary value problem (3.32) is given by therepresentation formula (3.33) or by its discretized version (4.11). To compute the missingDirichlet data g D := γ 0,int u we use the hypersingular equation(D κ g D )(x) = 1 2 g N(x) − (K ∗ κg N )(x) for x ∈ ∂Ωor−γ 1,int ∂Ωg D (y) ∂v κ(x, y) ds y = 1 ∂n y 2 g N(x) − g N (y) ∂v κ(x, y) ds y∂Ω ∂n xfor x ∈ ∂Ω