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

83with the unit direction vector d := √ 13[1, 1, 1] T . The scattered wave u s is depicted inFigure 5.7, in Figure 5.8 you can see the total wave u = u s + u i .5.5 Exterior Neumann Boundary Value ProblemNext we consider the exterior Neumann boundary value problem (3.48) with κ = 2 andthe testing solutions (5.7) for Ω being the sphere and the cube and (5.8) in the case of theelephant.To solve the exterior Neumann problem and to find the missing Dirichlet data weuse the discretized variational formulation related to the hypersingular boundary equation(3.51), i.e.,⟨D κ g D,h , ϕ i ⟩ ∂Ω =− 1 2 I − K∗ κ g N,h , ϕ i∂Ωfor all i ∈ {1, . . . , N}and the matrix formulationD κ,h g D =− 1 2 MT h − KT κ,h g N .E N Err D Err D,p Err ϑ320 162 1.92 · 10 −1 7.11 · 10 −2 1.92 · 10 −11240 622 6.36 · 10 −2 1.71 · 10 −2 5.34 · 10 −27432 3718 9.24 · 10 −3 4.39 · 10 −3 6.86 · 10 −3Table 5.12: Exterior Neumann BVP on the sphere.E N Err D Err D,p Err ϑ300 152 2.46 · 10 −1 2.32 · 10 −1 1.20 · 10 −11200 602 9.02 · 10 −2 4.18 · 10 −2 7.78 · 10 −27500 3752 1.86 · 10 −2 1.17 · 10 −2 1.11 · 10 −2Table 5.13: Exterior Neumann BVP on the cube.E N Err D Err D,p Err ϑ3962 1983 7.90 · 10 −2 5.59 · 10 −3 1.77 · 10 −27510 3757 7.14 · 10 −2 2.31 · 10 −3 1.78 · 10 −2Table 5.14: Exterior Neumann BVP on the elephant.