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

792002000.81500.81500.60.41000.60.41000.2500.2500−0.200−0.20−0.4−50−0.4−50−0.6−0.8−100−0.6−0.8−100−0.500.50.5(a) Real part.0−0.5−150−200−0.500.50.5(b) Imaginary part.0−0.5−150Figure 5.5: Solution to the interior Neumann BVP inside the cube with E = 1200.In Tables 5.4, 5.5, 5.6 we summarize the results. The error columns are now given byErr D := ∥g D − g D,h ∥ L 2 (∂Ω)∥g D ∥ L 2 (∂Ω), Err D,p := ∥g D − g D,p ∥ L 2 (∂Ω), Err ϑ := ∥u − u h∥ L 2 (ϑ)∥g D ∥ L 2 (∂Ω)∥u∥ L 2 (ϑ)(5.5)with g D , g D,h , g D,p denoting the exact, computed and L 2 projected Dirichlet data, respectively.Note that the error of the computed solution cannot be lower than the errorcorresponding to the L 2 projected function. Again, ϑ: [0, 1] → R 3 denotes the curve (5.4).See pictures 5.4, 5.5 for the solution on the cube and on a grid placed inside.5.3 Interior Mixed Boundary Value ProblemLet us now consider the interior mixed boundary value problem (3.37) with κ = 2 and thetesting solution (5.1). For Ω we choose the sphere {x ∈ R 3 : ∥x∥ ≤ 1} withΓ D := {x ∈ R 3 : ∥x∥ = 1 ∧ x 3 > 0}, Γ N := {x ∈ R 3 : ∥x∥ = 1 ∧ x 3 < 0}and the cube with three sides belonging to Γ D and the remaining part belonging to Γ N .To solve the mixed problem and to find the missing Cauchy data we use the discretizedGalerkin equations related to the symmetric formulation (3.41), i.e.,a(s h , t h , ψ k , ϕ l ) = F (ψ k , ϕ l )for all k ∈ E D , j ∈ N Nwith the matrix formulationVκ,h −K κ,h s=D κ,h tK T κ,h121−¯V ¯M κ,h 2 h + ¯K κ,h gN¯M T h − ¯KT κ,h−¯D κ,h g D.