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

52 4 Discretization and Numerical Realizationwhere ∆ k denotes the surface of τ k , we get for the piecewise constant approximationg k = 1 ∆ kτ kg(x) ds x .Let us now consider the case when g = Re g + i Im g is complex-valued. We thus searchfor coefficients g k = Re g k + i Im g k . For the L 2 error norm we get E g − 2g k ψ kk=1==∂Ω∂ΩL 2 (∂Ω) E = Re g + i Im g − 2(Re g k + i Im g k )ψ kk=1E Re g(x) − Re g k ψ k (x) + i Im g(x) −Re g(x) −k=1L 2 (∂Ω)E2Im g k ψ k (x) ds xk=12 ERe g k ψ k (x)+ Im g(x) −k=1 E = Re g − 2Re g k ψ kk=1L 2 (∂Ω) 2EIm g k (x)ψ k ds xk=1 E + Im g − 2Im g k ψ kk=1L 2 (∂Ω)and therefore, the problem is reduced to searching for appropriate coefficients of the tworeal-valued functions Re g and Im g as was described in the previous paragraph.The space T ψ (∂Ω) will be used for the approximation of the Neumann data, i.e., of thenormal derivatives on ∂Ω.4.2 Continuous Piecewise Affine Basis FunctionsLet N denote the number of all nodes of a given discretization. Then we define the familyof functions {ϕ k } N k=1continuous over the whole discretized boundary (see Figure 4.2b) as⎧⎪⎨ 1 for x = x k ,ϕ k (x) := 0 for x = x⎪⎩l ≠ x k ,piecewise affine otherwise.Furthermore, we define the linear space T ϕ (∂Ω) asT ϕ (∂Ω) := span{ϕ k } N k=1 .Obviously, dim T ϕ (∂Ω) = N. A restriction of a function ϕ k to an element τ l ⊂ supp ϕ kcan be identified with one of the functions ˆϕ 1 , ˆϕ 2 , ˆϕ 3 defined on the reference triangle asˆϕ 1 (ξ) := 1 − ξ 1 − ξ 2 , ˆϕ 2 (ξ) := ξ 1 , ˆϕ 3 (ξ) := ξ 2 for ξ ∈ ˆτ.