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

53k \ τ k x k 1x k 2x k 3local indices1 1 2 32 2 4 33 4 5 3....global indicesx 1 τ 1τ 2τ 3 x 5x 3 x 4x 2Figure 4.3: Connectivity table.We use the connectivity table (see Figure 4.3) to determine the mapping between globaland local indices of nodes building the triangular mesh. For example, the point x 4 is thesecond vertex of the triangle τ 2 , i.e., x 4 = x 2 2. Furthermore, for ϕ 4 | τ2 we haveϕ 4 | τ2 (x) = ϕ 4 | τ2 (R 4 (ξ)) = ˆϕ 2 (ξ) = ξ 1for x ∈ τ 2 , ξ ∈ ˆτ.As in the case of piecewise constant basis functions we can identify a functionT ϕ (∂Ω) ∋ g ϕ =Ng k ϕ kwith a vector [g 1 , . . . , g N ] T ∈ C N . For the approximation of a smooth enough function gwe can either use an interpolation, i.e.,k=1g k = g(x k ).For a function g ∈ L 2 (∂Ω) it is more appropriate to define the projection P ϕ : L 2 (∂Ω) →T ϕ (∂Ω) asP ϕ g := arg min ∥g − g ϕ∥ Lg 2 (∂Ω).ϕ∈T ϕ(∂Ω)The coefficients for real-valued functions can be found in the same way as for the piecewiseconstant functions as the solution to the system of linear equationsNg k ⟨ϕ k , ϕ j ⟩ L 2 (∂Ω) = ⟨g, ϕ j ⟩ L 2 (∂Ω)k=1for j ∈ {1, . . . , N}.In the case of complex-valued functions we can separately search for the real parts and theimaginary parts of the coefficients.The space T ϕ (∂Ω) will be used for the approximation of the Dirichlet data, i.e., thevalues of the solution on ∂Ω.4.3 Discretized Boundary Integral EquationsIn the following sections we describe the discretization of the boundary integral equationsderived in Section 3.5. Although the collocation method will be mentioned, we will prefer