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

494 Discretization and Numerical RealizationIn this section we will describe the discretization of the boundary integral equations coveredin the previous part. We will discuss the collocation method as well as the more complicatedGalerkin scheme. The solution will be sought in approximating function spaces introducedbelow.We restrict ourselves to triangular meshes approximating the corresponding boundary∂Ω, i.e.,E∂Ω ≈and assume that neighbouring triangular elements either share a whole edge or a singlevertex. Each element τ k ⊂ R 3 with nodes x k 1, x k 2, x k 3can be described via theparametrizationR k (ξ) := x k 1+ R k ξ for ξ ∈ ˆτ, (4.1)where R k denotes the matrixk=1⎡τ kx k 21 − xk 11 x k 31 − xk 11R k := x k 2− x k 1x k 3− x k 1= ⎣x k 22 − xk 12 x k 32 − xk 1⎦ 2∈ R 3×2 (4.2)x k 23 − xk 13 x k 33 − xk 13and ˆτ ⊂ R 2 represents the reference triangle (see Figure 4.1a)ˆτ := {ξ ∈ R 2 : 0 < ξ 1 < 1, 0 < ξ 2 < 1 − ξ 1 }.Remark 4.1. Note that a function f defined on τ k can be identified with a function ˆfdefined on ˆτ and vice versa, i.e.,⎤f(x) = f(R k (ξ)) =: ˆf(ξ)for x ∈ τ k , ξ ∈ ˆτ.In the following text we use the symbol ∂Ω to denote the discretized boundary.4.1 Piecewise Constant Basis FunctionsFor every element τ k we define the function ψ k (see Figure 4.2a) as1 for x ∈ τ k ,ψ k (x) :=0 otherwise.Following Remark 4.1, the function ψ k can alternatively be defined via the reference function1 for ξ ∈ ˆτ,ˆψ(ξ) :=0 otherwise.