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

63[8] (see also [16]) can be used. However, we will not describe this technique and will preferanalytic integration when possible.Due to the complexity of the integrands it seems impossible to compute both integralsanalytically. However, it is possible to integrate the inner integral analytically and for theremaining one use a suitable quadrature scheme. In this section we describe a combinationof analytic and numerical integration as proposed in [17].4.4.1 Numerical IntegrationIn the following text we assume the discretization of ∂Ω given by∂Ω ≈Eτ k ,where τ k are triangles in R 3 . Recall from (4.1) and (4.2), that for every element τ we havethe parametrisationk=1R(ξ) := x 1 + Rξ = x 1 + x 2 − x 1 ξ 1 + x 3 − x 1 ξ 2for ξ ∈ ˆτ,where ˆτ denotes the reference triangle (see Figure 4.1a)ˆτ := {ξ ∈ R 2 : 0 < ξ 1 < 1, 0 < ξ 2 < 1 − ξ 1 }.For an arbitrary function f defined on τ we thus havef(x) ds x = f(R(ξ)) x 2 − x 1 × x 3 − x 1 dsξ = 2∆ ττˆτˆτˆf(ξ) ds ξwith ˆf(ξ) := f(R(ξ)) and ∆ τ denoting the surface of the triangle τ, i.e.,∆ τ = 1 ds x .Hence, it is only sufficient to develop a quadrature scheme corresponding to the referencetriangle ˆτ. We approximate the integration asˆττˆf(ξ) ds ξ ≈ 1 2Mω k ˆf(ξ k )k=1with unknown parameters ω k and ξ k ∈ ˆτ for k ∈ {1, . . . , M}.In numerical examples presented in the last section we use a 7-point scheme, which isexact for polynomials up to order 5, i.e., for functions in the linear space P 5 (ˆτ) := span ξ1ξ a 2b , 0 ≤ a + b ≤ 5.