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

62 4 Discretization and Numerical Realizationand the system of linear equationsD κ,h g D =− 1 2 MT h − KT κ,h g N ,given by the matrices (4.14), (4.8) and (4.9). Again, the transpositions do not involve thecomplex conjugation.4.3.6 Exterior Mixed Boundary Value ProblemLastly, let us consider the exterior mixed boundary value problem (3.53). Similarly asin the case of the interior mixed problem we use the symmetric approach given by thevariational formulationa(s, t, q, r) = F (q, r) for all q ∈ H −1/2 (Γ D ), r ∈ H 1/2 (Γ N ) (4.22)with unknown functions s ∈ H −1/2 (Γ D ), t ∈ H 1/2 (Γ N ), the bilinear forma(s, t, q, r) := ⟨V κ s, q⟩ ΓD − ⟨K κ t, q⟩ ΓD + ⟨K ∗ κs, r⟩ ΓN + ⟨D κ t, r⟩ ΓNthe right-hand sideF (q, r) := − 1 2 I + K κ ˜g D , q − ⟨V κ˜g N , q⟩ ΓD + − 1 Γ D2 I − K∗ κ ˜g N , r − ⟨D κ˜g D , r⟩ ΓNΓ Nand some fixed prolongations of the Cauchy data ˜g D , ˜g N . Inserting the discretized functionss h , t h from (4.16) and the prolongations ˜g D , ˜g N from (4.17) into (4.22) we obtain the systemof Galerkin equationsa(s h , t h , ψ k , ϕ l ) = F (ψ k , ϕ l )for all k ∈ E D , l ∈ N Nand the related system of linear equationsVκ,h −K κ,h s −¯V κ,h − 1 ¯M 2 h + ¯K κ,h gN=D κ,h t ¯M T h − ¯KT κ,h−¯D κ,h g DK T κ,h− 1 2with the matrices defined in Section 4.3.3. The transpositions in the above given systemdo not involve the complex conjugation of the elements.4.4 Integration over Boundary ElementsThere are several approaches to the computation of the double integrals arising from theGalerkin discretization of the boundary integral equations. One way is to use a numericalquadrature for both integrals. It should be, however, noted, that numerical integrationintroduces an additional error and one has to be cautious when dealing with singularitiesin the integrands. As a workaround to this problem the Duffy transformation proposed in