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

60 4 Discretization and Numerical Realization⟨K κ˜g D , ψ k ⟩ ΓD = i∈N D˜g D i= ⟨V ˜g N , ψ k ⟩ ΓD = 1ϕ i (y)4πτ eiκ∥x−y∥k Γ + ∥x − y∥ 3 (1 − iκ∥x − y∥)⟨x − y, n(y)⟩ ds y ds xDi∈N D˜g D i ¯K κ,h [k, i],j∈E N˜g N j⟨˜g N , ϕ l ⟩ ΓN = 14πτ kτ je iκ∥x−y∥∥x − y∥ ds y ds x = τ jϕ l (x) ds x = j∈E N˜g N j ¯V κ,h [k, j],˜g jN ˜g j N ¯Mh [j, l],j∈E N j∈E N⟨Kκ˜g ∗ N , ϕ l ⟩ ΓN = ˜g N 1ej ϕ l (x)4πj∈EΓ Nτ iκ∥x−y∥j∥x − y∥ 3 (iκ∥x − y∥ − 1)⟨x − y, n(x)⟩ ds y ds xN= j∈E N˜g N j= ˜g j N ¯Kκ,h [j, l],j∈E N⟨D κ˜g D , ϕ l ⟩ ΓN = ˜g D 1i4πi∈N D1ϕ l (y)4πτ jΓ eiκ∥x−y∥N∥x − y∥ 3 (1 − iκ∥x − y∥)⟨x − y, n(y)⟩ ds y ds xΓ NΓ + De iκ∥x−y∥∥x − y∥ ⟨curl ∂Ω ϕ i (y), curl ∂Ω ϕ l (x)⟩ ds y ds x− κ2 e4πΓ iκ∥x−y∥N Γ + ∥x − y∥ ϕ i(y)ϕ l (x)⟨n(x), n(y)⟩ ds y ds xD= ˜g D ¯D i κ,h [l, i]i∈N Dwith Γ + Ddenoting the Dirichlet part of the boundary together with Neumann elementsadjacent to Γ D . Again, the matrices from the above relations are submatrices of (4.8),(4.9), (4.7) and (4.14) with dimensions¯M h ∈ R E D×N D, ¯Kκ,h ∈ C E D×N D, ¯Vκ,h ∈ C E D×E N,¯M h ∈ R E N×N N,¯K κ,h ∈ C E N×N N, ¯Dκ,h ∈ C N N×N D.The system of Galerkin equations can now be rewritten asVκ,hK T κ,h−K κ,h sD κ,h t=121−¯V ¯M κ,h 2 h + ¯K κ,h gN¯M T h − ¯KT κ,h−¯D κ,h g DNote that similarly as for the interior Neumann problem the transpositions do not involvethe complex conjugation..