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

59represented by vectors t ∈ C N Nand s ∈ C E D, respectively. Moreover, we define theprolongations ˜g D , ˜g N as˜g D ≈ ˜g D,h := ˜g i D ϕ i ∈ T ϕ (Γ D ),i∈N D˜g N ≈ ˜g N,h := ˜g j N ψ j ∈ T ψ (Γ N ),j∈E N(4.17)i.e., ˜g D,h | Γ− = 0 with Γ −NNdenoting the Neumann part of the boundary without elementsadjacent to Γ D and ˜g N,h | ΓD = 0. Inserting the approximate representations (4.16), (4.17)into the variational formulation (4.15) we obtain the system of Galerkin equationsa(s h , t h , ψ k , ϕ l ) = F (ψ k , ϕ l ) for all k ∈ E D , l ∈ N N . (4.18)For the left-hand side of (4.18) we obtain⟨V κ s, ψ k ⟩ ΓD = 1s j4πj∈E Dτ kτ je iκ∥x−y∥∥x − y∥ ds y ds x = j∈E Ds j Vκ,h [k, j],⟨K κ t, ψ k ⟩ ΓD = 1t i ϕ i (y)4πi∈Nτ kΓ eiκ∥x−y∥N∥x − y∥ 3 (1 − iκ∥x − y∥)⟨x − y, n(y)⟩ ds y ds xN= t iKκ,h [k, i],i∈N N⟨Kκs, ∗ ϕ l ⟩ ΓN = 1es j ϕ l (x)4πj∈EΓ Nτ iκ∥x−y∥j∥x − y∥ 3 (iκ∥x − y∥ − 1)⟨x − y, n(x)⟩ ds y ds xD= 1s j4πj∈E D= j∈E Ds j Kκ,h [j, l],τ jΓ Nϕ l (y) eiκ∥x−y∥∥x − y∥ 3 (1 − iκ∥x − y∥)⟨x − y, n(y)⟩ ds y ds x⟨D κ t, ϕ l ⟩ ΓN = 1 et i4πi∈NΓ NΓ iκ∥x−y∥N∥x − y∥ ⟨curl ∂Ω ϕ i (y), curl ∂Ω ϕ l (x)⟩ ds y ds xN− κ2 e4πΓ NΓ iκ∥x−y∥N∥x − y∥ ϕ i(y)ϕ l (x)⟨n(x), n(y)⟩ ds y ds x = t iDκ,h [l, i].i∈N NNote that the matrices from the previous formulae are submatrices of (4.7), (4.9) and(4.14) with dimensionsV κ,h ∈ C E D×E D, Kκ,h ∈ C E D×N N, Dκ,h ∈ C N N×N N.For the right-hand side of (4.18) we have⟨˜g D , ψ k ⟩ ΓD = ˜g iD ϕ i (x) ds x = ˜g iDi∈Nτ k D i∈N D¯M h [k, i],