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

694.4.5 Hypersingular Integral OperatorThe matrix corresponding to the hypersingular integral operator is given byD κ,h [i, j] := 1 4π− κ24π∂Ω∂Ω∂Ω∂Ωe iκ∥x−y∥∥x − y∥ ⟨curl ∂Ω ϕ j (y), curl ∂Ω ϕ i (x)⟩ ds y ds xe iκ∥x−y∥∥x − y∥ ϕ j(y)ϕ i (x)⟨n(x), n(y)⟩ ds y ds x .Since the unit outward normals are constant on individual elements, we have for the firstintegral D 1 1 e iκ∥x−y∥κ,h [i, j] :=4π ∂Ω ∂Ω ∥x − y∥ ⟨curl ∂Ω ϕ j (y), curl ∂Ω ϕ i (x)⟩ ds y ds x= ⟨curl ∂Ω ϕ j | τl (y), curl ∂Ω ϕ i | τk (x)⟩ 14πτ k ⊂supp ϕ i τ l ⊂supp ϕ i= ⟨curl ∂Ω ϕ j | τl (y), curl ∂Ω ϕ i | τk (x)⟩V κ,h [k, l]τ k ⊂supp ϕ i τ l ⊂supp ϕ iτ kτ le iκ∥x−y∥∥x − y∥ ds y ds xand thus the matrix entries are some linear combination of corresponding entries in V κ,h .Recall thatcurl ∂Ω ϕ(y) := n(y) × ∇ ϕ(y),where for the extensions of the basis functions we can choose functions constant along thenormal defined by the reference functionsˆϕ1 (ξ) := 1 − ξ 1 − ξ 2 , ˆϕ2 (ξ) := ξ 1 , ˆϕ3 (ξ) := ξ 2 for ξ = [ξ 1 , ξ 2 , ξ 3 ] T ∈ ˆτ × R.Hence, it is sufficient to compute the integralD 2 κ2κ,h [i, j] :=4π= κ24π+ κ24π∂Ω∂Ω∂Ωϕ i (x)ϕ i (x)ϕ i (x)∂Ω∂Ω∂Ωe iκ∥x−y∥∥x − y∥ ϕ j(y)⟨n(x), n(y)⟩ ds y ds x1∥x − y∥ ϕ j(y)⟨n(x), n(y)⟩ ds y ds xe iκ∥x−y∥ − 1∥x − y∥ ϕ j(y)⟨n(x), n(y)⟩ ds y ds x .The latter integrand is bounded and we can use the 7-point quadrature scheme. Theremaining integral is similar to the single layer potential matrix entries and although theanalytical formula is not given in [17], we will follow the computations in Section C.2.1