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

67with parametersp := αt x + s x1 + α 2 , q := u 2 x + (t x − αs x ) 21 + α 2 . (4.26)Notice that q ≥ 0 andq = 0 ⇔ [u x = 0 ∧ t x − αs x = 0].For α = α 1 and α = α 2 (see (4.24)) the above given situation corresponds to the point xlying on the line defined by the triangle sides x 1 x 2 and x 1 x 3 , respectively.The previous formulae provide an analytic computation of the integral for generalparameters. However, we need to discuss the singularities that may appear in (4.25).Although the arguments of the logarithms are non-negative, they may vanish. However,in both cases the situation is similar toln xlim x ln x = limx→0 + x→0 1 +xl ′ 1H = limxx→0 + − 1 x 2= limx→0 +−x = 0.In the case of u x = 0, i.e., the point x lying in the plane defined by the triangle τ wecan set the arctangent part equal to zero due to the jump property of V κ (3.5) and theboundedness of the arctangent function. Lastly, in the case when s = p, the last part ofthe function F becomes2u x arctan αqu x.4.4.4 Double Layer Potential OperatorThe entries in the double layer potential matrix are given byK κ,h [k, j] := 14π= 14π+ 14πτ k∂Ω τ k∂Ω τ k∂Ωϕ j (y) eiκ∥x−y∥∥x − y∥ 3 (1 − iκ∥x − y∥)⟨x − y, n(y)⟩ ds y ds x⟨x − y, n(y)⟩ϕ j (y)∥x − y∥ 3 ds y ds x⟨x − y, n(y)⟩ϕ j (y)∥x − y∥ 3e iκ∥x−y∥ (1 − iκ∥x − y∥) − 1 ds y ds x .Again, the latter integrand is bounded and we can use the 7-point scheme to compute thedouble surface integral. For the first integral it is sufficient to compute the local elementcontributionsD(τ, x) := 1 ⟨x − y, n(y)⟩ϕ 1 (y)4π∥x − y∥ 3 ds ywith the affine function ϕ 1 defined on τ by⎧⎪⎨ 1 for y = x 1 ,ϕ 1 (y) := 0 for y = x⎪⎩2 ,0 for y = x 3 .τ