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

66 4 Discretization and Numerical Realization4.4.3 Single Layer Potential OperatorTo compute the matrices arising from the Galerkin equations we need to evaluate thecorresponding double surface integrals. In the following three sections we provide analyticevaluations of the singular parts of the inner integrals related to the single layer potentialmatrix V κ,h , the double layer potential matrix K κ,h and the hypersingular operator matrixD κ,h . The outer integrals can be computed using the 7-point quadrature rule discussed inSection 4.4.1.The analytic formulae corresponding to the single layer potential operator and thedouble layer potential operator are given in [17], Section C.2, and we only provide the finalresults with some discussion. For the hypersingular operator we provide a more detailedcomputation.The entries of the single layer operator matrix are given byV κ,h [k, l] := 1 e4πτ kτ iκ∥x−y∥l∥x − y∥ ds y ds x= 14πτ kFor the latter integrand we have1τ l∥x − y∥ ds y ds x + 14πτ ke iκ∥x−y∥ − 1τ l∥x − y∥e iκ∥x−y∥ − 1 e iκa − 1 llim= lim′ H = lim iκe iκa = iκx→y ∥x − y∥ a→0 a a→0ds y ds x .and thus the integral is not singular and can be evaluated numerically. For the remainingsingular part of the integral we use an analytic computation based on the local coordinatesystem introduced in Section 4.4.2 combined with a quadrature scheme. For the innerintegral we obtain (see [17], Section C.1.2.)S V (τ, x) := 1 14π τ ∥x − y∥ ds y = 1 sτ α2 s1 dt ds4π 0 α 1 s (s − sx ) 2 + (t − t x ) 2 + u 2 x= 1 sτ ln t − t x + α2(s − s x )4π2 + (t − t x ) 2 + u 2 sx ds0α 1 s= 1 F V (s τ , α 2 ) − F V (0, α 2 ) − F V (s τ , α 1 ) + F V (0, α 1 ) ,4πwhereF V (s, α) := (s − s x ) ln αs − t x + (s − s x ) 2 + (αs − t x ) 2 + u 2 x − s+ αs x − t√ x ln 1 + α 2 (s − p) + (s − s x ) 2 + (αs − t x ) 2 + u 21 + α 2 x (sq − αsx−tx1+α −+ 2u x arctan2 sx ) 2 + (αs − t x ) 2 + u 2 x + (αs − t x − q)q(s − p)u x(4.25)