A new sigmoidal transformation for weakly singular integrals in the boundary element method

The power of the sigmoidal transformation in weakly singular integrals has been demonstrated by the recent works [A. Sidi, in Numerical Integration IV, H. Brass and G. Hämmerlin, eds., Birkhäuser-Verlag, Berlin, 1993, pp. 359-373; P. R. Johnston, Internat. J. Numer. Methods Engrg., 45 (1999), pp. 1333-1348; P. R. Johnston, Internat. J. Numer. Methods Engrg., 47 (2000), pp. 1709-1730; D. Elliott, Math. Methods Appl. Sci., 20 (1997), pp. 121-132; D. Elliott, J. Austral. Math. Soc. Ser. B, 40 (1998), pp. E77-E137]. Especially, application of this transformation is useful for efficient numerical evaluation of the singular integrals appearing in the usual boundary element method. In this paper, a new sigmoidal transformation containing a parameter b is presented. It is shown that the present transformation, with the Gauss-Legendre quadrature rule, can improve the asymptotic truncation error of the traditional sigmoidal transformations by controlling the parameter. For some examples, we compare the numerical results of the present method with those of the well-known Sidi- and Elliott-transformations to show the superiority of the former.