THE TRAPEZOIDAL RULE WITH A NONLINEAR COORDINATE TRANSFORMATION FOR WEAKLY SINGULAR INTEGRALS Yun, Beong-In;
It is well known that the application of the nonlinear coordinate transformations is useful for efficient numerical evaluation of weakly singular integrals. In this paper, we consider the trapezoidal rule combined with a nonlinear transformation (b;), containing a parameter b, proposed first by Yun . It is shown that the trapezoidal rule with the transformation (b;), like the case of the Gauss-Legendre quadrature rule, can improve the asymptotic truncation error by using a moderately large b. By several examples, we compare the numerical results of the present method with those of some existing methods. This shows the superiority of the transformation (b;).TEX>).
M. Cerrolaza and E. Alarcon, A bi-cubic transformation for the numerical evaluation of the Cauchy principal value integrals in boundary elements, Internat. J. Numer. Methods Engrg. 28 (1989), 987–999.
M. Doblare and L. Gracia, On non-linear transformations for the integration of weakly-singular and Cauchy principal value integrals, Internat. J. Numer. Methods Engrg. 40 (1997), 3325–3358.
D. Elliott, The cruciform crack problem and sigmoidal transformations, Math. Methods Appl. Sci. 20 (1997), 121–132.
D. Elliott, Sigmoidal transformations and the trapezoidal rule, J. Aust. Math. Soc. Ser. B 40(E) (1998), E77–E137.
P. R. Johnston, Application of sigmoidal transformations to weakly singular and neer singular boundary element integrals, Internat. J. Numer. Methods Engrg. 45 (1999), 1333–1348.
P. R. Johnston, Semi-sigmoidal transformations for evaluating weakly singular boundary element integrals, Internat. J. Numer. Methods Engrg. 47 (2000), 1709–1730.
P. R. Johnston and D. Elliott, Error estimation of quadrature rules for evaluating singular integrals in boundary element, Internat. J. Numer. Methods Engrg. 48 (2000), 949–962.
N. M. Korobev, Number –Theoretic Method of Approximate Analysis, GIFL, Moscow, 1963.
J. M. Sanz Serna, M. Doblare, and E. Alarcon, Remarks on methods for the computation of boundary-element integrals by co-ordinate transformation, Commun. Appl. Numer. Methods 6 (1990), 121–123.
M. Sato, S. Yoshiyoka, K. Tsukui and R. Yuuki, Accurate numerical integration of singular kernels in the two-dimensional boundary element method, in: C.A. Brebbia(ed.), Boundary Elements X, 1 (1988), Springer, Berlin, 279–296.
A. Sidi, A new variable transformation for numerical integration, in: H. Brass and G. Hammerlin(ed.), numerical Integration IV, ISNM Vol. 112, Birkhaiser-Verlag, Berlin, 1993, 359–373.
K. M. Singh and M. Tanaka, On non–linear transformations for accurate numerical evaluation of weakly singular boundary integrals, Internat. J. Numer. Methods Engrg. 50 (2001), 2007–2030.
J. C. F. Telles, A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals, Internat. J. Numer. Methods Engrg. 24 (1987), 959–973.
B. I. Yun, An efficient transformation with Gauss quadrature rule for weakly singular integrals, Comm. Numer. Methods Eng. 17 (2001), 881–891.
B. I. Yun and P. Kim, A new sigmoidal transformation for weakly singular integrals in the boundary element method, SIAM J. Sci. Comput. 24 (2003), 1203–1217.