Quadrature of integrands with a logarithmic singularity

Abstract

A quadrature rule is presented that is exact for integrands of the form π ( ξ ) + ϖ ( ξ ) log ⁡ ξ \pi (\xi ) + \varpi (\xi )\log \xi on the interval (0, 1), where π \pi and ϖ \varpi are polynomials. The computed weights and abscissae are given for one- through seven-point rules. In particular, the four-point rule is exact for integral operators with log-arithmically singular kernel on a cubic B-spline basis, and it is expected these results shall prove useful for numerical applications of weighted-residual finite element methods.