The modified trapezoidal rule for computing hypersingular integral on interval

The modified trapezoidal rule for the computation of hypersingular integrals in boundary element methods is discussed. when the special function of the error functional equals zero,the convergence rate is one order higher than the general case. a new quadrature rule is presented and the asymptotic expansion of error function is obtained. based on the error expansion,not only do we obtain a high order of accuracy,but also a posteriori error estimate is conveniently derived. some numerical results are also reported to confirm the theoretical results and show the efficiency of the algorithms. © 2013 jin li and xiuzhen li.