Superconvergence of Hermite rule for third order hypersingular integrals on interval

Abstract

The boundary element method has been widely applied to a lot of practical problems,such as fluid mechanics and fracture mechanics.As one of the important topics in boundary element method, the numerical calculation of hypersingular integrals is of great importance.This article deals with the composite Hermite rule of the third order hypersingular integrals .Based on the error expansion , the superconvergence result of the composite Hermite formula is obtained .We show that the convergence rate is $O(h^{3})$ when the local coordinate of the singular point $\tau=0$,which is one order higher than the global convergence.The accuracy of the result is verified by several numerical examples.