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(h3) when the
local coordinate of the singular point is ? = 0, which is one order higher
than the global convergence. The accuracy of the result is verified by
several numerical examples.