Splitting extrapolation algorithm for first kind boundary integral equations with singularities by mechanical quadrature methods

Abstract

The accuracy of numerical solutions near singular points is crucial for numerical methods. In this paper we develop an efficient mechanical quadrature method (MQM) with high accuracy. The following advantages of MQM show that it is very promising and beneficial for practical applications: (1) the $ O(h_{\rm {max}}^{3})$ convergence rate; (2) the $O(h_{\rm {max}}^{5})$ convergence rate after splitting extrapolation; (3) Cond = $O(h_{\rm {min}}^{-1})$ ; (4) the explicit discrete matrix entries. In this paper, the above theoretical results are briefly addressed and then verified by numerical experiments. The solutions of MQM are more accurate than those of other methods. Note that for the discontinuous model in Li et al. (Eng Anal Bound Elem 29:59---75, 2005), the highly accurate solutions of MQM may even compete with those of the collocation Trefftz method.