Abstract
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.
Highlights
Consider the following integral: I (f, s) := b ∫=a f (x) (x − s)p+1 dx, s ∈ (a, b), p = 1, 2, (1)where ∫=ba denotes a Hadamard finite-part integral (p = 1 is called hypersingular integral and p = 2 is called supersingular integral) and s is the singular point
One of the major problems arising from boundary element method, for solving such integral equations, is how to evaluate the hypersingular integrals on the interval or on the circle efficiently
In 1983, the series expansion of hypersingular integral kernel on circle was firstly suggested by Yu [19]
Summary
One of the major problems arising from boundary element method, for solving such integral equations, is how to evaluate the hypersingular integrals on the interval or on the circle efficiently. In 1983, the series expansion of hypersingular integral kernel on circle was firstly suggested by Yu [19] He solved the harmonic and biharmonic natural boundary integral equations successfully. The Newton-Cotes methods to compute the hypersingular integral on interval were firstly studied by Linz [20] with generalized trapezoidal and Simpson rules which fail altogether when the singular point s is close to a mesh point. Yu [21] gave new quadrature formulae to compute the case of singular point coinciding with the mesh point which presented that the error estimate is O(h| ln h|). For the case of singular point coinciding with the mesh point a new quadrature rule is introduced. Several numerical examples are given to validate our analysis
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
