The superconvergence of the composite midpoint rule for the finite-part integral

AbstractThe composite midpoint rule is probably the simplest one among the Newton–Cotes rules for Riemann integral. However, this rule is divergent in general for Hadamard finite-part integral. In this paper, we turn this rule to a useful one and, apply it to evaluate Hadamard finite-part integral as well as to solve the relevant integral equation. The key point is based on the investigation of its pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate of the midpoint rule is higher than what is globally possible. We show that the superconvergence rate of the composite midpoint rule occurs at the midpoint of each subinterval and obtain the corresponding superconvergence error estimate. By applying the midpoint rule to approximate the finite-part integral and by choosing the superconvergence points as the collocation points, we obtain a collocation scheme for solving the finite-part integral equation. More interesting is that the inverse of the coefficient matrix of the resulting linear system has an explicit expression, by which an optimal error estimate is established. Some numerical examples are provided to validate the theoretical analysis

On a novel numerical quadrature based on cycle index of symmetric group for the Hadamard finite-part integrals

To evaluate the Hadamard finite-part integrals accurately, a novel interpolatory-type quadrature is proposed in this article. In our approach, numerical divided difference is utilized to represent the high order derivatives of the integrated function, which make it possible to reduced the numerical quadrature into a concise formula based on the cycle index for symmetric group. In addition, convergence analysis is presented and the error estimation is given. Numerical results are presented on cases with different weight functions, which substantiate the performance of the proposed method

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

Numerical methods to compute hypersingular integral in boundary element methods

超奇异积分的近似计算是边界元方法,特别是自然边界元理论中必须面对的难题之一.经典的数值方法,如gAuSS求积公式和nEWTOn-COTES积分公式等数值方法,都不能直接用于超奇异积分的近似计算.本文将介绍超奇异积分基于不同定义的gAuSS积分公式、S型变换公式、nEWTOn-COTES积分公式和外推法近似计算超奇异积分的思路,重点阐述nEWTOn-COTES积分公式和基于有限部分积分定义的外推法近似计算超奇异积分的主要结论.The computation of hypersingular integral is one of the important subjects in boundary element methods especially in natural boundary element methods.Classical numerical methods such as Gauss methods,Newton-Cotes methods cannot be used to approximate the hypersingular integral directly.In this paper, we introduce the numerical methods such as Gauss methods, Newton-Cotes methods, S transformation methods and extrapolation methods which are based on different definitions; then we mainly present the results of Newton-Cotes methods and extrapolation methods which are used to compute the hypersingular integral.国家自然科学基金(批准号:11471195;11101247;11201209和91330106); 中国博士后科学基金(批准号:2013M540541)资助项

The Trapezoidal Rule for Computing Cauchy Principal Value Integral on Circle

The composite trapezoidal rule for the computation of Cauchy principal value integral with the singular kernel cot((x-s)/2) is discussed. Our study is based on the investigation of the pointwise superconvergence phenomenon; that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis

Numerical method for hypersingular integrals of highly oscillatory functions on the positive semiaxis

This paper deals with a quadrature rule for the numerical evaluation of hypersingular integrals of highly oscillatory functions on the positive semiaxis. The rule is of product type and consists in approximating the density function f by a truncated interpolation process based on the zeros of generalized Laguerre polynomials and an additional point. We prove the stability and the convergence of the rule, giving error estimates for functions belonging to weighted Sobolev spaces equipped with uniform norm. We also show how the proposed rule can be used for the numerical solution of hypersingular integral equations. Numerical tests which confirm the theoretical estimates and comparisons with other existing quadrature rules are presented