A High-Order Piecewise Quartic Spline Rule for Hadamard Integral and Its Application of the Cavity Scattering

We develop a fourth-order piecewise quartic spline rule for Hadamard integral. The quadrature formula of Hadamard integral is obtained by replacing the integrand function with the piecewise quartic spline interpolation function. We establish corresponding error estimates and analyze the numerical stability. The rule can achieve fourth-order convergence at any point in the interval, even when the singular point coincides with the grid point. Since the derivative information of the integrand is not required, the rule can be easily applied to solve many practical problems. Finally, the quadrature formula is applied to solve the electromagnetic scattering from cavities with different wave numbers, which improves the whole accuracy of the solution. Numerical experiments are presented to show the efficiency and accuracy of the theoretical analysis.     

Extrapolation methods to compute hypersingular integral in boundary element methods

The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel 1/sin 2(x-s) is discussed,and the main part of the asymptotic expansion of error function is obtained.Based on the main part of the asymptotic expansion,a series is constructed to approach the singular point.An extrapolation algorithm is presented and the convergence rate is proved.Some numerical results are also presented to confirm the theoretical results and show the efficiency of the algorithms.     

A sinc quadrature rule for Hadamard finite-part integrals

Summary A Sinc quadrature rule is presented for the evaluation of Hadamard finite-part integrals of analytic functions. Integration over a general are in the complex plane is considered. Special treatment is given to integrals over the interval (–1,1). Theoretical error estimates are derived and numerical examples are included.     

Superconvergence and ultraconvergence of Newton-Cotes rules for supersingular integrals

In this article, the general (composite) Newton-Cotes rules for evaluating Hadamard finite-part integrals with third-order singularity (which is also called “supersingular integrals”) are investigated and the emphasis is placed on their pointwise superconvergence and ultraconvergence. The main error of the general Newton-Cotes rules is derived, which is shown to be determined by a certain function . Based on the error expansion, the corresponding modified quadrature rules are also proposed. At last, some numerical experiments are carried out to validate the theoretical analysis.     

Integration of singular Galerkin-type boundary element integrals for 3D elasticity problems

Summary. A Galerkin approximation of both strongly and hypersingular boundary integral equation (BIE) is considered for the solution of a mixed boundary value problem in 3D elasticity leading to a symmetric system of linear equations. The evaluation of Cauchy principal values (v. p.) and finite parts (p. f.) of double integrals is one of the most difficult parts within the implementation of such boundary element methods (BEMs). A new integration method, which is strictly derived for the cases of coincident elements as well as edge-adjacent and vertex-adjacent elements, leads to explicitly given regular integrand functions which can be integrated by the standard Gauss-Legendre and Gauss-Jacobi quadrature rules. Problems of a wide range of integral kernels on curved surfaces can be treated by this integration method. We give estimates of the quadrature errors of the singular four-dimensional integrals. Received June 25, 1995 / Revised version received January 29, 1996     

The numerical evaluation of Hadamard finite-part integrals

Summary A quadrature rule is described for the numerical evaluation of Hadamard finite-part integrals with a double pole singularity within the range of integration. The rule is based upon the observation that such an integral is the derivative of a Cauchy principal value integral.     

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

The 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.     

Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations

The present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.     

The Superconvergence of the Composite Trapezoidal Rule for Hadamard Finite Part Integrals

The composite trapezoidal rule has been well studied and widely applied for numerical integrations and numerical solution of integral equations with smooth or weakly singular kernels. However, this quadrature rule has been less employed for Hadamard finite part integrals due to the fact that its global convergence rate for Hadamard finite part integrals with (p+1)-order singularity is p-order lower than that for the Riemann integrals in general. In this paper, we study the superconvergence of the composite trapezoidal rule for Hadamard finite part integrals with the second-order and the third-order singularity, respectively. We obtain superconvergence estimates at some special points and prove the uniqueness of the superconvergence points. Numerical experiments confirm our theoretical analysis and show that the composite trapezoidal rule is efficient for Hadamard finite part integrals by noting the superconvergence phenomenon. The work of this author was partially supported by the National Natural Science Foundation of China(No.10271019), a grant from the Research Grants Council of the Hong Kong Special Administractive Region, China (Project No. City 102204) and a grant from the Laboratory of Computational Physics The work of this author was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 102204).     

Superconvergence of the composite Simpson’s rule for a certain finite-part integral and its applications

The superconvergence phenomenon of the composite Simpson’s rule for the finite-part integral with a third-order singularity is studied. The superconvergence points are located and the superconvergence estimate is obtained. Some applications of the superconvergence result, including the evaluation of the finite-part integrals and the solution of a certain finite-part integral equation, are also discussed and two algorithms are suggested. Numerical experiments are presented to confirm the superconvergence analysis and to show the efficiency of the algorithms.     

THE DIFFERENTIAL INTEGRAL EQUATIONS ON SMOOTH CLOSED ORIENTABLE MANIFOLDS

1 IntroductionSillce tl1e limit value fOrlnula, viz. tl1e Plemelj fOrn1ula, of the Cauthe type integraJ withBochner-Martinelli kernel was proved in 1957[1], it has beell successfully used to the study Ofsingular i1ltegral equatious, solvi11g the 0b--equation, holomorphic extension, 0--closed exten-sion and C-R 111al1ifolds[2-51. Evideutly, the researcl1 of higher order singular integrals withBochuer-Martinelli kerllel itself also l1as important significallce. In 1952, J. Hadanmrd firstde…     

Numerical calculation of singular integrals appearing in three-dimensional potential flow problems

Two approximate methods for calculating singular integrals appearing in the numerical solution of three-dimensional potential flow problems are presented. The first method is a self-adaptive, fully numerical method based on special copy formulas of Gaussian quadrature rules. The singularity is treated by refining the partitions of the copy formula in the vicinity of the singular point. The second method is a semianalytic method based on asymptotic considerations. Under the small curvature hypothesis, asymptotic expansions are derived for the integrals that are involved in the calculation of the scalar potential, the velocity as well as the deformation field induced from curved quadrilateral surface elements. Compared to other methods, the proposed integration schemes, when applied to practical flow field calculations, require less computational effort.     

Numerical evaluation of cauchy principal values of integrals

Gaussian quadrature rules for the evaluation of Cauchy principal values of integrals are considered, their relation with Gauss-Legendre formulas is studied, and they are compared with other rules.     

A note on spline approximation for Hadamard finite-part integrals

In this paper we construct product quadrature rules, based on spline interpolation, for the numerical evaluation of singular integrals in the sense of Hadamard. We give a convergence result and examine the behaviour of the stability factor. We also present some numerical tests.     

An asymptotic analysis of two algorithms for certain Hadamard finite-part integrals

Two algorithms for the approximate evaluation of certain Hadamardfinite-part integrals, which date back to Grünwald (1867),have been analysed. It is shown that the quadrature sums areclosely related to the finite-part integrals of the Bernsteinpolynomials. The stability of the algorithms is considered andtwo numerical examples are given.     

基于切比雪夫节点的一类带显式系数的广义高斯求积公式(英文)

The goal here is to give a simple approach to a quadrature formula based on the divided diffierences of the integrand at the zeros of the nth Chebyshev polynomial of the first kind,and those of the（n-1）st Chebyshev polynomial of the second kind.Explicit expressions for the corresponding coefficients of the quadrature rule are also found after expansions of the divided diffierences,which was proposed in[14].     

Error estimate and extrapolation of a quadrature formula derived from a quartic spline quasi-interpolant

In this paper we analyze a quadrature rule based on integrating a C ³ quartic spline quasi-interpolant on a bounded interval which has been introduced in Sablonnière (Rend. Semin. Mat. Univ. Pol. Torino 63(3):107–118, 2005). By studying the sign structure of its associated Peano kernel we derive an explicit formula of the quadrature error with an approximation order O(h ⁶). A comparison of this rule with the composite Boole’s and the three-point Gauss-Legendre rules is given. We also compare the Nyström methods associated with the above quadrature formulae for solving the linear Fredholm integral equation of the second kind. Then, by combining the proposed rule with composite Boole’s rule, we construct a new quadrature rule of order O(h ⁷). All the obtained results are illustrated by several numerical tests.     

求解时间分布阶扩散方程的两个高阶有限差分格式

基于复化Simpson公式和复化两点Gauss-Legendre公式，构造了两个求解时间分布阶扩散方程的高阶有限差分格式.不同于以往文献中提出的时间一阶或二阶格式，这两种格式在时间方向都具有三阶精度，而在分布阶和空间方向可达到四阶精度.数值结果表明，两种算法都是稳定且收敛的，从而是有效的.两种格式的收敛速率也通过数值实验进行了验证，并且通过和文献中的算法对比可以得出其更为高效,     

A superconvergence result for the second-order Newton-Cotes formula for certain finite-part integrals

In this paper we investigate the superconvergence phenomenonof the second-order quadrature formula of Newton–Cotestype for the computation of finite-part integrals with a second-ordersingularity on an interval. Superconvergence points are foundand a superconvergence estimate is obtained. The validity ofthe theoretical result is demonstrated by numerical experiments.     

Estimates of the error in Gauss-Legendre quadrature for double integrals

Error estimates are a very important aspect of numerical integration. It is desirable to know what level of truncation error might be expected for a given number of integration points. Here, we determine estimates for the truncation error when Gauss-Legendre quadrature is applied to the numerical evaluation of two dimensional integrals which arise in the boundary element method. Two examples are considered; one where the integrand contains poles, when its definition is extended into the complex plane, and another which contains branch points. In both cases we obtain error estimates which agree with the actual error to at least one significant digit.