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Asymptotic error expansions and splitting extrapolation algorithm for two classes of two-dimensional Cauchy principal-value integrals

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APPLIED MATHEMATICS AND COMPUTATION 2019年 357卷 107-118页

作者： Ma, Yanying Huang, Jin Anhui Jianzhu Univ Sch Math & Phys Hefei 230601 Anhui Peoples R China Univ Elect & Sci Technol China Sch Math Sci Chengdu 611731 Sichuan Peoples R China

This paper proposes numerical quadrature rules for two-dimensional Cauchy principalvalue integrals of the forms integral integral(Omega)(x,y)/(x-s)(2) + (y-t)(2) dydx and integral integral(Omega)(x, y)/(x-s)(y-t)dydx. The derivationof these quadrature rules is based on the Euler-Maclaurin error expansion of a modified trapezoidal rule for one-dimensional Cauchy singular integrals. The corresponding error estimations are investigated, and the convergence rates O(h(m)(2 mu) + h(n)(2 mu)) are obtained for the proposed quadrature rules, where h(m) and h(n) are partition sizes in x and y directions, mu is a positive integer determined by integrand. To further improve accuracy, a splitting extrapolation algorithm is developed based on the asymptotic error expansions. Several numerical tests are performed to verify the effectiveness of the proposed methods. (c) 2019 Elsevier Inc. All rights reserved.

关键词： Two-dimensional Cauchy principal-value integral Asymptotic error expansion Quadrature rule splitting extrapolation algorithm

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extrapolation and splitting extrapolation algorithm for Multidimensional Weakly Singular Integral of Product Type

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IEEE ACCESS 2017年 5卷 10506-10514页

作者： Pan, Yubin Huang, Jin Univ Elect Sci & Technol China Sch Math Sci Chengdu 611731 Peoples R China

In this paper, we use mid-rectangular quadrature rules to solve multidimensional weakly singular integrals of product type (MWSIP) and construct corresponding multi-step and multi-parameter asymptotic expansions of errors. The convergence order can turn from O (h(i)(alpha i+1)) to O(h(0)(4)), -1 <= alpha(i) <= 0, h(0) = max(1 <= i <= s) {h(i)} by using extrapolation and splitting extrapolation algorithm. Moreover, we address a posterior error estimate and formulate self-adaptive algorithm. Numerical results illustrate that the method is of high accuracy and efficiency for solving MWSIP.

关键词： Multidimensional weakly singular integral Euler-Maclaurin error asymptotic expansions extrapolation algorithm splitting extrapolation algorithm posterior error estimate

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Mechanical quadrature method and splitting extrapolation algorithm for the boundary integral equation of the axisymmetric anisotropic Darcy's equation

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2017年第1期94卷 135-150页

作者： Li, Hu Huang, Jin Univ Elect Sci & Technol China Sch Math Sci Chengdu Sichuan Peoples R China

In this paper, we consider the numerical solution of the three-dimensional axisymmetric anisotropic Darcy's equation with Dirichlet boundary conditions. By the potential theory and the indirect boundary element methods, we transform the system to be a one-dimensional boundary integral equation. The mechanical quadrature method (MQM) is developed to solve the integral equation. Moreover, the errors of MQM have a multivariate asymptotic expansion with the order of O(h(3)) for all mesh widths h(i). Once discrete equations are solved in parallel, the accuracy of numerical approximations can be improved to be the order of O(h(max)(5)) by splitting extrapolation algorithm. Moreover, a posteriori asymptotic error estimate is derived to construct self-adaptive algorithms.

关键词： Mechanical quadrature method splitting extrapolation algorithm a posteriori estimate axisymmetric Darcy's equation

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High Accuracy Solutions of the Modified Helmholtz Equation

High Accuracy Solutions of the Modified Helmholtz Equation

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International Conference on Information Technology and Intelligent Transportation Systems (ITITS)

作者： Li, Hu Huang, Jin Univ Elect Sci & Technol China Sch Math Sci Chengdu 611731 Sichuan Peoples R China

ISBN: (纸本)9783319387895;9783319387871

We study the numerical solutions for modified Helmholz equation. Based on the potential theory, the problem can be converted into a boundary integral equation. Mechanical quadrature method (MQM) is presented for solving the equation, which possesses high accuracy order O (h(max)(3)) and low computing complexities. Moreover, the multivariate asymptotic error expansion of MQM accompanied with O(h(i)(3)) for all mesh widths hi is got. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at least O(h(max)(5)) by splitting extrapolation algorithm (SEA). The numerical examples support our theoretical analysis.

关键词： Mechanical quadrature method splitting extrapolation algorithm Modified Helmholtz equation

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splitting extrapolation algorithms for solving the boundary integral equations of anisotropic Darcy's equation on polygons by mechanical quadrature methods

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NUMERICAL algorithmS 2013年第1期62卷 27-43页

作者： Luo, Xin Huang, Jin Wang, Chuan-Long Univ Elect Sci & Technol China Sch Math Sci Chengdu 611731 Peoples R China Taiyuan Normal Univ Dept Math Taiyuan 030012 Peoples R China

In this paper we study the stability and convergence of the solution for the first kind integral equations of the anisotropic Darcy's equations by the mechanical quadrature methods on closed polygonal boundaries in a"e(2). Using the collectively compact theory, we construct numerical solutions which converge with the order , where is the mesh size. In addition, An a posteriori asymptotic error representation is derived by splitting extrapolation methods in order to construct self-adaptive algorithms, and the convergence rate can be achieved after using the splitting extrapolation methods once. Finally, the numerical examples show the efficiency of our methods.

关键词： Mechanical quadrature methods Anisotropic Darcy's equation splitting extrapolation algorithm A posteriori estimate

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