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Superconvergence of Newton–Cotes rule for computing hypersingular integral on a circle

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Abstract

In this paper, the composite Newton–Cotes rule for evaluating hypersingular integral on a circle is investigated, we focus on the emphasis of the pointwise superconvergence. Taking the density function approximated, with the hypersingular kernel calculated analysis, we obtain the main error estimation of the general Newton–Cotes rule. We have proved that the property of the special function have big influence on the numerical results. We also present the relationship between the order of singularity and order of polynomial interpolation: if the hypersingular integral is \(p+1\) order, the order of polynomial interpolation must be \(p-1\) at least. The modified quadrature rules are proposed to improve the convergence rate which can reach the superconvergence rate. At last, we also presented some numerical examples to test our theoretical analysis.

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References

  • Ainsworth M, Guo B (2000) An additive Schwarz preconditioner for p-version boundary element approximation of the hypersingular operator in three dimensions. Numer Math 85:343–366

    Article  MathSciNet  MATH  Google Scholar 

  • Andrews LC (1992) Special functions of mathematics for engineers, 2nd edn. McGraw-Hill, London

  • Bao G, Sun W (2005) A fast algorithm for the electromagnetic scattering from a cavity. SIAM J Sci Comput 27:553–574

    Article  MathSciNet  MATH  Google Scholar 

  • Carley M (2007) Numerical quadratures for singular and hyper singular integrals in boundary element methods. SIAM J Sci Comput 29:1207–1216

    Article  MathSciNet  MATH  Google Scholar 

  • Choi UJ, Kim SW, Yun BI (2004) Improvement of the asymptotic behaviour of the Euler-Maclaurin formula for Cauchy principal value and Hadamard finite-part integrals. Int J Numer Methods Eng 61:496–513

    Article  MathSciNet  MATH  Google Scholar 

  • Du QK (2001) Evaluations of certain hypersingular integrals on interval. Int J Numer Methods Eng 51:1195–1210

    Article  MathSciNet  MATH  Google Scholar 

  • Elliott D, Venturino E (1997) Sigmoidal transformations and the Euler–Maclaurin expansion for evaluating certain Hadamard finite-part integrals. Numer Math 77:453–465

    Article  MathSciNet  MATH  Google Scholar 

  • Frangi A, Bonnet M (1995) Singular boundary elements for the analysis of cracks in plain strain. Int J Numer Methods Eng 38:2389–2411

    Article  Google Scholar 

  • Frangi A, Bonnet M (1998) A Galerkin symmetric and direct BIE method for Kirchhoff elastic plates: formulation and implementation. Int J Numer Methods Eng 41:317–326

    Article  MathSciNet  MATH  Google Scholar 

  • Frangi A, Bonnet M (2000) A direct approach for boundary integral equations with high-order singularities. Int J Numer Methods Eng 49:871–898

    Article  MathSciNet  MATH  Google Scholar 

  • Hasegawa T (2004) Uniform approximations to finite Hilbert transform and its derivative. J Comput Appl Math 163:127–138

    Article  MathSciNet  MATH  Google Scholar 

  • Hui CW, Shia D (1999) Evaluations of hypersingular integrals using Gaussian quadrature. Int J Numer Methods Eng 44:205–214

    Article  MathSciNet  MATH  Google Scholar 

  • Ioakimidis NI (1985) On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives. Math Comput 44:191–198

    Article  MathSciNet  MATH  Google Scholar 

  • Karami G, Derakhshan D (1999) An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equation analysis. Eng Anal Bound Elem 23:317–326

    Article  MATH  Google Scholar 

  • Kim P, Jin UC (2003) Two trigonometric quadrature formulae for evaluating hypersingular integrals. Int J Numer Methods Eng 56:469–486

    Article  MathSciNet  MATH  Google Scholar 

  • Kolm P, Rokhlin V (2001) Numerical quadratures for singular and hyper singular integrals. Comput Math Appl 41:327–352

    Article  MathSciNet  MATH  Google Scholar 

  • Kress R (1995) On the numerical solution of a hypersingular integral equation in scattering theory. J Comput Appl Math 61:345–360

    Article  MathSciNet  MATH  Google Scholar 

  • Li J (2017) The extrapolation methods based on Simpson’s rule for computing supersingular intergral on interval. Appl Math Comput 310:204–214

    MathSciNet  MATH  Google Scholar 

  • Li J, Rui HX (2017) Error expansion of trapezoidal rule for certain two-dimensional Cauchy principal value integrals. Comput Math Appl 74(10):2608–2637

    Article  MathSciNet  MATH  Google Scholar 

  • Li J, Wang ZQ (2018) Simpsons rule to approximate the Hilbert integral and its application. Appl Math Comput 339:398–409

    MathSciNet  MATH  Google Scholar 

  • Li J, Yu DH (2011) The error estimate of Newton–Cotes methods to compute hypersingular integral. Math Numer Sin 31(1):77–87

    MathSciNet  MATH  Google Scholar 

  • Li J, Wu JM, Yu DH (2009) Generalized extrapolation for computation of hypersingular integrals in boundary element methods. Comput Model Eng Sci 42:151–175

    MathSciNet  MATH  Google Scholar 

  • Li J, Zhang XP, Yu DH (2013) Extrapolation methods to compute hypersingular integral in boundary element methods. Sci China Math 56(8):1647–1660

    Article  MathSciNet  MATH  Google Scholar 

  • Li J, Rui HX, Yu DH (2016) Trapezoidal rule for computing supersingular integral on a circle. J Sci Comput 219(4):1616–1624

    MathSciNet  MATH  Google Scholar 

  • Linz P (1985) On the approximate computation of certain strongly singular integrals. Computing 35:345–353

    Article  MathSciNet  MATH  Google Scholar 

  • Li J, Rui HX (2019) Extrapolation methods for computing Hadamard finite-part integral on interval. J Comput Math 37(2):261–277

  • Li J, Rui HX, Yu DH (2014) Composite Simpsons rule for computing supersingular integral on circle. CMES 97(6):463–481

  • Monegato G (1994) Numerical evaluation of hypersingular integrals. J Comput Appl Math 50:9–31

    Article  MathSciNet  MATH  Google Scholar 

  • Monegato G (2009) Definitions, properties and applications of finite-part integrals. J Comput Appl Math 229:425–439

    Article  MathSciNet  MATH  Google Scholar 

  • Moore MNJ, Gray LJ, Kaplan T (2007) Evaluation of supersingular integrals: second-order boundary derivatives. Int J Numer Methods Eng 69:1930–1947

    Article  MathSciNet  MATH  Google Scholar 

  • Sidi A (1989) Comparison of some numerical quadrature formulas for weakly singular periodic Fredholm integral equations. Computing 43:159–170

    Article  MathSciNet  MATH  Google Scholar 

  • Sidi A (2013) Compact numerical quadrature formulas for hypersingular integrals and integral equations. J Sci Comput 54:145–176

    Article  MathSciNet  MATH  Google Scholar 

  • Sidi A (2014) Analysis of errors in some recent numerical quadrature formulas for periodic singular and hypersingular integrals via regularization. Appl Numer Math 81:30–39

    Article  MathSciNet  MATH  Google Scholar 

  • Sidi A (2014) Richardson extrapolation on some recent numerical quadrature formulas for singular and hypersingular integrals and its study of stability. J Sci Comput 60:141–159

    Article  MathSciNet  MATH  Google Scholar 

  • Sidi A, Israeli M (1986) Quadrature methods for periodic singular and weakly singular Fredholm integral equations. J Sci Comput 3:201–231 (Originally appeared as Technical Report No. 384, Computer Science Dept., Technion. CIsrael Institute of Technology, (1985), and also as ICASE Report No. 86-50)

  • Sun WW, Wu JM (2005) Newton-Cotes formulae for the numerical evaluation of certain hypersingular integral. Computing 75:297–309

    Article  MathSciNet  MATH  Google Scholar 

  • Sun WW, Wu JM (2008) Interpolatory quadrature rules for Hadamard finite-part integrals and the superconvergence. IMA J Numer Anal 28:580–597

    Article  MathSciNet  MATH  Google Scholar 

  • Vainikko GM, Lifanov IK (2002) On the notion of the finite part of divergent integrals in integral equations. Differ Equ 38:1313–1326

    Article  MathSciNet  MATH  Google Scholar 

  • Wu JM, Lü Y (2005) A superconvergence result for the second order Newton–Cotes formula for certain finite part integrals. IMA J Numer Anal 25:253–263

    Article  MathSciNet  MATH  Google Scholar 

  • Wu JM, Sun WW (2005) The superconvergence of the composite trapezoidal rule for Hadamard finite part integrals. Numer Math 102:343–363

    Article  MathSciNet  MATH  Google Scholar 

  • Wu JM, Sun WW (2008) The superconvergence of Newton–Cotes rules for the Hadamard finite-part integral on an interval. Numer Math 109:143–165

    Article  MathSciNet  MATH  Google Scholar 

  • Wu JM, Yu DH (1999) The approximate computation of hypersingular integrals on interval. Chin J Numer Math Appl 21:25–33

    MathSciNet  Google Scholar 

  • Xia MH, Li J (2018) Extrapolation method for Cauchy principal value integral with classical rectangle rule on interval. CMES 115(3):313–326

    Google Scholar 

  • Yu DH (2002) Natural boundary integrals method and its applications. Kluwer Academic (2002)

  • Yu DH (1992) The approximate computation of hypersingular integrals on interval. Numer Math J Chin Univ (English Ser) 1:114–127

    MathSciNet  MATH  Google Scholar 

  • Zhang XP, Wu JM, Yu DH (2009) Superconvergence of the composite Simpson’s rule for a certain finite-part integral and its applications. J Comput Appl Math 223:598–613

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang XP, Wu JM, Yu DH (2009) The superconvergence of the composite Newton–Cotes rules for Hadamard finite-part integral on a circle. Computing 85:219–244

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang XP, Wu JM, Yu DH (2010) The superconvergence of the composite trapezoidal rule for Hadamard finite-part integral on a circle and its application. Int J Comput Math 87(4):855–876

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work of Jin Li was supported by Shandong Provincial Natural Science Foundation of China (Grant No. ZR2016JL006), Natural Science Foundation of Hebei Province (Grant No. A2019209533) and National Natural Science Foundation of China (Grant No.11771398). The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.

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  1. School of Science, Shandong Jianzhu University, Jinan, 250101, People’s Republic of China

    Jin Li & Yongling Cheng

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  1. Jin Li
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  2. Yongling Cheng
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Correspondence to Jin Li.

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Communicated by Armin Iske.

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Li, J., Cheng, Y. Superconvergence of Newton–Cotes rule for computing hypersingular integral on a circle. Comp. Appl. Math. 41, 269 (2022). https://doi.org/10.1007/s40314-022-01951-x

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