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

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Abstract

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.

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Authors and Affiliations

  1. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics,  , P. O. Box 8009, Beijing, 100088, P. R. China

    Jiming Wu

  2. Department of Mathematics, City University of Hong Kong, Hong Kong

    Weiwei Sun

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  1. Jiming Wu
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  2. Weiwei Sun
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Correspondence to Jiming Wu.

Additional information

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).

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Wu, J., Sun, W. The Superconvergence of the Composite Trapezoidal Rule for Hadamard Finite Part Integrals. Numer. Math. 102, 343–363 (2005). https://doi.org/10.1007/s00211-005-0647-9

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  • DOI: https://doi.org/10.1007/s00211-005-0647-9

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