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Fast and accurate solvers for weakly singular integral equations

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Abstract

Consider an integral equation \(\lambda \; u - T u = f\), where T is an integral operator, defined on C[0, 1],  with a kernel having an algebraic or a logarithmic singularity. Let \(\pi _m\) denote an interpolatory projection onto a space of piecewise polynomials of degree \(\le r - 1\) with respect to a graded partition of [0, 1] consisting of m subintervals. In the product integration method, an approximate solution is obtained by solving \(\lambda \; u_m - T \pi _m u_m = f.\) As in order to achieve a desired accuracy, one may have to choose m large, we find approximations of \(u_m\) using a discrete modified projection method and its iterative version. We define a two-grid iteration scheme based on this method and show that it needs less number of iterates than the two-grid iteration scheme associated with the discrete collocation method. Numerical results are given which validate the theoretical results.

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Funding

The third author was partially supported by CMUP, which is financed by national funds through FCT - Fundação para a Ciência e Tecnologia, I.P., under the project with reference UIDB/00144/2020.

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Authors and Affiliations

  1. Université de Lyon, Institut Camille Jordan, UMR5208, 23, rue du Dr Paul Michelon, Saint-Étienne Cedex 2, 42023, France

    Laurence Grammont

  2. Department of Mathematics, I.I.T. Bombay, Powai, Mumbai, 400076, India

    Rekha P. Kulkarni

  3. CMUP, Faculdade de Economia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-464, Porto, Portugal

    Paulo B. Vasconcelos

Authors
  1. Laurence Grammont
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  2. Rekha P. Kulkarni
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  3. Paulo B. Vasconcelos
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Correspondence to Paulo B. Vasconcelos.

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Grammont, L., Kulkarni, R.P. & Vasconcelos, P.B. Fast and accurate solvers for weakly singular integral equations. Numer Algor 92, 2045–2070 (2023). https://doi.org/10.1007/s11075-022-01376-x

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  • DOI: https://doi.org/10.1007/s11075-022-01376-x

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