A Fast Two-Grid Iteration for Integral Equations

Document Type : Original Article

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Mathematics and Engineering Physics Department, Engineering Faculty, Mansoura University, Mansoura 35516, Egypt

10.21608/jctae.2025.372264.1049

Abstract

In this paper, we introduce a novel two-level iterative approach for Nyström's method, aimed at the efficient solution of large systems of equations that arise from the discretization of Fredholm integral equations of the second kind. The developed two-level algorithm exhibits a computational cost advantage, demonstrating faster convergence than the Atkinson-Brakhage iterative method and providing convergence for a wider range of parameters, thus enhancing its stability and accuracy. Moreover, a derivation of the convergence of this new method is presented, providing a theoretical foundation for its use and guaranteeing its mathematical soundness. Illustrative numerical examples are included to showcase the method's effectiveness and its superior performance in comparison to existing methods. The practical efficacy of the algorithm is demonstrated through a detailed comparative analysis with the Atkinson-Brakhage method, with a focus on its improved computational efficiency, broader applicability, and reduced sensitivity to parameter selection. This new method leverages a two-stage iterative process where the first level provides a good initial guess for the second, leading to accelerated convergence. The analysis includes a rigorous examination of the spectral properties of the iteration matrix to establish the wider convergence range. The results indicate that this two-level approach provides a significant advancement in the numerical solution of Fredholm integral equations.

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