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Oscillation-preserving Legendre-Galerkin methods for second kind integral equations with highly oscillatory kernels

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Abstract

The original solutions of highly oscillatory integral equations usually have rapid oscillation, which means that conventional numerical approaches used to solve these equations have poor convergence. In order to overcome this difficulty, in this paper, we propose and analyze an oscillation-preserving Legendre-Galerkin method for second kind integral equations with highly oscillatory kernels. Concretely, we first incorporate the standard approximation subspace of Legendre polynomial basis with a finite number of oscillatory functions which capture the oscillation of the exact solutions. Then, we construct an efficient numerical integration scheme, yielding a fully discrete linear system. Making use of best approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces and the compactness operator theory, we establish that the fully discrete approximate equation has a unique solution and the approximate solution reaches an optimal convergence order without the influence of the wave number. In addition, we prove that for sufficiently large wave number, the spectral condition number of the corresponding linear system is uniformly bounded. At last, we use numerical examples to demonstrate the effectiveness of the proposed method.

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Funding

The research of this author is supported by the National Natural Science Foundation of China (12171278, 11971259) and by National Science Foundation of Shandong Province (ZR2020MA047).

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  1. School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, 250014, China

    Haotao Cai

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  1. Haotao Cai
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Correspondence to Haotao Cai.

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Cai, H. Oscillation-preserving Legendre-Galerkin methods for second kind integral equations with highly oscillatory kernels. Numer Algor 90, 1091–1115 (2022). https://doi.org/10.1007/s11075-021-01223-5

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  • DOI: https://doi.org/10.1007/s11075-021-01223-5

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