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Multilevel augmentation methods for eigen-problems of compact integral operators

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Abstract

In this paper, we combine the ideas of multilevel augmentation methods for solving integral equations and the shifted-inverse power method to develop a new multilevel augmentation method for solving eigen-problem of compact integral operators with smooth kernels. We first solve an eigen-problem in a suitable initial coarse level and then seek a more accurate approximation from solving a linear system on a finer mesh. Moreover, we need only to deal with a small linear system corresponding to the initial coarse level when we solve the linear system on a finer mesh. The method exhibits to be convenient for implementing adaptivity, and reduces the computational cost greatly and leads to the method faster. Numerical examples are presented to illustrate the theoretical estimates for the error of these methods.

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Funding

This study is supported in part by the Natural Science Foundation of Guangxi in China under grant AD20238065, the key project of Guangxi Provincial Natural Science Foundation of China under grants 2017GXNSFDA198014 and 2018GXNSFDA050014, and the Natural Science Foundation of China under grant 11761015.

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  1. Department of Mathematics, Nanning Normal University, Nanning, 530001, People’s Republic of China

    Guangqing Long, Rong Jie & Li-bin Liu

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  1. Guangqing Long
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  2. Rong Jie
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  3. Li-bin Liu
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Correspondence to Guangqing Long.

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Long, G., Jie, R. & Liu, Lb. Multilevel augmentation methods for eigen-problems of compact integral operators. Numer Algor 88, 1523–1540 (2021). https://doi.org/10.1007/s11075-021-01084-y

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