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Tailored Finite Point Methods for Solving Singularly Perturbed Eigenvalue Problems with Higher Eigenvalues

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Abstract

We study tailored finite point methods (TFPM) for solving the singularly perturbed eigenvalue (SPE) problems. We first provide an asymptotic analysis for the eigenpairs and show that for some special potential functions when \(\varepsilon \) approaches to zero the square of eigenfunction converges to a Dirac delta function weakly, and the eigenvalue converges to the minimum value of the potential function. For computing the eigenfunction with higher eigenvalue we propose two variants of TFPM for one-dimensional SPE problems and a nonlinear least square TFPM for two-dimensional problems. The eigenfunction with higher eigenvalue can be easily computed on a related coarse mesh on numerical tests, and suggests that the proposed schemes are accurate and efficient for the SPE problems.

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Acknowledgements

The authors thank the anonymous referees for their kind comments and valuable suggestions. Research by the first author is supported by the National Science Foundation of China (NSFC) under Grant numbers 11371218 and 91330203. Research by the second author is supported by the Ministry of Science and Technology of Taiwan under Grant MOST 103-2115-M005-004-MY2. Research by the third author is supported by NSFC under Grant numbers 11571196 and 60873252, and by a Grant from the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions.

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Authors and Affiliations

  1. Department of Mathematical Science, Tsinghua University, Beijing, 100084, China

    Houde Han & Dongsheng Yin

  2. Department of Applied Mathematics, National Chung Hsing University, 250 Kuo-Kuang Road, Taichung, 40227, Taiwan

    Yintzer Shih

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  1. Houde Han
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  2. Yintzer Shih
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  3. Dongsheng Yin
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Correspondence to Yintzer Shih.

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Han, H., Shih, Y. & Yin, D. Tailored Finite Point Methods for Solving Singularly Perturbed Eigenvalue Problems with Higher Eigenvalues. J Sci Comput 73, 242–282 (2017). https://doi.org/10.1007/s10915-017-0411-1

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  • DOI: https://doi.org/10.1007/s10915-017-0411-1

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