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Two-scale finite element discretizations for nonlinear eigenvalue problems in quantum physics

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Abstract

In this paper, some two-scale finite element discretizations are introduced and analyzed for a class of nonlinear elliptic eigenvalue problems on tensor product domains. It is shown that the solution obtained by the standard finite element method on a one-scale fine grid can be numerically replaced with a combination of some solutions on a coarse grid and some univariate fine grids by two-scale finite element discretizations. Compared with the standard finite element solution, the two-scale finite element approximations save computational cost significantly while achieving the same accuracy.

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Acknowledgements

The authors thank Professor Huajie Chen and Professor Aihui Zhou for fruitful discussions and assistance on numerical computations.

Funding

This work was supported by the National Natural Science Foundation of China (grant 11971066 and 11771467) and the disciplinary funding of Central University of Finance and Economics.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, 100875, Beijing, China

    Pengyu Hou

  2. School of Statistics and Mathematics, Central University of Finance and Economics, 102206, Beijing, China

    Fang Liu

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  1. Pengyu Hou
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  2. Fang Liu
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Correspondence to Fang Liu.

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Communicated by: Long Chen

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Hou, P., Liu, F. Two-scale finite element discretizations for nonlinear eigenvalue problems in quantum physics. Adv Comput Math 47, 59 (2021). https://doi.org/10.1007/s10444-021-09883-6

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