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An Asymptotics-Based Adaptive Finite Element Method for Kohn–Sham Equation

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Abstract

In Radovitzky and Ortiz (Comput Methods Appl Mech Eng 172(1–4):203–240, 1999), an error estimation technique for nonlinear PDEs is presented to adaptively generating mesh, based on the reduction of the order of the approximate polynomial. In this paper, following a similar analysis framework, we propose an a posteriori error estimation for Kohn–Sham equation by coarsening mesh. An upper bound for the difference of the total energies on two successively refined meshes is derived by the numerical solutions on two meshes through an asymptotic analysis, which finally generates an a posteriori error estimation. A variety of numerical tests show that such an a posteriori error estimation works very well in our h-adaptive finite element method framework. In addition, to further improve the efficiency, we solve a Poisson equation instead of the Kohn–Sham equation on the coarsened mesh. The effectiveness of this improvement is analyzed and numerically examined.

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Acknowledgements

This work was partially supported by FDCT 029/2016/A1 from Macao SAR, MYRG2017-00189-FST from University of Macau, and National Natural Science Foundation of China (Grant No. 11401608).

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Authors and Affiliations

  1. Department of Mathematics, University of Macau, Macau, Macao SAR, China

    Yedan Shen, Yang Kuang & Guanghui Hu

  2. Zhuhai UM Science and Technology Research Institute, Zhuhai, China

    Guanghui Hu

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  1. Yedan Shen
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  2. Yang Kuang
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  3. Guanghui Hu
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Corresponding author

Correspondence to Guanghui Hu.

Additional information

The authors would like to thank the support from FDCT 029/2016/A1 of Macao S.A.R., MYRG2017-00189-FST from University of Macau, and and National Natural Science Foundation of China (Grant No. 11401608).

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Shen, Y., Kuang, Y. & Hu, G. An Asymptotics-Based Adaptive Finite Element Method for Kohn–Sham Equation. J Sci Comput 79, 464–492 (2019). https://doi.org/10.1007/s10915-018-0861-0

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  • DOI: https://doi.org/10.1007/s10915-018-0861-0

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