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Numerical Solution of the Kohn-Sham Equation by Finite Element Methods with an Adaptive Mesh Redistribution Technique

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Abstract

A mesh redistribution method is introduced to solve the Kohn-Sham equation. The standard linear finite element space is employed for the spatial discretization, and the self-consistent field iteration scheme is adopted for the derived nonlinear generalized eigenvalue problem. A mesh redistribution technique is used to optimize the distribution of the mesh grids according to wavefunctions obtained from the self-consistent iterations. After the mesh redistribution, important regions in the domain such as the vicinity of the nucleus, as well as the bonding between the atoms, may be resolved more effectively. Consequently, more accurate numerical results are obtained without increasing the number of mesh grids. Numerical experiments confirm the effectiveness and reliability of our method for a wide range of problems. The accuracy and efficiency of the method are also illustrated through examples.

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Acknowledgements

The authors would like to thank Prof. Chao Yang (Lawrence Berkeley National Laboratory) for his helpful comments, suggestions on this work. The research was supported in part by the NSF Focused Research Group grant DMS-0968360. The research of G. Bao was supported in part by the NSF grants DMS-0908325, CCF-0830161, EAR-0724527, DMS-0968360, DMS-1211292, the ONR grant N00014-12-1-0319, a Key Project of the Major Research Plan of NSFC (No. 91130004), and a special research grant from Zhejiang University.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Gang Bao

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Gang Bao, Guanghui Hu & Di Liu

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  1. Gang Bao
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  2. Guanghui Hu
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  3. Di Liu
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Correspondence to Guanghui Hu.

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Bao, G., Hu, G. & Liu, D. Numerical Solution of the Kohn-Sham Equation by Finite Element Methods with an Adaptive Mesh Redistribution Technique. J Sci Comput 55, 372–391 (2013). https://doi.org/10.1007/s10915-012-9636-1

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