Journal of Scientific Computing

, Volume 79, Issue 1, pp 464–492 | Cite as

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

  • Yedan Shen
  • Yang Kuang
  • Guanghui HuEmail author


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.


Electronic structure calculation Kohn–Sham density functional theory Adaptive finite element method Ground state energy Coarsening mesh 



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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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MacauMacauChina
  2. 2.Zhuhai UM Science and Technology Research InstituteZhuhaiChina

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