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An asymptotics-based adaptive finite element method for Kohn-Sham equation. (English) Zbl 1419.65122

Summary: In [R. Radovitzky and M. Ortiz, Comput. Methods Appl. Mech. Eng. 172, No. 1–4, 203–240 (1999; Zbl 0957.74058)], 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.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
82-08 Computational methods (statistical mechanics) (MSC2010)
35Q82 PDEs in connection with statistical mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)

Citations:

Zbl 0957.74058
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References:

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