Convergence study of the h-adaptive PUM and the hp-adaptive FEM applied to eigenvalue problems in quantum mechanics

被引：15
作者：
Davydov D. [1 ]
Gerasimov T. [2 ]
Pelteret J.-P. [1 ]
Steinmann P. [1 ]
机构：
[1] Chair of Applied Mechanics, University of Erlangen-Nuremberg, Egerlandstr. 5, Erlangen
[2] Institute of Applied Mechanics, Technische Universitaet Braunschweig, Bienroder Weg 87, Braunschweig
来源：
Advanced Modeling and Simulation in Engineering Sciences | / 4卷 / 1期
基金：
欧洲研究理事会;
关键词：
Adaptive finite element method; Density functional theory; Error estimators; Local interpolation error estimates; Partition-of-unity method; Schrödinger equation;
D O I：
10.1186/s40323-017-0093-0
中图分类号：
学科分类号：
摘要：
In this paper the h-adaptive partition-of-unity method and the h- and hp-adaptive finite element method are applied to eigenvalue problems arising in quantum mechanics, namely, the Schrödinger equation with Coulomb and harmonic potentials, and the all-electron Kohn–Sham density functional theory. The partition-of-unity method is equipped with an a posteriori error estimator, thus enabling implementation of error-controlled adaptive mesh refinement strategies. To that end, local interpolation error estimates are derived for the partition-of-unity method enriched with a class of exponential functions. The efficiency of the h-adaptive partition-of-unity method is compared to the h- and hp-adaptive finite element method. The latter is implemented by adopting the analyticity estimate from Legendre coefficients. An extension of this approach to multiple solution vectors is proposed. Numerical results confirm the theoretically predicted convergence rates and remarkable accuracy of the h-adaptive partition-of-unity approach. Implementational details of the partition-of-unity method related to enforcing continuity with hanging nodes are discussed. © 2017, The Author(s).
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