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

Davydov, D., Gerasimov, T., Pelteret, J.-P. and Steinmann, P. (2017) Convergence study of the h-adaptive PUM and the hp-adaptive FEM applied to eigenvalue problems in quantum mechanics. Advanced Modeling and Simulation in Engineering Sciences, 4, 7. (doi: 10.1186/s40323-017-0093-0)

Abstract

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.

• Item Type: Articles
• Additional Information: Funding: The support of this work by the ERC Advanced Grant 289049 MOCOPOLY (DD, JP, PS) and German Science Foundation (Deutsche Forschungs-Gemeinschaft, DFG), Grant DA 1664/2-1 (DD) is gratefully acknowledged. Second author (TG) is supported by the European Research Council (ERC) Starting Researcher Grant INTERFACES, Grant Agreement No. 279439.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Steinmann, Professor Paul
• Authors: Davydov, D., Gerasimov, T., Pelteret, J.-P., and Steinmann, P.
• College/School: College of Science and Engineering > School of Engineering > Infrastructure and Environment
• Journal Name: Advanced Modeling and Simulation in Engineering Sciences
• Publisher: Springer Open
• ISSN: 2213-7467
• ISSN (Online): 2213-7467
• Published Online: 12 December 2017
• Copyright Holders: Copyright © 2017 The Authors
• First Published: First published in Advanced Modeling and Simulation in Engineering Sciences 4:7
• Publisher Policy: Reproduced under a Creative Commons License