On the h-adaptive PUM and the hp-adaptive FEM approaches applied to PDEs in quantum mechanics

by jppelteret

Authors: D. Davydov, T. Gerasimov, J-P. V. Pelteret, and P. Steinmann

In this paper the h-adaptive partition-of-unity method and the h- and hp-adaptive finite element method are applied to partial differential equations arising in quantum mechanics, namely, the Schrodinger equation with Coulomb and harmonic potentials, and the Poisson problem. Implementational details of the partition-of-unity method related to enforcing continuity with hanging nodes and the degeneracy of the basis are discussed. 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 results are the same as for the finite element method and thereby admit the usage of standard residual error indicators. The efficiency of the h -adaptive partition-of- unity method is compared to the h – and h p -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 remarkable accuracy of the h -adaptive partition-of-unity approach. In case of the Hydrogen atom, the h -adaptive linear partition-of-unity method was found to be comparable to the hp -adaptive finite element method for the target eigenvalue accuracy of 10-3. [1]

[1] D. Davydov, T. Gerasimov, J-P. Pelteret, and P. Steinmann, “On the h-adaptive PUM and the hp-adaptive FEM approaches applied to PDEs in quantum mechanics,” Journal of Scientific Computing, 2017.
[Bibtex]
@Article{davydov2016a-preprint,
author = {Davydov, D. and Gerasimov, T. and Pelteret, J-P. and Steinmann, P.},
title = {On the h-adaptive PUM and the hp-adaptive FEM approaches applied to PDEs in quantum mechanics},
journal = {Journal of Scientific Computing},
year = {2017},
note = {Submitted},
abstract = {In this paper the h-adaptive partition-of-unity method and the h- and hp-adaptive finite element method are applied to partial differential equations arising in quantum mechanics, namely, the Schrodinger equation with Coulomb and harmonic potentials, and the Poisson problem. Implementational details of the partition-of-unity method related to enforcing continuity with hanging nodes and the degeneracy of the basis are discussed. 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 results are the same as for the finite element method and thereby admit the usage of standard residual error indicators. The efficiency of the h -adaptive partition-of- unity method is compared to the h - and h p -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 remarkable accuracy of the h -adaptive partition-of-unity approach. In case of the Hydrogen atom, the h -adaptive linear partition-of-unity method was found to be comparable to the hp -adaptive finite element method for the target eigenvalue accuracy of 10-3.},
comment = {Submitted},
file = {davydov2016a-preprint.pdf:Users/jp/Documents/Academic/My_articles/Website/davydov2016a-preprint.pdf:PDF},
keywords = {Adaptive finite element method, partition-of-unity method, error estimators, hp-adaptive finite element method, Schrodinger equation, local interpolation error estimates},
owner = {Jean-Paul Pelteret},
timestamp = {2016.05.19},
url = {https://arxiv.org/abs/1612.02305},
}