In this paper, we investigate a class of nonlinear eigenvalue problems resulting from quantum physics. We first prove that for any open set G, there exists an eigenfunction that cannot be a polynomial on G, which may be reviewed as a refinement of the classic unique continuation property. Then we apply the non-polynomial behavior of the eigenfunction to show that the adaptive finite element approximations are convergent even if the initial mesh is not fine enough. We finally remark that similar arguments can be applied to a class of linear eigenvalue problems that improve the relevant existing results.
Keywords: Adaptive finite element approximation, complexity, convergence, nonlinear eigenvalue problem, non-polynomial behavior, unique continuation property
@article{M2AN_2021__55_1_209_0,
author = {Yang, Bin and Zhou, Aihui},
title = {Eigenfunction behavior and adaptive finite element approximations of nonlinear eigenvalue problems in quantum physics},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {209--227},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {1},
doi = {10.1051/m2an/2020078},
mrnumber = {4216829},
zbl = {1476.35250},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020078/}
}
TY - JOUR AU - Yang, Bin AU - Zhou, Aihui TI - Eigenfunction behavior and adaptive finite element approximations of nonlinear eigenvalue problems in quantum physics JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 209 EP - 227 VL - 55 IS - 1 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020078/ DO - 10.1051/m2an/2020078 LA - en ID - M2AN_2021__55_1_209_0 ER -
%0 Journal Article %A Yang, Bin %A Zhou, Aihui %T Eigenfunction behavior and adaptive finite element approximations of nonlinear eigenvalue problems in quantum physics %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 209-227 %V 55 %N 1 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020078/ %R 10.1051/m2an/2020078 %G en %F M2AN_2021__55_1_209_0
Yang, Bin; Zhou, Aihui. Eigenfunction behavior and adaptive finite element approximations of nonlinear eigenvalue problems in quantum physics. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 1, pp. 209-227. doi: 10.1051/m2an/2020078
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