Volume 55, Number 1, January-February 2021
|Page(s)||209 - 227|
|Published online||18 February 2021|
Eigenfunction behavior and adaptive finite element approximations of nonlinear eigenvalue problems in quantum physics
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China
2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P.R. China
* Corresponding author: firstname.lastname@example.org
Accepted: 10 November 2020
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.
Mathematics Subject Classification: 35Q55 / 65N15 / 65N25 / 65N30 / 81Q05
Key words: Adaptive finite element approximation / complexity / convergence / nonlinear eigenvalue problem / non-polynomial behavior / unique continuation property
© EDP Sciences, SMAI 2021
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