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PreprintCC BY 4.0arXiv

Numerical analysis of linear finite element methods for the Gross--Pitaevskii eigenvalue problem

  • Tianyang Chu
  • Mingyan Zhang
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Abstract

In this paper, we analyze the standard and the mass-lumped linear finite element methods for the Gross--Pitaevskii eigenvalue problem. For the standard method, we give a new proof of the uniqueness of the discrete ground state up to sign for sufficiently small mesh sizes, which only requires the H1-convergence of the discrete ground states. For both methods, we establish optimal convergence rates in the Lp- and W1,p-norms for 1 ≤ p < ∞. Under additional regularity assumptions on the potential, we further obtain an optimal convergence rate in the W1,∞-norm and an almost optimal convergence rate, up to a logarithmic factor, in the L∞-norm. Moreover, we prove that the discrete ground states superconverge to the Ritz projection of the exact ground state in the W1,p-seminorm. The analysis is based on a perturbation argument. Numerical experiments confirm the theoretical results.

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Cite

BibTeXRIS
BibTeX
@misc{chu2026numerical,
  title = {{Numerical analysis of linear finite element methods for the Gross--Pitaevskii eigenvalue problem}},
  author = {Tianyang Chu and Mingyan Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41715},
  eprint = {2610.41715},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41715},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Numerical analysis of linear finite element methods for the Gross--Pitaevskii eigenvalue problem
AU  - Tianyang Chu
AU  - Mingyan Zhang
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41715
UR  - https://arxiv.org/abs/2610.41715
AB  - In this paper, we analyze the standard and the mass-lumped linear finite element methods for the Gross--Pitaevskii eigenvalue problem. For the standard method, we give a new proof of the uniqueness of the discrete ground state up to sign for sufficiently small mesh sizes, which only requires the H^1-convergence of the discrete ground states. For both methods, we establish optimal convergence rates in the L^p- and W^1,p-norms for 1 ≤ p < ∞. Under additional regularity assumptions on the potential, we further obtain an optimal convergence rate in the W^1,∞-norm and an almost optimal convergence rate, up to a logarithmic factor, in the L^∞-norm. Moreover, we prove that the discrete ground states superconverge to the Ritz projection of the exact ground state in the W^1,p-seminorm. The analysis is based on a perturbation argument. Numerical experiments confirm the theoretical results.
ER  -