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Peer-reviewedCC BY 4.0Axioms

An Energy-Stable Variable-Step SAV Finite Element Method for a Droplet Thin-Film Coarsening Model with a Lennard–Jones Singular Potential

  • Maoqin Yuan1
  • Juan Zhang2
  • Mingyang Li3
  • Lixiu Dong4
4 affiliations
  1. 1School of Mathematics, Physics and Artificial Intelligence, China University of Petroleum-Beijing at Karamay, Karamay 834000, China
  2. 2School of Sciences, Lanzhou University of Technology, Lanzhou 730050, China
  3. 3School of Optoelectronics, University of Chinese Academy of Sciences, Beijing 100049, China
  4. 4Department of Mathematics, Faculty of Arts and Sciences, Beijing Normal University, Zhuhai 519087, China
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Abstract

We develop an energy-stable variable-step BDF2-based scalar auxiliary variable (SAV) finite element scheme for a droplet thin-film coarsening model with a Lennard–Jones singular potential. The method uses continuous piecewise linear finite elements in space and retains the original linear SAV update. We prove discrete mass conservation, modified-energy decay under an admissible step-ratio condition, and solvability of the coupled algebraic system. A Sherman–Morrison realization exploits the rank-one coupling and reduces each accepted time step to two solves with the same sparse base matrix. Numerical experiments assess temporal and spatial convergence, SAV-shift sensitivity, two-droplet mesh dependence, long-time coarsening, and solver efficiency. The film-height variable exhibits second-order temporal convergence and the expected spatial rates, whereas the auxiliary variable is first-order accurate; the consistency audit shows no hidden cancellation. The computations also verify mass conservation, modified-energy decay, positive film heights in the reported runs, and the efficiency of the sparse solver.

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Cite

BibTeXRIS
BibTeX
@article{yuan2026energystable,
  title = {{An Energy-Stable Variable-Step SAV Finite Element Method for a Droplet Thin-Film Coarsening Model with a Lennard–Jones Singular Potential}},
  author = {Maoqin Yuan and Juan Zhang and Mingyang Li and Lixiu Dong},
  journal = {Axioms},
  year = {2026},
  month = oct,
  doi = {10.3390/axioms15100751},
  url = {https://www.mdpi.com/2075-1680/15/10/751},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - An Energy-Stable Variable-Step SAV Finite Element Method for a Droplet Thin-Film Coarsening Model with a Lennard–Jones Singular Potential
AU  - Maoqin Yuan
AU  - Juan Zhang
AU  - Mingyang Li
AU  - Lixiu Dong
PY  - 2026
DA  - 2026/10/09
JO  - Axioms
DO  - 10.3390/axioms15100751
UR  - https://www.mdpi.com/2075-1680/15/10/751
AB  - We develop an energy-stable variable-step BDF2-based scalar auxiliary variable (SAV) finite element scheme for a droplet thin-film coarsening model with a Lennard–Jones singular potential. The method uses continuous piecewise linear finite elements in space and retains the original linear SAV update. We prove discrete mass conservation, modified-energy decay under an admissible step-ratio condition, and solvability of the coupled algebraic system. A Sherman–Morrison realization exploits the rank-one coupling and reduces each accepted time step to two solves with the same sparse base matrix. Numerical experiments assess temporal and spatial convergence, SAV-shift sensitivity, two-droplet mesh dependence, long-time coarsening, and solver efficiency. The film-height variable exhibits second-order temporal convergence and the expected spatial rates, whereas the auxiliary variable is first-order accurate; the consistency audit shows no hidden cancellation. The computations also verify mass conservation, modified-energy decay, positive film heights in the reported runs, and the efficiency of the sparse solver.
SN  - 2075-1680
ER  -