@Article{NMTMA-18-1,
author = {Yuan, Maoqin and Lixiu, Dong and Zhang, Juan},
title = {A Second Order Accurate in Time, Energy Stable Finite Element Scheme for a Liquid Thin Film Coarsening Model},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2025},
volume = {18},
number = {1},
pages = {127--156},
abstract = {<p style="text-align: justify;">In this paper, we propose and analyze a second order accurate (in time)
mass lumped mixed finite element numerical scheme for the liquid thin film coarsening model with a singular Leonard-Jones energy potential. The backward differentiation formula (BDF) stencil is applied in the temporal discretization, and
a convex-concave decomposition is derived, so that the concave part corresponds
to a quadratic energy. In turn, the Leonard-Jones potential term is treated implicitly
and the concave part is approximated by a second order Adams-Bashforth explicit
extrapolation. An artificial Douglas-Dupont regularization term is added to ensure
the energy stability. Furthermore, we provide a theoretical justification that this numerical algorithm has a unique solution, such that the positivity property is always
preserved for the phase variable at a point-wise level, so that a singularity is avoided
in the scheme. In fact, the singular nature of the Leonard-Jones potential term
around the value of 0 and the mass lumped FEM approach play an essential role in
the positivity-preserving property in the discrete level. In addition, an optimal rate
convergence estimate in the $ℓ^∞(0, T ; H^{−1}_h
)∩ ℓ^2
(0, T ; H^1_h
)$ norm is presented. Finally,
two numerical experiments are carried out to verify the theoretical properties.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2024-0081},
url = {https://global-sci.com/article/91753/a-second-order-accurate-in-time-energy-stable-finite-element-scheme-for-a-liquid-thin-film-coarsening-model}
}