TY - JOUR
T1 - A Second Order Accurate in Time, Energy Stable Finite Element Scheme for a Liquid Thin Film Coarsening Model
AU - Yuan , Maoqin
AU - Dong , Lixiu
AU - Zhang , Juan
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 127
EP - 156
PY - 2025
DA - 2025/04
SN - 18
DO - http://doi.org/10.4208/nmtma.OA-2024-0081
UR - https://global-sci.org/intro/article_detail/nmtma/23944.html
KW - Liquid thin film model, second order accuracy, mass lumped mixed FEM, positivity preserving, energy stability, optimal rate convergence analysis.
AB - <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>