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Volume 18, Issue 1
A Second Order Accurate in Time, Energy Stable Finite Element Scheme for a Liquid Thin Film Coarsening Model

Maoqin Yuan, Lixiu Dong & Juan Zhang

Numer. Math. Theor. Meth. Appl., 18 (2025), pp. 127-156.

Published online: 2025-04

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  • Abstract

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.

  • AMS Subject Headings

60F10, 60J75, 62P10, 92C37

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-18-127, author = {Yuan , MaoqinDong , Lixiu 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 = {

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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2024-0081}, url = {http://global-sci.org/intro/article_detail/nmtma/23944.html} }
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 -

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.

Yuan , MaoqinDong , Lixiu and Zhang , Juan. (2025). A Second Order Accurate in Time, Energy Stable Finite Element Scheme for a Liquid Thin Film Coarsening Model. Numerical Mathematics: Theory, Methods and Applications. 18 (1). 127-156. doi:10.4208/nmtma.OA-2024-0081
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