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arXiv · 2609.21328

History-Compatible Energy-Stable Finite Element Schemes for Variable-Density Cahn--Hilliard--Navier--Stokes Flows on Evolving Meshes

Wenbin Wang·Yunqing Huang·Yin Yang·Huayi Wei
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Abstract

We consider variable-density Cahn--Hilliard--Navier--Stokes (CHNS) discretizations on finite element meshes that may change between accepted time levels through fixed-topology motion or topology-changing remeshing. When the discrete spaces vary in time, the phase, kinetic, and pressure histories entering a multistep scheme are measured in different discrete structures and cannot, in general, be transferred by a single operator. We develop decoupled backward Euler (BE) and second-order backward differentiation formula (BDF2) schemes by combining exact physical cross-mesh pairings with history representations compatible with the corresponding phase-energy, kinetic-energy, and pressure-gradient storages. The phase update also determines an Abels--Garcke--Grün-consistent mass flux used in the momentum transport. A scalar capillary-exchange equation separates the phase and fluid solves while retaining the discrete energy exchange. The resulting field subproblems are linear, the scalar equation has a unique positive solution, and the schemes satisfy modified energy balances without a time-step restriction under the stated admissibility assumptions. Numerical experiments confirm second-order temporal convergence under both mesh updates, phase-mass conservation, modified-energy decay in the unforced tests, and comparable Rayleigh--Taylor and rising-bubble dynamics.

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Wenbin Wang, Yunqing Huang, Yin Yang, Huayi Wei. 2026-09-18. History-Compatible Energy-Stable Finite Element Schemes for Variable-Density Cahn--Hilliard--Navier--Stokes Flows on Evolving Meshes. https://arxiv.org/abs/2609.21328

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