Global constraints preserving Lagrange multiplier approach for the incompressible flow-coupled phase-field vesicle membrane models

This paper presents a novel model for lipid vesicles that overcomes the limitations of classical phase-field elastic bending models, which utilize penalty terms to approximate volume and surface area conservation. Our approach achieves accurate volume conservation and employs a Lagrange multiplier to ensure precise surface area conservation. For the incompressible flow-coupled system, we introduce two efficient, linear, and energy-stable schemes that integrate the scalar auxiliary variable (SAV) approach with the Lagrange multiplier technique. These schemes maintain the efficiency of the SAV approach for unconstrained gradient flows, requiring only the solution of linear equations with constant coefficients at each time step alongside a negligible-cost nonlinear algebraic system. The proposed schemes guarantee unconditional energy stability while effectively preserving global surface area conservation. Furthermore, the numerical schemes are fully decoupled, linear, unconditionally energy-stable, and exhibit second-order accuracy in time. A significant innovation enabling this full decoupling is the introduction of an ordinary differential equation to address nonlinear coupling terms while satisfying the “zero-energy-contribution” property. Extensive 2D and 3D simulations demonstrate the accuracy and stability of our proposed schemes. The code is published as open-source and can be accessed here: https://github.com/HYQ688/CAC_New_Lagrange_method.