Efficient energy-stable numerical methods for phase-field vesicle membrane models with strict volume and surface area constraints

We propose a new phase-field vesicle model that rigorously enforces physical constraints and admits efficient, energy-stable numerical discretizations. The model couples a conserved Allen–Cahn gradient flow, which guarantees exact preservation of the enclosed volume, with a single Lagrange multiplier that imposes the global surface area constraint. Based on this formulation, we design two classes of numerical schemes. The first is a linear SAV-based framework that is simple to implement and reduces each time step to constant-coefficient linear solves. The second is a new Lagrange multiplier scheme that retains the efficiency of SAV while exactly preserving the physical constraints and strictly dissipating the original free energy, requiring only two inexpensive nonlinear solves per step. For both schemes, we establish discrete energy dissipation laws, provide implementation details, and validate their performance through extensive 2D and 3D simulations. Numerical results confirm accuracy, unconditional stability, exact constraint preservation, and computational efficiency, demonstrating the effectiveness of the proposed approaches for simulating vesicle dynamics.