An efficient discontinuous Galerkin method for the hydro-dynamically coupled phase-field vesicle membrane model

Abstract

This paper presents a linear, fully-decoupled discontinuous Galerkin (DG) method for the flow-coupled phase-field vesicle membrane model. The fully discrete scheme is implemented by combining several reliable numerical techniques. Firstly, we employ the DG method for spatial discretization, which does not require that the numerical solutions are continuous across different grid cells, enabling adaptive treatment of complex vesicle membrane shapes and fluid flows. Secondly, to cope with nonlinear and coupling terms, scalar auxiliary variables (SAV) and implicit-explicit approaches are used, which improve numerical stability and computational efficiency. Finally, for the momentum equation, the pressure-correction method is used to assure the decoupling of velocity and pressure. Through rigorous mathematical proofs, this paper demonstrates the unconditional energy stability of the proposed scheme and derives its optimal error estimation. Numerical examples also show the method's effectiveness in terms of precision, energy stability, and simulated vesicle deformation in thin channels. The results highlight the method's potential application in fluid-coupled phase-field vesicle membrane models, while also providing new methodologies and technological support for numerical simulation and computational research in related domains.