Discontinuous Galerkin Finite Element Methods For The Flow-Coupled Vesicle Model

Biofilm constitutes a natural barrier between cells and organelles,enabling some important life activities to be carried out in a relatively independent space,resulting in the processes of material transport,information transmembrane transmission,and energy conversion between cells and organelles.The material transport between membranous organelle in cells is mainly completed by vesicles.Studying vesicles can deepen people’s understanding of biofilms,therefore studying vesicle models has important practical significance.In this paper,a phase-field vesicle model coupled with an incompressible viscous fluid is used to describe the hydrodynamic mechanism of vesicles.We use the discontinuous Galerkin(DG)method to spatially discretize the model.By combining the scalar auxiliary variable(SAV)method,appropriate implicit-explicit techniques,and the pressure projection method,we construct linear,decoupled,fully discrete DG schemes with first and second order time accuracy,respectively.We strictly prove energy stability and optimal error estimates,especially for the pressure term.It should be emphasized that designing efficient and accurate numerical schemes for such a complex and highly nonlinear coupled model with physical constraints such as volume and surface area conservations is a very challenging issue.The main difficulties arise from:(1)how to discretize nonlinear energy potentials to obtain linear properties;(2)how to discretize the coupling terms of phase-field variable and fluid velocity,such as advection and stress terms;(3)how to discretize the momentum equation to decouple the fluid pressure and velocity fields under the constraint of incompressible conditions.In this paper,the SAV method is used to linearize the nonlinear energy potentials,and suitable implicit-explicit techniques are used to discretize the nonlinear coupling terms,thereby the linear schemes are obtained.Secondly,standard pressure projection methods are used to decouple fluid velocity and pressure.In addition,the solution to the above problems must be based on the premise of ensuring that the proposed numerical schemes can satisfy the energy dissipation law at the discrete level.In this paper,the DG methods are used for spatial discretization,compared to the traditional finite element methods,the DG methods have some significant advantages,including arbitrary order accuracy,local mass conservation,adaptive ease of implementation,and excellent ability in capturing interface motion.It should be noted that the main difficulty in analyzing the stability and error results of the DG methods is the lack of control over some jump and penalty terms at the boundaries of the mesh elements,which can affect the energy stability and error estimation of the schemes.In this paper,we introduce an additional step to decouple velocity and pressure,then fully discrete DG pressure projection schemes are constructed.By selecting appropriate test functions,the energy stabilities of the proposed schemes are successfully demonstrated in theory.Then,by constructing suitable operators and using appropriate numerical techniques,the optimal error estimates including pressure are obtained.In addition,the second-order decoupling scheme is more difficult to construct.In existing works,it requires solving an additional ODE system,and due to the presence of a large number of nonlocal operators,it can bring difficulties to practical calculations.In this paper,we construct an energy-stable second-order linear decoupling scheme,which only needs to solve a series of decoupled elliptic equations at each time step,greatly improving computational speed.Finally,numerical experiments show that the numerical results are consistent with the theoretical analysis,which verifies the accuracy and stability of the proposed schemes.