Numerical Methods For The 2D Incompressible Navier-Stokes-Fokker-Planck Equation

In this paper,we talk about the numerical analysis for the Fokker-Planck equation of two-dimensional Brownian particles in an incompressible viscous fluid,i.e.,a convection-diffusion equation where the velocity is governed by the Navier-Stokes equation.With the help of the grids locally refined in temporal direction,discrete Gronwall inequality and discrete Ladyzhenskaya inequality,we obtain the almost unconditionally stability and convergence results of Euler SAV finite element scheme for the Navier-Stokes-Fokker-Planck equation with non-smooth initial data.And the numerical experiments are also provided to support the theoretical results.Last but not least,in the outlook part,we construct two kinds SAV schemes with low-regularity exponential integrators,and then analyze their consistency.However,the complete stability and convergence analysis needs to be given later.