Numerical analysis of a discontinuous Galerkin method for time-fractional Navier-Stokes-Fokker-Planck equations with weakly singular solutions

In this paper, a class of time-fractional Navier-Stokes-Fokker-Planck equations (TF-NSFPEs) describing position of particles with anomalous diffusion in viscous incompressible fluids are proposed. We develop the symmetric interior penalty discontinuous Galerkin (IPDG) method for TF-NSFPEs. The L1 method in the time on graded mesh is used for the reason that the solution of the time fractional Fokker-Planck equation (TFFPE) usually has a weak singularity near the initial time. The stability and the optimal error estimates of the IPDG semi-discrete scheme are proved by using the discrete fractional Grönwall inequality. Further, based on these results, the fully discrete optimal error estimates are obtained. Finally, some numerical experiments are performed to justify the effectiveness of the theoretical results.