Characteristic discontinuous Galerkin method for fractional diffusion equations with local memory in nonstatic media

In this paper, a class of fractional diffusion equations with local memory in nonstatic media (viscous incompressible fluids) are proposed, and then, we numerically study the equations. The characteristic method and L1 scheme are used to discretize the time derivatives, and the interior penalty discontinuous Galerkin (IPDG) method is applied for the spatial operators. It is worth to emphasize that a hybrid characteristic discontinuous Galerkin method (CDGM) with continuous and discontinuous polynomials is designed in this paper to overcome the challenge of strictly obtaining the theoretical results of the CDGM designed according to the traditional approach. By estimating the differences between the above-mentioned polynomials and some theoretical results of characteristic finite element method (CFEM), one can get the theoretical results of the CDGM. The stability and optimal error estimates of the CDGM are achieved by using the Gronwall inequality and induction method. Finally, some numerical experiments are performed to justify the effectiveness of the theoretical results. Meanwhile, it indicates that the scheme still has robust simulation performance under the condition of small viscosity coefficients.