The virtual element method for the time fractional convection diffusion reaction equation with non-smooth data

In this paper, we develop an efficient virtual element method to solve the two-dimension time fractional convection diffusion reaction equation involving the Caputo fractional derivative with non-smooth solutions in the time direction. The equation exhibits a weak singularity near the initial time. We first use the exponential transformation to eliminate the convection terms, and obtain the time fractional diffusion reaction equation. By utilizing properties of the energy projection operator and continuous Grönwall inequality, we derive estimates of the approximation error in the L2-norm and H1-norm under semi-discrete on polygonal meshes. Based on the L1 scheme on graded mesh in time, the fully-discrete scheme is proved to be unconditionally stable and the optimal convergence results are derived with regards to the L2-norm and H1-norm. Finally, some numerical experiments are implemented to verify the theoretical results.