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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 L 2 -norm and H 1 -norm under semi-discrete on polygonal meshes. Based on the L 1 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 L 2 -norm and H 1 -norm. Finally, some numerical experiments are implemented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

• A local projection stabilization term is introduced to keep the scheme stable for time-fractional Burgers equation with high Reynolds number. • Construct the virtual element scheme for time-fractional Burgers equation on polygonal meshes in spatial. • Based on scheme on a graded mesh in time direction, the error estimates are -robust. • Numerical experiments are effective for time fractional Burgers equation with high Reynolds number. We propose and analyze a local projection stabilization virtual element method for time-fractional Burgers equation on polygonal meshes, whose solutions display a weak singularity at the initial time. Based on the L 1 scheme on a graded mesh in time and virtual element method in space, the stability and existence and uniqueness of the fully discrete scheme are proved; and the priori error estimates in L 2 norm is derived with local projection stabilization term. Numerical experiments show that our method is effective for time fractional Burgers equation with high Reynolds number. [ABSTRACT FROM AUTHOR]

• A new local projection stabilization term is introduced to keep the scheme stable in the convective-dominated regime. • Compared with residual stabilization, the local projection stabilization is simpler in format and saves computation. • Error estimates do not depend on the inverse of ϵ , which means that the method is effective for small viscosity coefficient. • We have carried out numerical simulations for variable coefficients and constant coefficients, and obtained ideal results. In this paper, we propose and analyze a local projection stabilization virtual element method for steady scalar convection-diffusion-reaction problem in the convective-dominated regime. Stability and optimal convergence property in the convective-dominated regime are proven in a suitable norm. Numerical results show that the proposed method is stable at very large Reynolds number and is in good agreement with the expected rate of convergence. [ABSTRACT FROM AUTHOR]