Virtual element stabilization for the system of time-dependent nonlinear convection-diffusion-reaction equations.

In this paper, we develop the Virtual Element Method for a system of time-dependent nonlinear convection-diffusion-reaction equations on polygonal meshes. We present a fully nonlinear discrete scheme with a proof of existence and uniqueness of the numerical solution. In order to overcome the complexity of solving nonlinear discrete system, we propose an equivalent iterative linearized system of equations. We perform the stability analysis and then derive the optimal convergence estimates in the H1 semi-norm. Finally, we investigate the performance of the fully discrete scheme through a set of numerical tests. To improve computational efficiency, we have performed the simulations using a two-grid method.