A two-grid combined finite element-upwind finite volume method for a nonlinear convection-dominated diffusion reaction equation

In this paper, we first present a combined finite element-upwind finite volume method for a fully nonlinear convection-dominated diffusion reaction equation and derive its a priori optimal error estimate in H1H1-norm and sub-optimal error estimate in L2L2-norm for piecewise linear finite element combining with the first order upwind finite volume scheme. Then we study a type of two-grid method for the nonlinear convection-dominated transport equation together with the combined finite element-upwind finite volume method on the fine grid ThTh and the streamline diffusion finite element scheme on the coarse grid THTH, which not only significantly reduces the computational cost on nonlinear iterations but also remains the numerical computation stabilized and the approximation accuracy unchanged. A priori error estimate of such designed two-grid method in H1H1-norm is proved to be O(h+H32), showing that the two-grid method achieves the optimal approximation as long as the mesh sizes satisfy h=O(H32). Finally, a numerical example is carried out to verify the accuracy and efficiency of the present numerical method.