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 H 1 -norm and sub-optimal error estimate in L 2 -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 T h and the streamline diffusion finite element scheme on the coarse grid T H , 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 H 1 -norm is proved to be O ( h + H 3 2 ) , showing that the two-grid method achieves the optimal approximation as long as the mesh sizes satisfy h = O ( H 3 2 ) . Finally, a numerical example is carried out to verify the accuracy and efficiency of the present numerical method. [ABSTRACT FROM AUTHOR]