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. (C) 2015 Elsevier B.V. All rights reserved.