Two-Grid Scheme For Solving Nonlinear Convection-Diffusion Equations

The convection-diffusion equations are a kind of basic equations, which can be used to describe the distribution of pollutants in the water or in the air, the flow of liquid and etc. So it is significative in theory or in practice to study the numerical solution for convection-diffusion equations and to seek a quick, practical and stable numerical method.In this paper, two-grid method is constructed .combined with the characteristics method, finite element method and mixed finite element method, for solving nonlinear convection-dominated equations .which can reduce the numerical oscillations of the convection-dominated equations and improve the computational efficiency. In this approach, the full nonlinear system is solved on a coarse grid. The nonlinearities are expanded about the coarse grid solution, and the resulting linear system is solved on a fine grid. Convergence analysis and error estimates are also established.An outline of this paper is as follows. In chapter 1, some fundamental conceptions and their physical meanings are given, as well as the background and introduction of convection-diffusion equations. In chapter 2, severalnumerical methods existent are introduced for solving convection-diffusion equations. In chapter 3, a two-grid scheme of characteristic finite-element solution or a two-grid scheme of characteristic mixed finite-element solution for one nonlinear convection dominated diffusion equation is constructed and convergence analysis and error estimates are also established. In chapter 4, for 2D nonlinear convection-dominated diffusion equation ,a two-grid scheme of characteristic finite-element solution is constructed. Numerical examples are presented and computed with MatLab software. The numerical results show that these algorithms are effective and feasible.