Two-Grid Mixed Finite Element Method For Nonlinear Problems

Nonlinear problem is an important part in the study of differential equations. Many nonlinear problems come up with the development of production and scientific research. In some case, the derivative is needed as well as the unknown function value, as it is also an important physical quantity and high accuracy is required. So it is necessary to construct a scheme to satisfy such requirement. Reaction diffusion equations and convection diffusion equations are common mathematic models in many domains, such as hydrology, physics, chemistry and biology, etc. It is valuable to study numerical method for those equations.In this paper, a scheme of two-grid mixed finite element method is constructed to solve nonlinear reaction diffusion equations and convection diffusion equations, by combining the two-grid method with mixed finite element method. With mixed finite element method, the unknown function value is got, at the same time more accurate derivative is come up. The two-grid method divides the domain for two times. As a result, nonlinear iteration is devoted into coarse grid which keeps quite less nodes. On the fine grid, it becomes a linear problem profiting from the Taylor expansion. So this scheme keeps the both advantages of mixed finite element method for getting high accurate derivative and of the two-grid method for less computation with nonlinear problems.Algorithm construction, error estimate, program design and numerical examples are included in the paper. Convergence analysis and the numerical examples show that this scheme provides greater efficiency than the standard FEM, with no loss in order of accuracy. And, a more accurate derivative comes up at the same time. It is an efficient method for solving nonlinear problems.