The Finite Volume Two-Grid Method For Nonlinear Problems

In the nature, many things are related with nonlinear elliptic problems, such as nonlinear diffusion theory, burning of gas theory, and the law of gravitational equilibrium between planets, etc. So it has attracted much interest in the study of nonlinear problems. Generalized finite difference method that is called finite volume method in the international mathematic field, is a new part in PDE theories. Many scholars have made study in it, and gotten some valuable results. Finite volume method has the character of accuracy of FEM, while it keeps the simpleness of finite difference method at most degree.The domain is divided twice in two-grid method. At first, divide the domain into coarse gird. A nonlinear problem is solved on this coarse grid. Then, refine the coarse grid into fine grid. On the fine grid, only a linear problem needs to solve. With the help of two-grid method, the efficiency of nonlinear elliptic problem calculation is greatly improved but no accuracy loses.In the paper, finite volume method is constructed for linear elliptic problems firstly. Then, two-grid finite element method is constructed for nonlinear chord-balance problems. Theory analysis and numerical calculation are provided. At last, two-grid finite volume method is constructed for solving 2D nonlinear elliptic problems by combining finite volume method with two-grid method. Numerical example shows that two-grid finite volume method is an efficient, accurate and stable method for solving nonlinear elliptic problems, which keeps the advantages of every method in the algorithm.With two-grid finite volume method, the nonlinear calculation is devoted into coarse grid which has quite less nodes, and time of calculation is decreased greatly. The analysis and numerical calculation show that time decrease is not at the cost of accuracy losing.