An Efficient Three-step Scheme Two-grid Method For Nonlinear Parabolic Equations

All along, fluid mechanics has always been an important subject for many scien-tists to study. Most people are more concerned about the fluid movement in porous media, it is a very complex physical movement, and its corresponding mathematical model is also very complex. Because the reaction-diffusion equation contains two variables: velocity and pressure. Therefore, many researchers usually use the hybrid finite element method to study the equation, we know that the system of equations obtained after the dispersion is generally nonlinear. For solving nonlinearity prob-lem, two-grid method is a lot of researchers love the subject. Nonlinear parabolic equations are very non-linear sexual partial differential equations, is widely used.Many researchers have used a variety of two-grid method to solve the numerical solution of the reaction-diffusion equation with linearity of the compression coeffi-cient. For this kind of compression coefficient, the research on the nonlinear term is less. This subject is based on the extended mixed finite element method and the two-grid algorithm, and uses the two-grid method with three-step scheme to solve the nonlinear parabolic equation with the compression coefficient as nonlinear term.In this paper, we study the two-grid method network of nonlinear parabolic equations studied by Professor Yanping Chen and Luoping Chen dimensional two-step scheme,a two-grid method with three-step scheme is used to solve the numerical solution of the nonlinear parabolic equation. We refer to the two-step scheme, usu-ally refers to the first nonlinear equations on the coarse grid solution, and then in the fine grid to solve the linearization of the linear system after the group. And we get two of the three-step scheme,The first scheme is in the original two-step scheme after the fine grid on the correction, unlike the first scheme, the second scheme re-quires that we do a correction on the coarse grid. In the proof analysis process, We mainly use the superconvergence of mixed finite element, eventually we found for the two schemes, We can construct the algorithm to optimize the solution of the mixed finite element method, as long as we select the coarse grid steps H were satisfied H = O(h1/4) (or H = O(h1/3)) for the nonlinearity problem.