A New Two Grid Algorithm For Nonlinear Reaction-Diffusion Equations

Some practical problems,such as groundwater pollution,petroleum reservoir simulation,semiconductor devices,and biogeochemical phenomena,have been widely concerned.These problems can be simulated by semilinear or nonlinear reaction-diffusion models,which further increases the tractability of the problems.The traditional mixed element method would produce saddle point problem and inflexible space selection,which would lead to very difficult numerical solution.Based on the above situation,the split positive definite mixed finite element method was adopted in this paper to discretize.Solving large matrix requires large computation cost,In order to solve the problem that solving large matrix requires large computation cost,in this paper,we adopt two-grid technique to improve the computation speed while maintaining the convergence order.The main content of this paper is divided into three parts,which are interrelated and progressive:Firstly,the novel mixed element method of two-grid algorithm for elliptic model is studied.In this section,we give the scheme of the novel mixed element method for elliptic models and prior error estimates of the scheme on traditional fine grids: the scheme is order convergent in the sense of norm and norm.Then,the two-grid technique is used to reduce the computation time,and the priori error estimate is the same as that obtained on the traditional fine mesh.These results are optimal when the size of coarse mesh and fine mesh is satisfied.In addition,numerical experiments are carried out to verify the results.Secondly,the novel mixed element method of double mesh algorithm for semilinear parabolic models is studied.First,the format of the novel mixed element method for semilinear problems is given.The first order Euler method is used in time,and the prior error estimates of the two-grid technique are given.Finally,the theory is verified by numerical experiments,and the results show that the computing speed can be increased by about 20 times by using the two-grid technology.Finally,we study the two-grid algorithm of the novel mixed element method for nonlinear parabolic models.The main content of this part is the theoretical derivation of the novel and mixing element method for nonlinear problems,which is also calculated by using the two grid technology and verified by numerical experiments.