Two-grid Finite Volume Element Method For Semilinear Parabolic Problems And Schr(?)dinger Problems

With the expansion of physics research objects in breadth and depth,the application of partial differential equations is more and more extensive.Semilinear parabolic equations and Schr(?)dinger equations are important partial differential equations,which are widely used in the fields of plasma physics,nonlinear optics,bimolecular dynamics and protein chemistry.However,it is difficult to obtain their exact solutions in practical problems,so the research of efficient numerical algorithm for these problems has important practical application value.In view of the above two problems,this paper mainly carries out the following research work:A finite volume element discretization based on Crank-Nicolson scheme is proposed for semilinear parabolic equations.The optimal error estimates of the method in L~2 and H~1 norms are proved.At the same time,a two-grid algorithm is constructed for the problem.The main idea of two-grid method is to obtain a rough approximation through solving a nonsymmetric and nonlinear problem on the coarse grid space,and then solve a linearized equation on the basis of the approximate solution to get a more accurate modified solution on the fine grid.The error estimate of the two-grid algorithm is also given.Finally,numerical examples are given to verify the results of theoretical analysis and the efficiency of the algorithm.The backward Euler finite volume element scheme is used to discretize the time-dependent Schr(?)dinger problem,and the corresponding error analysis of this algorithm is made.At the same time,a two-grid decoupling algorithm is proposed.In this algorithm,the original coupling problem is simplified to solving the same problem on the coarse grid and two Poisson problems on the fine grid.Afterwards,the error estimate of decoupling algorithm in H~1 norm is proved.Finally,numerical examples are given to verify the results of theoretical analysis and the efficiency of the algorithm.