Researches On Two-grid Finite Element Method For Two Classes Of Nonlinear Hyperbolic Equations

The main work of this paper is studying two classes of nonlinear hyperbolic equations by finite element method.The superclose and the global superconvergence results of two-grid methods(TGMs)are obtained.Firstly,by use of the bilinear finite element,the superconvergence analysis of a second-order fully discrete scheme with TGM for the(2+1)-dimensional nonlinear hyperbolic equation is obtained.The existence and uniqueness of the solution for the scheme are proved rigorously.By use of the combination technique of the interpolation and Ritz projection,the superclose estimate of order O(h2+-H4+?2)of the variable u in the H1-norm is deduced and the global superconvergence estimate of order O(h2+H4+?2)of the variable u in the H1-norm is derived through the interpolated postprocessing approach.Secondly,by use of the nonconforming EQ1rot finite element,a second order TGM for hyperbolic Allen-Cahn equation is developed.The stability of the fully discrete scheme is proved.Based on the special properties of the element,the derivative transfer technique and the interpolated postprocessing approach,the superclose and superconvergence results of order O(h2+H4+?2)in the discrete H1-norm are deduced.Finally,numerical examples of the above-mentioned two schemes are given to verify the correctness of the theories,respectively.The results show that the proposed TGMs are indeed very effective numerical methods for the given examples and the computing cost is only half of the traditional finite element methods.