Researches On Finite Element Methods For Two Classes Of Unsteady And Nonlinear Partial Differential Equations

The main work of this paper is to study two classes of unsteady and non-linear partial differential equations by finite element methods and to derive the superclose and the global superconvergence properties.Firstly,a new conforming expanded mixed finite element scheme is proposed for nonlinear Sobolev equation with Q₁₁/Q₀₁×Q₁₀element.Based on the high accuracy analysis of these two elements,through the technique of combination of interpolation and projection,mean-value technique and the interpolated post-processing approach,the superclose and global superconvergence results of the relevant variables are derived.Secondly,by use of Q₁₁element,we study the BDF2 and TGM fully-discrete schemes for nonlinear wave equation.By introducing the auxiliary variable,the equation is splitted into two parabolic equations.Based on the technique of combination of interpolation and projection,the superclose and superconvergence results of the original variable and auxiliary variable are derived for BDF2 and TGM fully-discrete schemes.Finally,numerical results of the above-mentioned schemes which verify con-sistency of the theoretical analysis and numerical results are provided.Further-more,it shows that the TGM is indeed a very effective numerical method for solving the considered partial differential equation and the CPU cost is about half of the traditional finite element method.