Researches On Unconditional High Accuracy Finite Element Methods For Nonlinear Evolution Equations

In this paper some nonlinear evolution equations?such as nonlinear parabolic equations,nonlinear Schr¨odinger equations,nonlinear Sobolev equations,nonlin-ear Ginzburg-Landau equations and nonlinear hyperbolic equations?have been studied deeply by different angles and different techniques from conforming finite element method,nonconforming finite element method and mixed finite element method to get the unconditional super-convergence results and the corresponding experiments under different fully discrete schemes.The main innovations in the paper are as follows:?1?High accuracy error analysis requires high regularities of the equations.However if the region is polygonal?such as rectangle?,high regularities are hard to derive.Thus we use some special and different skills to get the unconditional super-convergence properties under the lower regularities;Using Taylor expansion to deal with the nonlinear term in order to ensure the desired order of the time step.?2?Due to the linearized fully discrete schemes,the mathematical induction method is used.The results about the n-1th time are utilized to prove the results about nth.the The key is to control each time layer with a uniform coefficient.?3?Construct new second-order scheme for nonlinear hyperbolic equations to reduce the unconditional super-convergence results and even the unconditional convergent properties for nonlinear hyperbolic equations have never been reported before.?4?Abandoning the splitting technique,we use some new techniques to get rid of the ratio for some special nonlinear evolution equations.First of all,low order nonconforming finite element and conforming ele-ment are considered for the nonlinear parabolic equations by using linearized CN?Crank-Nicolson?schemes and linearized linearized BE?Backward-Euler?schemes,respectively.Both of them are analyzed for the unconditional super-close properties.By introducing auxiliary equations?i.e.the time-discrete sys-tem?,the error is split into two parts—the time error and the spatial error.The time error gives the regularities of time discrete equations.Avoiding the fact that the solution of time discrete equation can only belongs to H²???not H³???,we adopt the mathematical induction method to unified the coefficient at each time step.Taylor expansion is exploited novelty to solve the difficulties causing by the nonlinear term of the equations and the order ofis not reduced.As an attempt,we use more sharp estimation to estimate the space error.Rather than the traditional error result O?h²?,the error result about O?h?h+???h is the space parameter andis the time step?is reduced.Further,if the right hand of the nonlinear parabolic equation is restricted to the local Lipschitz continuous,the boundedness of the numerical solution in L^????-norm is needed,based on which,the unconditional super-closeness is derived.Secondly,the linearized full discrete schemes?BE scheme and CN scheme?for the nonlinear Schr???dinger equation are given.In order to overcome the difficulties caused by the imaginary unit i?,the skills of subtracting the two adjacent time layers is utilized to derive the higher order of the time error.As a result,the better regularities is derived too.The combination of the projection operator and the interpolation operator is used to obtain the unconditional super-convergence of the original variable Uⁿ in H¹???-norm.On one hand,the projection operator successfully reduced the strict regularity of Uⁿ and numerical solution is bounded to ensure the existence and uniqueness of the solution about each layer.On the other hand,based on the interpolation operator,the interpolation post-processing technology is used to get the global super-convergence.The above two points also fully show the importance of the combination about the two operators in numerical analysis.Thirdly,the unconditional super-convergence of the nonlinear Sobolev equa-tions and nonlinear Ginzburg-Landau equations are studied by H¹-Galerkin finite element method.On one hand,an important lemma for the pair of nonconform-ing finite elements(EQ₁^rot element and zero order Raviart-Thomas element)is given.A linearized CN scheme is constructed for the nonlinear Sobolev equa-tions.Different from the splitting method,some new techniques are used to get the super-close results with the special characteristics of the nonlinear Sobolev equations themselves.On the other hand,the pair of conforming finite elements?the bilinear element and the zero-order Raviart-Thomas element?is exploited to get the unconditional super-closeness with the CN scheme for the nonlinear Ginzburg-Landau equations.Finally,in view of the nonlinear hyperbolic equation,a linearized new second-order scheme is constructed for the first time and the truncation error is proven hard.By introducing the corresponding time-discrete system,the regularities are obtained and the unconditional super-closeness are derived with the nonconform-ing element EQ₁^rot.In this paper,the corresponding numerical examples are given for each of the above parts.The numerical results further demonstrate that the proposed method is efficient and feasible.