Researches On The Mixed Finite Element Methods For A Type Of Nonlinear Fourth-order Parabolic Equation

This paper mainly includes three parts.In the first part, based on the bilinear element Q11, a lower order conforming mixed finite element approximation scheme is proposed for a type of nonlinear fourth-order hyperbolic equation. With the help of the known high accuracy results of the element and derivative delivery techniques of time t, the error estimate between the interpolation and Ritz projection and splitting skills, the superclose properties for both original variable u and intermediate variable v =-?u in H1 norm are derived for semi-discrete and fully-discrete schemes, respectively. Meanwhile, through interpolated postprocessing approach, the global superconvergence results are proved.In the second part, with the help of bilinear element Q11 and N′ed′elec s element,an expanded conforming mixed finite elements approximation scheme is proposed for the above equation. Firstly, the existence and uniqueness of approximation solution are proved.Secondly, based on the high analysis results of above two elements, an error estimate is established between interpolation and Riesz projection. Moreover, by use of the same techniques and methods as the first part, the superconvergence results of original variable u and intermediate variable v =-?u in H1-norm and intermediate variable q =- u, σ =-(?u) in(L2)2-norm are derived for semi-discrete and fully-discrete schemes, which can’t be derived by interpolation and Riesz projection alone.In the third part, with the help of EQrot1 and zero order Raviart-Thomas elements,a new expanded nonconforming mixed finite elements approximation scheme is proposed.By use of two special properties of EQrot1element:(i) the consistence error is one order higher than its interpolation error;(ii) the interpolation operator is equivalent to its Ritz-projection operator, the superclose properties and superconvergence results of corresponding variable the same as the second part are derived for semi-discrete scheme with high-accuracy analysis results and interpolation post-processing technique, respectively.Finally, through the asymptotic expansion formula of EQrot1 element and a suitable extrapolation scheme, the extrapolation solutions are obtained.