Unconditional Superconvergence Analysis Of Mixed Finite Element Methods For Two Classes Of Nonlinear Equations

In this paper, the unconditionally superconvergent analysis of the mixed finite element methods(MFEMs) is discussed for the nonlinear Sobolev and parabolic equations.Firstly, the semi-discrete and linearized fully-discrete systems of the new MFEM are proposed for the nonlinear Sobolev equation, and the existence and uniqueness of the solutions are proved. Through the combination of interpolation and projection skill, theτ-independent superclose estimates of the original variable u in H1-norm and the flux variable q =-(a(u) ut+ b(u) u) in L2-norm are obtained under weaker hyphethesis of ut∈ H2(?). At the same time, the new interpolated postprocessing operators are construted which have smaller degrees of freedom compared with the original literature,the corresponding identical global superconvergence results are derived.Secondly, the new mixed finite element scheme for nonlinear parabolic equation is studyed. By the splitting technique, the error is divided into two parts: the time-discrete error and the spatial-discrete error. For the time-discrete system, the boundedness of the original variable u and the flux variable q =-a(u) u in L∞norm are deduced. For the spatial-discrete system, by the above bounded estimates, the spatial superclose and superconvergent estimates are presented unconditionally, which compensate for the only convergent results of the previous document.Finally, the corresponding numerical results are provided for the two classes of nonlinear equations, which show the efficiency of the methods and skills and then confirm our theoretical analysis.