Mixed Finite Element Methods For Nonlinear Sobolev And BBM Equations

The paper mainly includes two parts:In the first part,the error analysis of a new nonconforming mixed finite element method for nonlinear Sobolev equation is studied.Firstly,the nonconforming CNQ₁^rotot element and piecewise constant element Q₀×Q₀are used to approximate the original variable u and intermediate variable p,and the semi-discrete and fully discrete schemes are developed.Secondly,by use of the special characters of CNQ₁^rot,the skill of combin-ing of interpolation and projection,as well as the interpolation post-processing technique,the superclose estimates and superconvergence results are obtained.Finally,we carry out a numerical example to confirm the theoretical analysis,and show the efficiency of the method.In the second part,a new conforming mixed finite element scheme is proposed for nonlinear BBM equation with bilinear element Q₁₁and Q₀₁×Q₁₀element.Based on the high accruracy analysis of these two elements,the superclose and global superconver-gence results of the relevant variables are derived through the technique of combination of interpolation and projection.Finally,numerical results verify the theoretical analysis.