Mixed Finite Element Methods Based On Second-order Time Discrete Scheme For Two Classes Of Nonlinear Partial Differential Equations

In this paper, two second-order time discrete mixed finite element methods are stud-ied and discussed for symmetric regularized-long-wave (SRLW) equation and nonlinear Sobolev equation.In the part Ⅰ, the numerical method of mixed finite element combined with second-order Crank-Nicolson scheme in time is proposed for SRLW equation. The spatial direc-tion is approximated by using mixed element method and the second-order Crank-Nicolson scheme is used to discretize temporal derivative, some error estimates based on second-order fully discrete scheme are derived, whose result is one order higher than the one in the literatures.In the part Ⅱ, the nonlinear Sobolev equation is solved by making use of a mixed element method. Here, the fully discrete scheme based on the high-order spatial approx-imation and nonlinear second-order two-step backward Euler scheme in time is studied, then some a priori error estimates in both L2 and H1-norms are derived and proved.