The Finite Element Analysis For Evolution Nonlinear Equations

This article mainly discusses the finite element approximations for three classes of nonlinear evolution equations. Firstly, the computable error bounds of a nonconforming finite element for the parabolic equation are presented. Without the traditional Ritz pro-jection, the convergence analysis and the corresponding sharp estimation are derived for right triangle meshes:Secondly, under the uniform meshes,the semi-discrete finite element approximation for the nonlinear convection-diffusion equation is investigated. Meanwhile, based on the interpolated postprocessing technique, the global superconvergence result is obtained. Finally,we give the convergence analysis of the generalized nerve conductive equation with a nonconforming finite element of the least degrees of freedom-the con-strained rotation Q1 element. Simultaneously, by use of the special construction of the element,Bramble-Hilbert lemma and the interpolation trick, the optimal error estimation and super-approximation result are deserved without the Ritz projection and modification.