Research On Superconvergence And Extrapolation For Three Kinds Of Nonlinear Differential Equations

This paper studies the convergence, superclose, superconvergence and extrapolation schemes of the approximate solutions of three types nonlinear equations with strong ap-plied backgrounds:nonlinear viscoelasticity type equations, a class of generalized neure conductive equations and nonlinear Sobolev equations with bilinear element, Hermite-type rectangular element and a noncforming element respectively. For the former two elements, by use of the integral identity and its asymptotic expansion, and constructing an auxiliary problem, corresonding the suitable extrapolation schemes so as to improve the accuracy of the finite element approximations. For the nonconforming element, by means of some special properties of the element:for example, the difference between the exact solution and its finite element interpolation is orthogonal in the sense of energy norm, compati-bility error is one order higher than the interpolation error, the derivative transformation techniques and so on, the same superconvergence results are derived. The study of this paper shows that using the element interpolation operator can not only simpify the proof directly, but also can reduce the computing cost comparing with the traditional Ritz pro-jection, and sometimes can get better results than the previous literature, which further extends the application of finite element methods.