【作者】 杨朝晖；

【导师】 石东洋；

【作者基本信息】 郑州大学 ， 计算数学， 2011， 硕士

【摘要】 本文主要研究具有很强实际应用背景的三类非线性方程：非线性粘弹性方程,一类广义神经传播方程和非线性Sobolev方程.分别用双线性元、Hermite型矩形元及一个非协调元解逼近时的收敛性、超逼近性、超收敛性及外推格式.其中对前两个单元来说,主要利用积分恒等式及其渐进展开式,通过构造辅助问题,提出相应地适当外推格式以提高有限元的逼近精度.而对于非协调元来说,主要借助于单元的特殊性质：例如精确解与其有限元插值之差在能量意义下正交、相容性误差比插值误差高一阶、导数转嫁等技巧导出了与使用协调元时完全相同的超收敛结果.研究表明直接利用单元上插值算子的性质对某些单元来说不仅可以简化证明过程,比使用传统的Ritz投影减少工作量,有时还可以得到比以往文献更好的结果,从而进一步扩展了有限元方法的使用范围.更多还原

【Abstract】 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.更多还原