两类发展型方程的新混合元格式

A New Mixed Finite Element Schemes for Two Classes of Evolutional Differential Equations

作者: 专业:计算数学 导师:石东洋 年度:2010 学位:硕士  院校: 郑州大学

Keywords

The generalized nerve conduction type equation, The hyperbolic type integro-differential equation, Anisotropic meshes, Nonconforming, New mixed finite element

        本文包含两部分,首先将一个Crouzeix-Raviart型非协调三角形元应用到双曲型积分微分方程,给出了这类方程的新混合元格式,证明了传统Riesz-Volterra投影与有限元插值的一致性,得到了与协调元方法完全相同的L2-模最优误差估计,同时还得到了一般混合元格式得不到的H1-模最优误差估计.其次,将一个各向异性线性三角形元应用到广义神经传播方程,并建立了一个新混合元格式,利用平均值和导数转嫁技巧,在不需要传统的Ritz投影条件下得到了L2-模最优误差估计,最后,我们还给出了广义神经传播方程任意阶格式的误差估计.
    This paper is composed of two parts. Firstly, We use a Crouzeix-Raviart type non-conforming trianglular element to approximate the hyperbolic type integro-differential equation, and a new mixed finite element scheme is established for this kind problem. The uniformity of the traditional Riese projection and the finite element interpolation is proved. Then, We get the same optimal error estimates as the conforming finite elements, At the same time, the H1-norm is derived,which can not be obtained for the usual mixed finite element schemes. Secondly, A linear anisotropic triangular element is applied to the generalized nerve conduction type equation. Under the new formulation, by use of the mean-value and derivative delivery technique and instead of the generalized Ritz projec-tion, We get the optimal error estimates in L2-norm. At last, The optimal error estimates of any order finite element method also is given for the generalized nerve conduction type equation.
        

两类发展型方程的新混合元格式

摘要4-5
Abstract5
引言7-10
第一章 预备知识10-20
第二章 双曲型积分微分方程在新混合元格式下非协调有限元分析20-31
第三章 广义神经传播方程三角形元的低阶和任意格式的收敛性分析31-41
参考文献41-45
附录 个人简历与硕士期间的主要研究成果45-46
致谢46
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