混合有限元方法的非标准格式分析

Analysis of Nonstandard Schemes for Mixed Finite Element Methods

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

Least-Squares mixed finite element methods, H~1-Galerkin mixed finite element methods, Anisotropic meshes, Nonconforming, Superclose and supercon-vergence

        本文主要讨论了两个非标准混合元方法的收敛性及超收敛性分析.首先我们讨论了一个二阶椭圆问题的最小二乘非协调混合元方法的收敛性以及超收敛性分析,针对该格式的特殊性,我们把真解的逼近空间取成五节点非协调有限元空间,而把流量函数的逼近空间取为最低阶的Raviart-Thomas空间,通过引入新的方法和技巧,我们得到了与传统方法相同的收敛性结果.并且我们通过一些新的技和利用插值后处理方法,得到了真解的整体超收敛结果.其次,我们又研究了发展方程的非协调H1-Galerkin混合元方法.我们利用和第二章相同的非协调混合元逼近空间,在不引入Ritz投影和Ritz-Volterra投影的情况下,仍然得到了与传统方法相同的误差估计结果,并利用插值后处理技巧得到了整体超收敛结果.最后我们讨论了Sobolev方程全离散格式的H1-Galerkin混合有限元方法的超逼近性质.
    In this paper, we focus on convergence and superconvergence analysis of two nonstan-dard mixed finite element methods. Firstly, we consider the applications of Least-Squares nonconforming mixed finite element methods for the second elliptic problems. Since the speciality of the scheme, we discuss the case that the approximation space of the exact solution is the so-called five-nodes nonconforming space and one of flux is the lowest order Raviart-Thomas space. By means of the novel techniques, we obtain the same convergence as the traditional methods. And we also derive the global superconvergence of the exact solution using the novel skill and the postprocessing trick.Secondly, we study the nonconforming H1-Galerkin mixed finite element methods of evolution equations. Similarly, using the same nonconforming mixed finite element spaces as ones of the second chapter, the same convergence as the traditional methods is obtained without the Ritz or Ritz-Volterra projection. And we also derive the global superconvergence of the exact solution using the postprocessing technique. Finally, we discuss the superclose of the full discrete scheme of the Sobolev equation using the H1-Galerkin mixed finite element methods.
        

混合有限元方法的非标准格式分析

摘要4-5
Abstract5
引言7-9
第一章 预备知识9-18
第二章 二阶椭圆问题的最小二乘非协调混合有限元分析18-32
第三章 发展方程的非协调H~1-Galerkin混合有限元分析32-41
参考文献41-45
附录:个人简历及在学期间发表的学术论文与研究成果45-46
致谢46
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