【作者】 刘鸣放；

【导师】 陈绍春；

【作者基本信息】 郑州大学 ， 计算数学， 2011， 博士

【摘要】 本文针对三维四阶椭圆方程本文构造了两个新的非协调四面体单元,一个是十四参数单元,单元自由度个数较少；另一个为十六参数单元,具有能量正交的形函数空间。证明了两个新单元对三维四阶椭圆方程收敛。文章的第二部分,关于三维四阶椭圆方程我们构造出了一个20参数三棱柱单元,并分析了该单元的收敛性。对于定常Stokes问题,我们在压力空间取Q={q∈H1(Ω)；∫Ωqdx=0}情形下得到了新的变分形式,关于新的变分问题我们构造出了基于泡函数的任意阶混合元格式,给出了误差估计。同时,还给出了压力空间取二次元时基于泡函数新的一阶、二阶和三阶格式,所得结果中对压力的误差估计比已有的相应阶格式收敛阶分别高出一阶。关于Darcy-Stokes问题我们运用宏元技巧探讨了P2-P1元在不规则十字网格、扭曲十字网格以及重心剖分网格上的一致稳定性和逼近性。更多还原

【Abstract】 In this paper, two new nonconforming tetrahedral elements are presented for the fourth order elliptic partial differential operators in three spatial dimensions. Of the two newly constructed elements one is the 14-parameter tetrahedral element which has lower degree, the other is the 16-parameter energy-orthogonal tetrahedral element which has an energy-orthogonal shape function space. These two elements are proved to be convergent for a model biharmonic equation in three dimensions.In the second part of this paper, we constructed a 20-parameter triangular prism element for the fourth order elliptic equation and analysed the approximation of this element.For the stationnary Stokes problem, when the pressure space is taken as Q = {q∈H1 (Ω);∫Ωqdx= 0}, we get the new variational form. The mixed element formats of any order based on bubble func-tions are presented for the new variational problem in triangular and tetrahedral meshes. and the convergence of any order mixed element formats are proved. Furthermore, when the pressure finite element space consists piecewise polynomials of degree two, we propose the new mixed element formats of one order, two order and three order based on bubble functions for the new variational problem, the error estimates of the pressure in L2-norm for the new formats are one degree higher than that of the existing formats.At the end of this paper, by the macroelement partition theorem, we study the uniform stability and approximation properties of P2-P1 element for Darcy-Stokes problem on irregular crisscross meshes, on distorted crisscross meshes and on barycentric trisected meshes respectively.更多还原