高阶非协调有限元的构造、收敛性分析及其应用

项目名称： 高阶非协调有限元的构造、收敛性分析及其应用

项目编号： No.11301053

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 孟兆良

作者单位： 大连理工大学

项目金额： 22万元

中文摘要： 本项目针对二阶椭圆问题、Stokes问题及平面弹性问题中的非协调元构造问题开展理论和应用研究，主要考虑二维和三维网格上的高收敛阶非协调元的构造。非协调元构成的混合元对更容易满足LBB条件，求解Stokes问题能得到稳定的数值解；同时它还可以有效避免平面弹性问题中的数值死锁现象。相比低阶非协调元，高阶非协调元的自由度的选取更为复杂，并且对应的自由节点通常在一条低次代数曲线（面）上；对于四边形网格或三维网格，自由度的个数通常要大于相应全次数多项式空间的维数。本项目首先根据实际需要给出自由度的选取方法（特别是对三维网格），然后拟采用代数几何和计算几何中的技巧来选取形函数空间，使得与自由度的选取相匹配，并通过（广义）分片检验。利用本项目构造的非协调元求解二阶椭圆问题、Stokes问题及平面弹性问题，给出相应的误差估计和大量的数值实验。另外也考虑在构造过程中产生的实际数数值计算问题。

中文关键词： 非协调元；四边形网格；六面体网格；混合元；LBB条件

英文摘要： This project will study nonconforming finite element defined on two or three dimensional meshes which can solve second-order elliptic problems, Stokes problems and plane elasticity problems, etc. Nonconforming element pairs can produce stable numerical solution for Stokes problems since in this case the discrect LBB condition can be satisfied easily. Also nonconforming element can avoid numerical locking efficiently for plane elasticity problems. Compared to lower-order nonconforming element, it is difficult to decide their DOFs for higher-order noconconforming element. Besides, usually all the points lie in a lower-degree algebraical curve. For quadrilateral meshes or three-dimensional meshes,the numbers of DOFs are usually bigger than the dimension of polynomial space of total degree considered. In this project, we first develop a method to select DOFs and construct the corresponding shape function space by the method of algebraical geometry and computational geometry. We will enlarge the space of the shape function by rational functions or spline functions to match the degrees of freedom and pass the patch test. Besides, we will give the error estimates for second-order elliptic problems, Stokes problems and plane elasticity problems. Moreover, we also consider the computation aspect.

英文关键词： nonconforming elements；quadrilateral meshes；hexahedral meshes；mixed element；LBB condition