有限元超收敛新论

本文从三个方面讨论二阶椭圆问题有限元超收敛．1．一致网格上的新超收敛结果．利用新的“投影型插值”,我们解决了高次三角形元的超收敛问题．2．一般网格的超收敛性．利用局部插值处理和局部磨光处理我们获得了整体超收敛性结果．3．关于当前的两种超收敛技巧．Cornell学派利用一个精致的内估计和网格的点对称性,获得了一个“普遍”的结果,中国学派利用两个基本估计和离散Green函数理论获得了令人满意的结果,两者均很复杂．本文综合了两个学派的方法,简洁地证得上述普遍结果．     

对称矩阵分解因子的直接换元修正法

§1.引言 考虑非线性方程组 F(x)=0, (1)其中F:Ω?R~n→R~n使F′(x)对称.本文给出求解(1)的一种分解修正法,这种方法始于Jacobian F′(x)的初始对称三角分解,然后利用换元技巧直接修正上三角分解因子,进而前代与回代求迭代点.本文分析了分解修正法的运算量,证明了这个算法不用重新启动仍具有局部超线性收敛性和大范围收敛性.此外,这个算法自然保持分解因子的稀疏传递性和修正矩阵的对称传递性,特别当Jacobian正定时,还具有正定传递性.由此本文完成了[1]和[2]无法完成的工作.本算法特别适于大规模带状方程组和最优化问题,数值例子也表明了这一点.     

对称锥互补问题的一种非精确光滑牛顿算法

基于一个光滑函数,就单调对称锥互补问题,给出了一种解决高维对称锥互补问题的非精确光滑牛顿算法.在适当条件下,证明了该算法具有全局收敛性和局部二次收敛性.数值试验证实了算法对大规模对称锥互补问题的可行性和有效性.     

大步长非单调线搜索规则的Lampariello修正对角稀疏拟牛顿算法的全局收敛性

本文在ZhangH．C．的非单调线搜索规则基础上，结合ShiZ．J．大步长线搜索技巧提出了新的大步长的非单调线搜索规则，设计了求解无约束最优化问题的大步长非单调线搜索规则的Lampariello修正对角稀疏拟牛顿算法，在△f（x）一致连续的条件下给出了算法的全局收敛性和超线性收敛性分析．数值例子表明算法是有效的，适合求解大规模问题．     

求解广义非线性互补问题的光滑化拟牛顿法

利用光滑对称扰动Fischer-Burmeister函数将广义非线性互补问题转化为非线性方程组,提出新的光滑化拟牛顿法求解该方程组.然后证明该算法是全局收敛的,且在一定条件下证明该算法具有局部超线性(二次)收敛性.最后用数值实验验证了该算法的有效性.     

非线性方程的数值迭代法及其半局部收敛性

本文主要探讨非线性(算子)方程的数值迭代法及其半局部收敛性.在迭代方法部分,讨论了迭代法的构造技巧,主要可分为线性逼近、积分插值、Adomian级数分解、Taylor展开以及多步迭代等;在半局部收敛性部分,讨论了半局部收敛性的收敛条件以及证明收敛性的方法,包括递归法和优界序列法,同时还讨论了优界序列法所使用的优界函数.     

非线性方程的数值迭代法及其半局部收敛性北大核心CSCD

本文主要探讨非线性(算子)方程的数值迭代法及其半局部收敛性.在迭代方法部分,讨论了迭代法的构造技巧,主要可分为线性逼近、积分插值、Adomian级数分解、Taylor展开以及多步迭代等;在半局部收敛性部分,讨论了半局部收敛性的收敛条件以及证明收敛性的方法,包括递归法和优界序列法,同时还讨论了优界序列法所使用的优界函数.     

关于多元非线性方程的Broyden方法

本文提出了求解多元非线性方程的Broyden方法，讨论了该方法的局部与半局部收敛性，并估计了其超线性收敛速度．数值实验表明，新方法是可行有效的，并且其计算效率高于方向Newton法和方向割线法．     

关于奇次矩形元恢复导数的强超收敛性的进一步研究

通过推广林群等的超收敛结果及Green函数估计, 对矩形元利用SPR技巧给出了一种强超收敛方法, 证明了在局部对称点上导数具有$O(h^{k+3})$~($k\geq 3$ 为奇数)的强超收敛阶及位移具有$O(h^{k+4})$~($k\geq 4$ 为偶数)的强超收敛性.     

一个低阶各向异性有限元的超收敛分析

针对二阶椭圆问题,在各向异性网格上得到了由Park和Sheen提出的一个低阶非协调单元的收敛性分析,并给出了相应的误差估计．进一步利用插值后处理技巧,得到了后处理后的离散解与真解本身的整体超收敛性质．最后的数值试验验证了理论的可靠性．     

Some superconvergence results of high-degree finite element method for a second order elliptic equation with variable coefficients

In this paper, some superconvergence results of high-degree finite element method are obtained for solving a second order elliptic equation with variable coefficients on the inner locally symmetric mesh with respect to a point x ₀ for triangular meshes. By using of the weak estimates and local symmetric technique, we obtain improved discretization errors of O(h ^p+1 |ln h|²) and O(h ^p+2 |ln h|²) when p (≥ 3) is odd and p (≥ 4) is even, respectively. Meanwhile, the results show that the combination of the weak estimates and local symmetric technique is also effective for superconvergence analysis of the second order elliptic equation with variable coefficients.     

The local superconvergence of the trilinear element for the three-dimensional Poisson problem

In this paper, we shall combine the finite element theory of Green?s function presented in this paper, the extrapolation technique and the local symmetric technique to investigate the local superconvergence of the trilinear element for the three-dimensional Poisson equation.     

The local superconvergence of the quadratic triangular element for the poisson problem in a polygonal domain

It is the first time for us to combine the local symmetric technique and the weak estimates to investigate the local superconvergence of the finite element method for the Poisson equation in a bounded domain with polygonal boundary where a uniform family of partitions is not required or the solution need not have high global smoothness. Combining a uniform family of triangulations in the interior of domain with a quasiuniform family of triangulations at the boundary of domain, we present a special family of triangulations. By the finite element theory of the derivative of the Green's function presented in this article, we combine the local symmetric technique and the weak estimates to obtain the local superconvergence of the derivative for the quadratic elements. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1854–1876, 2014     

Superconvergence of new mixed finite element spaces

In this paper we prove some superconvergence of a new family of mixed finite element spaces of higher order which we introduced in [ETNA, Vol. 37, pp. 189-201, 2010]. Among all the mixed finite element spaces having an optimal order of convergence on quadrilateral grids, this space has the smallest unknowns. However, the scalar variable is only suboptimal in general; thus we have employed a post-processing technique for the scalar variable. As a byproduct, we have obtained a superconvergence on a rectangular grid. The superconvergence of a velocity variable naturally holds and can be shown by a minor modification of existing theory, but that of a scalar variable requires a new technique, especially for k=1. Numerical experiments are provided to support the theory.     

Superconvergence for triangular finite elements

Based on two classes of the orthogonal expansions in a triangle, superconvergence of m-degree triangular finite element solution (for evenm) and its average gradient (for oddm) at symmetric points for a second order elliptic problem are studied. There are no other superconvergence points independent of the coefficients of elliptic equation. Project supported by the National Natural Science Foundation of China (Grant No. 19331021).     

Superconvergence for rectangular serendipity finite elements

Based on an orthogonal expansion and orthogonality correction in an element, superconvergence at symmetric points for any degree rectangular serendipity finite element approximation to second order elliptic problem is proved, and its behaviour up to the boundary is also discussed.     

A local parallel superconvergence method for the incompressible flow by coarsening projection

In this article, we investigate a local parallel superconvergence method by coarsening projection for the incompressible Stokes flow. The method is a combination of the local superconvergence technique and the given framework of local parallel method. For the smooth subdomains, the local superconvergence method is applied in a higher order finite dimensional space corresponding to an appropriate coarse mesh on interior domain. Moreover, a useful and flexible local parallel method is designed to obtain the local parallel superconvergence results of presented method, which offset theoretical limitation of the model without the smoothness of the exact solution and a priori regularity of the underlying problem over the whole domain. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1209–1223, 2015     

High accuracy analysis of two nonconforming plate elements

We prove the superconvergence of Morley element and the incomplete biquadratic nonconforming element for the plate bending problem. Under uniform rectangular meshes, we obtain a superconvergence property at the symmetric points of the elements and a global superconvergent result by a proper postprocessing method. The research is supported by the Special Funds For Major State Basic Research Project (No. 2005CB321701).     

有限元网格的超收敛测度及其应用

给出线性有限元求解二阶椭圆问题的有限元网格超收敛测度及其应用.有限元超收敛经常是在具有一定结构的特殊网格条件下讨论的,而本文从一般网格出发,导出一种网格的范数用来描述超收敛所需要的网格条件以及超收敛的程度.并且通过对这种网格范数性质的考察,可以证明对于通常考虑的一些特殊网格的超收敛的存在性.更进一步,我们可以通过正则细分的方式在一般区域上也可以自动获得超收敛网格.最后给出相关的数值结果来验证本文的理论分析.