高次三角形有限元的超收敛问题

关于二维区域二阶线性椭圆问题的有限元求解,[1,2]各自独立地对低次奇妙族矩形元采用单元合并技巧,获得能量的近似正交性(或称插值误差的第一弱估计),从而获得应力佳点定理.若获得更佳形式的能量正交性(或称插值误差的第二弱估计),则可获得位移佳点定理.运用以上方法,[1—8]解决了奇妙族矩形任意次元及三角形线元、二次元     

三三次长方体有限元的超收敛

本文首先介绍了三维投影型插值算子,并通过这个算子导出了三三次长方体有限元的弱估计．然后,利用离散导数Green函数的W^2,1半范估计和弱估计证明了有限元uh的梯度和三三次投影型插值Пh^2u的梯度在逐点意义下有超逼近．最后,将这种超逼近用于超收敛分析并导出了有限元的整体超收敛估计．     

三角形有限元的超收敛性 *

基于三角形上的两类正交展开 ,对二阶椭圆问题研究了任意m次三角形有限元解(对偶数m)及平均梯度(对奇数m)在对称点上的超收敛性 .除此之外 ,再没有其他与方程系数无关的超收敛点 .     

超收敛中的双P猜想续篇——离散格林函数的权模估计

探讨超收敛猜想中p=4的情形.为此目的我们推导了离散格林函数的权模估计.     

有限元的一个局部超收敛结果

１．引 言 设 是一个有界开域，具充分光滑的边界 且设 是 上的一族拟一致的三角剖分，用 表示定义在Ｔｈ上的分片线性有限元空间，并置考虑模型问题 用 分别表示的有限元解及内插，那么有插值估计：（见［１］）一般地，如ｕ为问题（１．１）的解，我们有有限元逼近误差估计（见［３］） 命题１．设 并设 分别表示按定义的Ｇｒｅｅｎ函数及其有限元逼近，那么有其中 Ｃ与 ｚ，ｈ无关．（参见［３］） 注意．如 且 ，那么至少存在一个点 ，使即ｘ０是ｆ的奇点，例如其中 为常数， ，显然如果。，如果故我们假定 本文将证明，误差与ｆ的奇性…     

二阶常微分方程初值问题C^0有限元的超收敛

基于在一个单元上的改进的单元正交展开，得到二阶常微分方程C^0有限元Uh的误差表示式，以及节点和单元内部一些特征点上的超收敛结果。     

各向异性网格下Bogner-Fox-Schmit元的超收敛

1引言 有限元超收敛的研究已有三十多年的历史,至今为止已取得了丰富的成果,可见[3],[18],[10],[6],[5]以及[17].1981年,陈传淼(见[2]345-372页)考虑了四阶板问题有限元解的超收敛性,得到了高一阶的超收敛结果.1995年,林群和罗平[8]用积分恒等式技巧再次研究这个问题,在均匀矩形网格的条件下,得到了更好的结论,有限元解与有限元插值函数之间的误差在H2范数下,有高二阶的超收敛.     

三角形线元与二次元L~2投影的整体超收敛

Based on an orthogonal expansion in a triangle, superconvergence for L^2-projection to linear and quadratic triangular elements is studied.Assume that Ω is a polygonal domain,triangulation is uniform and Th is a set of all vertexesand side midpoints.Then, on Th,the average gradient ^-D(u-uh)=O(h^2) for linear element uh,and u-uh=O(h^4) for quadratic element uh. Under someboundary conditions,these properties upto the boundary are valid.     

粘弹性方程ACM有限元的超收敛分析和外推

本文通过积分恒等式技巧和插值后处理技术,得到了粘弹性方程ACM有限元的超逼近和超收敛性质.进一步利用Bramble-Hilbert引理构造了一个合适的外推格式,得到了比以往文献高一阶的外推结果.     

板问题三次样条有限元的超收敛及高精度组合公式

1．引言与问题提出双三次样条有限元是应用比较广泛的矩形板元之一，与其它协调元相比恻双三次Her－mite矩形元），除达到饱和精度O（h‘）外，计算量却小得多［‘］，因而受到人们的重视．目前关于这方面的研究工作已有许多［1，并但关于板弯曲问题的双三次样条有限元的超收敛结果，却尚未见到．本文应用双三次样条插值的逐项渐近展式，通过对有限元的双线性形式进行展开，得到了双三次样条有限元一、二阶偏导数的渐近展式及超收敛结果．考虑如下矩形板弯曲模型问题上述方程对应的双线性形式为对VV，VE”，满足其中，M，。为正常数，11…     

Global superconvergence for Maxwell's equations

In this paper, the global superconvergence is analysed on two schemes (a mixed finite element scheme and a finite element scheme) for Maxwell's equations in . Such a supercovergence analysis is achieved by means of the technique of integral identity (which has been used in the supercovergence analysis for many other equations and schemes) on a rectangular mesh, and then are generalized into more general domains and problems with the variable coefficients. Besides being more direct, our analysis generalizes the results of Monk.

     

Local superconvergence of the derivative for tensor‐product block FEM

In this article, we shall discuss local superconvergence of the derivative for tensor‐product block finite elements over uniform partition for three‐dimensional Poisson's equation on the basis of Liu and Zhu (Numer Methods Partial Differential Eq 25 (2009) 999–1008). Assume that odd m ≥ 3, x₀ is an inner locally symmetric point of uniform rectangular partition \begin{align*}\mathcal{T}_{h}\end{align*} and ρ(x₀,?Ω) means the distance between x₀ and boundary ?Ω. Combining the symmetry technique (Wahlbin, Springer, 1995; Schatz, Sloan, and Wahlbin, SIAM J Numer Anal 33 (1996), 505–521; Schatz, Math Comput 67 (1998), 877–899) with weak estimates for tensor‐product block finite elements of degree m ≥ 3 [see Liu and Zhu, Numer Methods Partial Differential Eq 25 (2009) 999–1008] and the finite element theory of Green function in ??³ presented in this article, we propose the \begin{align*}O(h^{m+3}|\ln h|^{\frac{4}{3}}+h^{2m+2}|\ln h|^{\frac{4}{3}}\rho(x_{0},\partial\Omega)^{-m})\end{align*} convergence of the derivatives for tensor‐product block finite elements of degree m ≥ 3 on x₀. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 457–475, 2012     

Analysis of Linear Triangular Elements for Convection-diffusion Problems by Streamline Diffusion Finite Element Methods

This paper is devoted to studying the superconvergence of streamline diffusion finite element methods for convection-diffusion problems.In [8],under the condition thatε≤h~2 the optimal finite element error estimate was obtained in L~2-norm.In the present paper,however,the same error estimate result is gained under the weaker condition thatε≤h.     

A review of two different approaches for superconvergence analysis

In 1995, Wahbin presented a method for superconvergence analysis called Interior symmetric method, and declared that it is universal. In this paper, we carefully examine two superconvergence techniques used by mathematicians both in China and in America. We conclude that they are essentially different.     

一类非线性双曲型方程的非协调有限元方法

研究了一类非线性双曲型方程的非协调有限元方法,在不需要传统的Ritz投影的情况下,得到了半离散格式下的误差估计及超收敛结果.     

Lagrange interpolation and finite element superconvergence

We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d‐dimensional Q_k‐type elements with d ≥ 1 and k ≥ 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H¹ norm. For d‐dimensional P_k‐type elements, we consider the standard Lagrange interpolation—the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d ≥ 2 and k ≥ d + 1 that such interpolation and the finite element solution are not superclose in both H¹ and L² norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33–59, 2004.     

Superconvergence of tetrahedral quadratic finite elements for a variable coefficient elliptic equation

For a variable coefficient elliptic boundary value problem in three dimensions, using the properties of the bubble function and the element cancelation technique, we derive the weak estimate of the first type for tetrahedral quadratic elements. In addition, the estimate for the W^1,1‐seminorm of the discrete derivative Green's function is also given. Finally, we show that the derivatives of the finite element solution u_h and the corresponding interpolant Π₂u are superclose in the pointwise sense of the L^∞‐norm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Finite element superconvergence approximation for one‐dimensional singularly perturbed problems

Superconvergence approximations of singularly perturbed two‐point boundary value problems of reaction‐diffusion type and convection‐diffusion type are studied. By applying the standard finite element method of any fixed order p on a modified Shishkin mesh, superconvergence error bounds of (N^?1 ln (N + 1))^p+1 in a discrete energy norm in approximating problems with the exponential type boundary layers are established. The error bounds are uniformly valid with respect to the singular perturbation parameter. Numerical tests indicate that the error estimates are sharp; in particular, the logarithmic factor is not removable. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 374–395, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10001     

Global superconvergence of finite element methods for parabolic inverse problems

In this article we transform a large class of parabolic inverse problems into a nonclassical parabolic equation whose coefficients consist of trace type functionals of the solution and its derivatives subject to some initial and boundary conditions. For this nonclassical problem, we study finite element methods and present an immediate analysis for global superconvergence for these problems, on basis of which we obtain a posteriori error estimators. This research was supported in part by the Shahid Beheshti University, the National Basic Research Program of China (2007CB814906), the National Natural Science Foundation of China (10471103 and 10771158), Social Science Foundation of the Ministry of Education of China (Numerical methods for convertible bonds, 06JA630047), Tianjin Natural Science Foundation (07JCYBJC14300).