多尺度函数与多小波的构造

在多小波和单小波的基础上利用矩阵卷积构造出了一类多尺度函数与多小波,并通过实例对构造算法加以说明.     

基于重数延长法提升加细向量函数的逼近阶

基于任意给定的伸缩因子为a的正交多尺度函数, 给出一种提升其逼近阶的算法. 设Φ(x)=[φ₁(x),x)=[φ₂(x),…,φ_r(x)]^T是伸缩因子为a,逼近阶为m的正交多尺度函数,则可以构造出一个重数为r+s,逼近阶为m+L(LÎZ₊)的新正交多尺度函数Φ^new(x)=Φ^T(x),φ_r+1(x), φ_r+2(x),…, φ_r+s(x)^T. 换言之, 通过增加多尺度函数的重数提升了它的逼近阶. 另外, 讨论了一个特殊情形：如果所给的正交多尺度函数Φ(x)=[φ₁(x),φ₂(x),…,φ_r(x)] ^T是对称的,则新构造的多尺度函数 Φ^new(x)不仅能提升其逼近阶, 而且还保持对称性. 给出了若干构造算例.     

对称尺度函数的参数化设计

In this paper, the factorization for filters with length Km of scaling functions intosimple blocks is considered.     

有限元多尺度小波

利用有限元插值和多尺度分析理论构造出了有限元多尺度小波.这些小波函数集许多优良性质于一身,如固定的短支集、高阶的消失矩、半正交性及正则性等.     

尺度函数的两个性质

Two properties are given in this paper about the scaling function: suppose Vj; j ∈ Z is a multiresolution analysis with a continuous scaling function φ which have compact support set and that φ the Fourier transform of φ is a continuous real function, compactly supported, then φ(0) ≠ 0 and when supp φ = [a1,b1]∪[a2,b2](b1 < a2,0 < a2), then we havea1 ≤ 0, 0 < b1, a1 < b2/2 ≤ b1, 2π < b2 - a1 ≤ 8π.     

任意多连通截面扭转应力函数的有限元解法

杆件扭转问题的求解,主要有基于扭转理论翘曲函数的边界元法和有限元法、基于薄壁杆件理论的数值解法和基于扭转理论应力函数的有限元法．根据任意多连通截面直杆扭转问题的应力函数理论,讨论并改进了与微分方程及定解条件等效的泛函,在此基础上推导了求解多连通截面扭转应力函数的有限元列式,将扭转问题的翘曲位移单值条件转化为边界节点上的集中荷载．采用主从节点法满足孔洞边界上应力函数的同值条件,实现了任意多连通复杂截面扭转应力函数的有限元直接求解,通过应力函数积分获得截面的扭转常数．算例验证了方法的可行性和有效性．     

多元代数函数逼近的存在性与局部性质

The existing scheme of rational polynomial approximants, defined by multivariate power series, is extended to define approximants with branch points. The existence theorem is obtained. The basic properties used to define the rational approximants can be preserved almost intactly. Especially, the local behavior of the diagonal bivariate quadratic algebraic function approximation is analysed.     

绝对值函数的一致光滑逼近函数

绝对值函数是一个非光滑函数,研究了绝对值函数的光滑逼近函数.给出了绝对值函数的上方一致光滑逼近函数和下方一致光滑逼近函数,分别研究了其性质,并通过图像展示了逼近效果.     

M进制超函数的构造

该文把M进制尺度函数的逼近阶与超函数联系起来，证明超函数可以显式构造出来，并给出利用Strang Fix条件构造超函数的一般方法，就M=2情形，对由G.Strang和V.Strela提供的例子，利用文中算法，得到它的超函数和一些新的性质。     

三角域上一类正交函数系的构造

V系统是作者2005年构造的一类L2[0,1]空间上的正交完备函数系. κ次V系统由κ次分片多项式组成,具有多分辨特性,是Haar小波函数的推广.基于V系统的正交表达,可以对CAGD中常见的几何模型用有限项V-级数做到精确重构,完全消除Gibbs现象,这是有限项Fourier级数或连续小波级数不能做到的.针对多变量情形,给出了三角域上的κ次正交V系统的构造方法.三角域上的V系统的重要应用显现在对3D复杂几何图组的整体频谱分析上.     

A Stabilized Low Order Finite Element Method for Three Dimensional Elasticity Problems

We introduce a low order finite element method for three dimensional elasticity problems. We extend Kouhia-Stenberg element [12] by using two nonconforming components and one conforming component, adding stabilizing terms to the associated bilinear form to ensure the discrete Korn's inequality. Using the second Strang's lemma, we show that our scheme has optimal convergence rates in $L^2$ and piecewise $H^1$-norms even when Poisson ratio $\nu$ approaches $1/2$. Even though some efforts have been made to design a low order method for three dimensional problems in [11,16], their method uses some higher degree basis functions. Our scheme is the first true low order method. We provide three numerical examples which support our analysis. We compute two examples having analytic solutions. We observe the optimal $L^2$ and $H^1$ errors for many different choices of Poisson ratios including the nearly incompressible cases. In the last example, we simulate the driven cavity problem. Our scheme shows non-locking phenomena for the driven cavity problems also.     

Maximum Norm Resolvent Estimates for Elliptic Finite Element Operators

We study a finite element approximation A _h, based on simplicial Lagrange elements, of a second order elliptic operator A under homogeneous Dirichlet boundary conditions in two and three dimensions, where h is thought of as a meshsize. The main result of the paper is a new resolvent estimate for the operator A _h in the L -norm. This estimate is uniform with respect to h for the case with at least quadratic elements. In the case with linear elements, the estimate contains on the right a factor proportional to (log log ), where = 1 or = in two or three dimensions, respectively.This revised version was published online in October 2005 with corrections to the Cover Date.     

Stokes方程非协调混合元的特征值下界

通过利用Crouzeix-Raviart元({1,x,y}),旋转元({1,x,y,x~2-y~2}),拓广旋转元({1,x,y,x~2,y~2})以及拓广Crouzeix-Raviart元({1,x,y,x~2+y~2})这四种混合有限元(参看正文中示图)来提供求Stokes特征值下界的方法.并找到恰当的理论框架,重要的是证明不仅统一,而且出奇的短,仅需几行.最后给出相关的数值结果来验证本文的理论分析.     

三角形二次插值系数有限元法解半线性椭圆问题的超收敛性

基于均匀三角形的剖分求解一类二阶半线性椭圆问题,用插值系数有限元方法比经典有限元法更容易实现,与经典二次有限元一样,二次插值系数有限元方法在对称点处也有四阶超收敛精度,数值计算表明这些结论是正确的.     

非协调元特征值渐近下界

利用有限元收敛速度下界的结果获得某些非协调元方法新的Aubin-Nitsche估计形式,然后再结合非协调元特征值的展开式获得不需要额外条件下非协调元特征值渐近下界的结果.     

Sums of Element Orders in Finite Groups

Given a finite group G, write ψ(G) to denote the sum of the orders of the elements of G. Our main result is that if C is a cyclic group and G is a noncyclic group of the same order, then ψ(G) < ψ(C).     

Finite Element Approximation of Vector-Valued Hemivariational Problems

In this paper we develop a finite element approximationfor vector-valuedhemivariational inequalities.This class of hemivariational problems wasintroducedin [12],[13]. We study two differentproblems: unconstrained oneand constrained one witha nonempty, closed, convex constraint set K.We shall show firstly that the discrete problemsare solvable by usingconsequences of Kakutanifixed point theorem and secondly that the solutionsof the discrete problemsare close on subsequences to the continuous ones.     

四阶椭圆方程的一个十六参数能量正交四面体元(英文)

In this paper,the 16-parameter nonconforming tetrahedral element which has an energy-orthogonal shape function space is presented for the discretization of fourth order elliptic partial differential operators in three spatial dimensions.The newly constructed element is proved to be convergent for a model biharmonic equation.     

一类基于小波基函数插值的有限元方法

在分析具有大的梯度问题中,将具有紧支集的小波基函数引入到传统的有限元插值函数的构造中,对传统的插值方法进行修正。对新的插值模式进行了数值稳定性（解的唯一存在性）分析并通过分片分析讨论了解的收敛性,新的插值模式所引入的附加自由度通过静力凝聚法来消除,最后得到了基于变分原理的小波有限元列式。     

Raviart-Thomas混合元的超收敛

考虑二阶椭圆方程Dirichlet边值问题在正则矩形网格上k阶RaviartThomas混合有限元的超收敛.对有限元解经插值处理后,与通常的有限元最优误差估计相比,收敛速度提高了两阶.