二元插值的一般框架

本文通过引进多参数建立了二元插值的一般框架.这样,许多著名的经典插值格式,如Newton插值、分叉连分式插值、对称连分式插值等均可视为本文的特殊情形．     

一种广义插值法

本文考虑一种广义插值问题，插值条件为小区间上的积分值，以弥补现有的插值方法在L2空间不再适用的不足，除了多项式插值外，还讨论了两种一次样条插值方法。     

分段有理三次保凸插值

给定插值数据{(x_i,y_i)}_(1=0)~n,在许多实际应用中(如VLSI、CAD/CAM等),要求插值函数除满足一定的光滑性条件外,还必须反映插值点集的整体几何性质。例如,通常要求单调(凸)数据产生的插值函数也是单调(凸)的,用标准插值技术象多项式或三次样条,这些要求     

广义Vandermonde行列式及其应用

1 广义Vandermonde行列式的定义 1966年,I.J.Schoenberg在文[1]中明确提出具有一般性的Hermite-Birkhoff插值及其插值适定性问题.而一般的Hermite-Birkhoff插值问题则未必是适定的,关于这方面目前已有许多工作,见[2]—[7].我们知道,Hermite-Birkhoff插值问题是 Hermite插值问题的推     

多项式插值余项定理的一个自然证明

通过对插值多项式函数性质进行分析,多项式插值余项的基本形式得到诱导,再从该基本形式出发,获得了多项式插值余项定理的新证明.整个证明过程无需借助辅助函数的构造,因而显得较为自然.这种自然证明的方式也可用于Hermite切触型插值多项式余项的证明.     

关于球面上的一种Hermite插值

本文旨在提出单位球面上的一种Hermite插值格式.为此,本文首先研究了沿球面同轴圆周组上的Hermite插值问题,给出了三种适定的插值泛函组.然后研究了球面上的Hermite插值问题,给出了球面上Hermite插值的一种叠加插值法,即添加圆周组法.进一步将二者结合,导出了一类球面上Hermite插值的适定插值泛函组.为了说明这类适定插值泛函组的构造方法,在本文最后还给出了构造球面上低次Hermite适定插值泛函组的一些具体例子和数值算例.     

关于广义逆的向量连分式插值样条

本文首次引入了关于广义逆的向量有理插值样条的概念．这类插值样条具有Thiele型连分式的截断分式的表现形式．在它的构造过程中,不必用到连分式的三项递推关系,本文得到的新的有效的系数算法具有递推运算的特点．存在性的一个充分条件得以建立．包括唯一性在内的有关插值问题的某些结果得到证明．最后,本文给出了一个精确的插值误差公式．     

三元重心Thiele型混合有理插值

有理插值比多项式插值有更好的近似,但有理插值一般很难控制极点的产生.基于Thiele型连分式插值与重心有理插值,构造三元重心Thiele型混合有理插值,当选取适当的权后能避免部分极点的产生.文章最后通过数值例子验证了这种方法的正确性和有效性.     

C^1有理三次可控保形插值曲线

一、引言给定插值数据点集{(x_i,y_i)}_(i-0)~n,在许多实际应用中(VLSI,CAD/CAM等),要求插值曲线除满足一定的光滑性条件外,还必须反映插值点集的整体几何性质。例如,通常要求单调(凸)数据产生的插值曲线是单调(凸)的。分段三次Hermite插值多项式是外形     

关于多项式保形插值理论及应用

保形插值在CAGD中有着重要的应用,本综述了多项式保形插值的理论及应用。     

A CLASS OF NEW NONCONFORMING FINITE ELEMENTS

A class of new nine-parameter nonconforming finite elements with convergent prop-erties are constructed in terms of the approach of interpolation methods in this paper. All of those finite elements can be viewed as the generalization of Zienkiewicz's finite element by dis-lurbing its parameters.     

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.     

半线性椭圆问题Petrov-Galerkin逼近及亏量迭代

本文考虑了二阶半线性椭圆问题的Petrov-Galerkin逼近格式,用双二次多项式空间作为形函数空间,用双线性多项式空间作为试探函数空间,证明了此逼近格式与标准的二次有限元逼近格式有同样的收敛阶.并且根据插值算子的逼近性质,进一步证明了半线性有限元解的亏量迭代序列收敛到Petrov-Galerkin解.     

Piecewise constructions of inverses of cyclotomic mapping permutation polynomials

Given a permutation polynomial of a large finite field, finding its inverse is usually a hard problem. Based on a piecewise interpolation formula, we construct the inverses of cyclotomic mapping permutation polynomials of arbitrary finite fields.     

THE DEFECT ITERATION OF THE FINITE ELEMENT FOR ELLIPTIC BOUNDARY VALUE PROBLEMS AND PETROV-GALERKIN APPROXIMATION

1.IntroductionFranketc.of.[l]establishedtheiterateddefectcorrectionschemeforfiniteelemelltofellipticboundaryproblems.FOrlinearellipticboundaryvalueproblem[2--5]havediscllssedtheefficiencyoftheschemebyusillgsuperconvergenceandasymptoticexpansion"lidertheco…     

有限元近似可计算的误差界

本文研究了有限元近似可计算的误差界，利用“二次插值过渡”方法，获得二维线性、双线性有限元和三维三线性有限元的新的插值常数估计值．理论分析和数值实验表明该结果是有效的，发展了P．Arbenz等人的工作．     

非协调有限元与本征值下界

与协调有限元相比,非协调有限元常常给出本征值下界.这种现象已经由很多数值例子观察到.但是理论上的证明最近才做的更多.以Morley非协调元作为更简单的范例,来说明这种理论为何成功.     

变系数抛物方程的有限元强校正格式

本文利用投影型插值和Ritz-Volterra投影研究一维变系数抛物方程的有限元方法,直接得到导数和位移的一个强校正格式.对于有限元解,分别对应力和位移获得整体的hk＋2和hk＋3阶的强结果.     

Design of Interpolating Biorthogonal Multiwavelet Systems with Compact Support

In this paper we characterize all totally interpolating biorthogonal finite impulse response (FIR) multifilter banks of multiplicity two, and provide a design framework for corresponding compactly supported multiwavelet systems with high approximation order. In these systems, each component of the analysis and synthesis portions possesses the interpolating property. The design framework is based on scalar filter banks, and examples with approximation order two and three are provided. We show that our multiwavelet systems preserve almost all of the desirable properties of the generalized interpolating scalar wavelet systems, including the dyadic-rational nature of the filter coefficients, equality of the flatness degree of the low-pass filters and the approximation order of the corresponding functions, and equality between the uniform samples of a signal and its projection coefficients for a given scale. This last property allows us to avoid the cumbersome prefiltering associated with standard multiwavelet systems. We also show that there are no symmetric totally interpolating biorthogonal multifilter banks of multiplicity two. Finally, we point out that our design framework incorporates a simple relationship between the multiscaling functions and multiwavelets that substantially simplifies the implementation of the system.