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.     

求解双曲守恒律方程的三阶修正模板WENO格式北大核心CSCD

为了降低经典的三阶加权本质无振荡(WENO)格式的数值耗散,提出了一种新的三阶WENO格式的修正模板近似方法.改进了经典WENO-JS格式中各候选模板上数值通量的一阶多项式逼近,通过加入二次项使模板逼近达到三阶精度.计算了相应的候选通量,并且通过引入可调函数φ(x),使得新的格式具有ENO性质.最后给出了一系列数值算例,证明了该方法的有效性.     

多项式基函数法

提出一种新型的数值计算方法--基函数法.此方法直接在非结构网格上离散微分算子,采用基函数展开逼近真实函数,构造出了导数的中心格式和迎风格式,取二阶多项式为基函数,并采用通量分裂法及中心格式和迎风格式相结合的技术以消除激波附近的非物理波动,构造出数值求解无粘可压缩流动二阶多项式的基函数格式,通过多个二维无粘超音速和跨音速可压缩流动典型算例的数值计算表明,该方法是一种高精度的、对激波具有高分辨率的无波动新型数值计算方法,与网格自适应技术相结合可得到十分满意的结果.     

改进Lagrange插值多项式误差上界的系数

当用Lagrange插值多项式逼近函数时,重要的是要了解误差项的性态.本文研究具有等距节点的Lagrange插值多项式,估计了Lagrange插值多项式逼近函数误差项的上界,改进了小于5次Lagrange插值多项式逼近函数误差界的系数.     

单隐层神经网络与最佳多项式逼近

研究单隐层神经网络逼近问题．以最佳多项式逼近为度量,用构造性方法估计单隐层神经网络逼近连续函数的速度．所获结果表明:对定义在紧集上的任何连续函数,均可以构造一个单隐层神经网络逼近该函数,并且其逼近速度不超过该函数的最佳多项式逼近的二倍．     

二元Lagrange三角插值多项式在Orlicz空间内的逼近

研究了二元函数用一种组合型的三角插值多项式算子逼近的问题.借助连续模这一工具,给出了这类三角插值多项式在Orlicz空间内的逼近定理.     

Lagrange四边形单位分解有限元法的最优误差分析

用构造最优局部逼近空间的方法对Lagrange型四边形单位分解有限元法进行了最优误差分析.单位分解取Lagrange型四边形上的标准双线性基函数,构造了一个特殊的局部多项式逼近空间,给出了具有2阶再生性的Lagrange型四边形单位分解有限元插值格式,从而得到了高于局部逼近阶的最优插值误差.     

K 泛函与逼近阶(Ⅰ)

<正> K 泛函的概念首先是由 Peetre 引入的.DeVore 等利用 K 泛函这一工具成功地得到了各种用函数的光滑模来表示的逼近阶.本文的目的是要推广这一工具,使它可以用来考察函数及其导函数的同时逼近(以下简称同时逼近)问题.我们利用这一推广了的工具,对于用样条、三角多项式以及代数多项式线性同时逼近的阶,得到了比较完善的结果.     

渐近于Hermite多项式的双正交系统

本文利用渐近于Gauss函数的函数类?,给出渐近于Hermite正交多项式的一类Appell多项式的构造方法,使得该序列与?的n阶导数之间构成了一组双正交系统.利用此结果,本文得到多种正交多项式和组合多项式的渐近性质.特别地,由N阶B样条所生成的Appell多项式序列恰为N阶Bernoulli多项式.从而,Bernoulli多项式与B样条的导函数之间构成了一组双正交系统,且标准化之后的Bernoulli多项式的渐近形式为Hermite多项式.由二项分布所生成的Appell序列为Euler多项式,从而,Euler多项式与二项分布的导函数之间构成一组双正交系统,且标准化之后的Euler多项式渐近于Hermite多项式.本文给出Appell序列的生成函数满足的尺度方程的充要条件,给出渐近于Hermite多项式的函数列的判定定理.应用该定理,验证广义Buchholz多项式、广义Laguerre多项式和广义Ultraspherical(Gegenbauer)多项式渐近于Hermite多项式的性质,从而验证超几何多项式的Askey格式的成立.     

简支板边界条件下四阶问题基于降阶格式的一种有效的谱Galerkin逼近

本文研究了简支板边界条件下四阶问题基于降阶格式的一种有效的谱Galerkin逼近.通过引入一个辅助函数和适当的Sobolev空间,将原问题化为两个耦合的二阶问题,建立其弱形式和相应的离散格式,利用Lax-Milgram定理和投影算子的逼近性质,我们证明了弱解和逼近解的存在唯一性以及它们之间的误差估计.再利用Legendre多项式的正交性质构造了一组适当的基函数,推导了离散格式基于张量积的矩阵形式.最后,我们给出了一些数值算例,数值结果验证了算法的有效性和理论结果的正确性.     

Superconvergence of a finite element method for the biharmonic equation

Ciarlet‐Raviart's scheme is a finite element method for solving the mixed formulation of the biharmonic equation. So far, there has been no superconvergence for the vorticity from this method if a general rectangular mesh is used. In this article, we deal with the biquadratic elements under the uniform rectangular mesh and prove for the first time a superconvergence for the vorticity. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 420–427, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10010     

各向异性双二次元的自然超收敛性结果

The paper studies the convergence and the superconvergence of the biquadratic finite element for Poisson' problem on anisotropic meshes.By detailed analysis,it shows that the biquadratic finite element is anisotropically superconvergent at four Gauss points in the element.     

ADI quadratic finite volume element methods for second order hyperbolic problems

We use the biquadratic elements to develop an alternating direction implicit (ADI) finite volume element method for second order hyperbolic problems in two spatial dimensions. The optimal H ¹-norm error estimate of second order accuracy is proved. Numerical experiments that corroborate the theoretical analysis are also presented.     

A symmetric characteristic FVE method with second order accuracy for nonlinear convection diffusion problems

The finite volume element (FVE) methods used currently are essentially low order and unsymmetric. In this paper, by biquadratic elements and multistep methods, we construct a second order FVE scheme for nonlinear convection diffusion problem on nonuniform rectangular meshes. To overcome the numerical oscillation, we discretize the problem along its characteristic direction. The choice of alternating direction strategy is critical in this paper, which guarantees the high efficiency and symmetry of the discrete scheme. Optimal order error estimates in H¹$H^1$-norm are derived and a numerical example is given at the end to confirm the usefulness of the method.     

Biquadratic finite volume element method based on optimal stress points for second order hyperbolic equations

Based on optimal stress points, we develop a full discrete finite volume element scheme for second order hyperbolic equations using the biquadratic elements. The optimal order error estimates in L^∞(H¹), L^∞(L²) norms are derived, in addition, the superconvergence of numerical gradients at optimal stress points is also discussed. Numerical results confirm the theoretical order of convergence. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles

The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.     

A Finite-element Front-tracking Enthalpy Method for Stefan Problems

A finite-element front-tracking method, which avoids explicittreatment of the jump condition on the phase boundary, is proposed.This method is based on the discretization of a weak enthalpyformulation with isoparametric space—time finite elements.A one-dimensional two-phase problem and a two-dimensional one-phaseproblem are solved by this method. Then the method is appliedto a generalized Stefan problem—the spot-welding problemand extended to the alloy-solidification problem. Numerical experiments show that this method is convergent andunconditionally stable and that second-order convergence canbe expected if the biquadratic elements are used for one-dimensionalproblems. The effectiveness of this method is, in particular,shown in solving the last two problems.     

解Poisson方程的基于应力佳点的双二次元有限体积法

本文提出了求解Poisson方程的一种新的双二次元有限体积法.新方法与通常的双二次元有限体积法作对偶剖分的方式不同,其主要特点是取应力佳点(Gauss点)作为对偶单元的节点,试探函数空间取双二次有限元空间,检验函数空间取相应于对偶剖分的分片常数函数空间.证明了新方法具有最优的H~1模和L~2模误差估计,讨论了在应力佳点数值梯度的超收敛性估计,并通过数值实验验证了理论分析的结果.     

Elements with prime and small indices in bicyclic biquadratic number fields

We give necessary and sufficient conditions for the existence of primitive algebraic integers with index A in totally complex bicyclic biquadratic number fields where A is an odd prime or a positive rational integer at most 10. We also determine all these elements and prove that there are infinitely many totally complex bicyclic biquadratic number fields containing elements with index A.     

关于Ciarlet-Raviart混合有限元法的一个注记

本文推广解双调合方程的Ciarlet-Raviart混合有限元方案：用二次元逼近流函数φ．一次元逼近涡度-Δφ．在拟一致三角形剖分的条件下,证明了推广方案具有φ和-Δφ都用二次元逼近的标准Ciarlet－Raviart方案同样的精度阶．