非线性Sobolev方程Galerkin解法的后处理与超收敛

研究非线性Sobolev方程Galerkin解法的后处理与超收敛．对半离散及全离散格式,证明了当有限元空间次数,r≥2时,有限元解经过后处理,H1-模和L2-模误差估计可分别提高一阶．     

Cahn-Hilliard方程的有限元分析

建立了求解非线性发展型Cahn-Hilliard方程的有限元方法,借助于一个双调和问题的有限元投影逼近,给出了最优阶L_2模误差估计。特别对于3次Hermite型有限元,导出了L_∞模和W_∞~1模的最优阶误差估计和导数逼近的超收敛结果。     

四阶椭圆问题有限元导数恢复技术与超收敛性

1引言有限元导数恢复技术是近年来发展起来的计算有限元导数并获得导数逼近超收敛性的一种新的后处理技术.对于一维和二维区域上的二阶椭圆边值问题,文[1,2]提出了Z-Z小片插值技术,得到了有限元导数逼近在小片恢复区域上的一阶超收敛结果和剖分节点处二阶强超收敛性;文[3,4]则建立了更为实用的小片插值恢复技术并得到与文[1,2]相平行的超收敛结果;文[5]对两点边值问题构造了一种积分形式的导数恢复公式,利用这个公式可获得剖分节点处有限元导数逼近的O(h~(2k))阶超收敛估计.本文将对一维四阶椭圆     

Sobolev方程的最小二乘Galerkin有限元法

本文对于Sobolev方程提出并分析了两种新型数值方法：最小二乘Galerkin有限元法．这种方法的优越性在于不需要验证LBB条件,可以更好的选择有限元空间．误差估计表明在L~2(Ω))~2×L~2(Ω)范数意义下,这两种方法均具有最优收敛阶,并且关于时间分别具有一阶精确度和二阶精确度．     

非线性湿气迁移方程的超收敛分析

本文用混合有限元方法研究一般的非线性湿气迁移方程.利用双线性元Q₁₁和零阶Raviart-Thomas元(Q₁₀×Q₀₁)证明方程的超收敛性.利用这两个单元插值算子的性质和平均值技巧,得到了方程半离散格式的O(h²)阶超收敛结果.对于方程线性化的Crank-Nicolson(C-N)全离散格式,得到了具有O(h²+τ²)阶的超收敛结果,这里h是空间剖分参数,τ是时间步长.该方法说明如果线性化问题有超收敛性,那么对应的非线性问题有同样的超收敛性.最后,给出数值算例,证实了理论分析的正确性和方法的有效性.     

分数阶波方程的数值解法

首先,把分数阶波方程转换成等价的积分-微分方程;然后,利用带权的分数阶矩形公式和紧差分算子分别对时间和空间方向进行离散.证明了当权重为1/2时,时间方向的收敛阶为α,其中α(1α2)为Caputo导数的阶数.利用Gronwall不等式,证明了数值格式的收敛性和稳定性.数值例子进一步表明了数值格式的有效性.     

等周等积定理的两个再推广

文 [1 ]证明了“对任意直角三角形总存在一个矩形 ,使得矩形与三角形的周长和面积比均等于常数 k( k≥ 1 )”.本文作如下推广 :定理 1　对任意梯形 ,总存在一个矩形 ,使得矩形与这梯形的周长和面积比均等于常数 k( k≥ 1 ) .证明　在梯形 ABCD中 ,设 AD∥ BC,且 AD =a,BC=b,两腰 AB=c,CD =d,要求长为 x,宽为 y的矩形 ,使得方程组2 ( x + y) =k( a + b + c + d) ,xy =k .12 ( a + b) dsin C.有正解 ,仅需证明方程t2 - k( a + b + c + d)2 t+ 12 k( a + b) dsin C= 0有正解 .事实上 ,由于 k≥ 1 ,　 0 相似文献   

各向异性EQrot1非协调元高精度分析的一般格式

本文在各向异性网格下讨论了一般二阶椭圆方程的EQrot1非协调有限元逼近.利用Taylor展开,积分恒等式和平均值技巧导出了一些关于该元新的高精度估计.再结合该元所具有的二个特殊性质:(a)当精确解属于H3(Ω)时,其相容误差为O(h2)阶比它的插值误差高一阶;(b)插值算子与Ritz投影算子等价,得到了在能量模意义下O(h2)阶的超逼近性质.进而,借助于插值后处理技术给出了整体超收敛的一般估计式.     

拟线性Sobolev方程Carey元解的高精度分析

在一种半离散格式下讨论了拟线性Sobolev方程Carey元的超收敛及外推.根据Carey元的构造证明了其有限元解的线性插值与三角形线性元的解相同,再结合线性元的高精度分析和插值后处理技巧导出了超逼近和整体超收敛及后验误差估计.与此同时,根据线性元的误差渐近展开式,构造了一个新的辅助问题,得到了比传统的有限元误差高三阶的外推结果.     

Sobolev方程的等参有限元法

用等参有限元去逼近曲边区域可以使误差阶不受损失,达到和凸多边形区域上同样的收敛阶.本文研究Sobolev方程的等参有限元方法,在半离散和向后欧拉全离散格式下,分别讨论真解和有限元解之间的误差估计,最后用数值算例进一步验证了理论结果.     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

Qualitative Analysis of Bifurcating Solutions in the Lengyel-Epstein Model

One of the most fundamental problems in theoretical biology is to explain the mechanisms by which patterns and forms are created in the＇living world. In his seminal paper ＂The Chemical Basis of Morphogenesis＂, Turing showed that a system of coupled reaction-diffusion equations can be used to describe patterns and forms in biological systems. However, the first experimental evidence to the Turing patterns was observed by De Kepper and her associates（1990） on the CIMA reaction in an open unstirred reactor, almost 40 years after Turing＇s prediction. Lengyel and Epstein characterized this famous experiment using a system of reaction-diffusion equations. The Lengyel-Epstein model is in the form as follows     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

(0,1 ;0)-INTERPOLATION ON INFINITE INTERVAL (-∞, +∞)

In this paper, we study the explicit representation and convergence of (0, 1; 0)-interpolation on infinite interval, which means to determine a polynomial of degree ≤ 3n - 2 when the function values are prescribed at two set of points namely the zeros of Hn(x) and H′n(x) and the first derivatives at the zeros of H′n(x).     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

CONTENTS OF VOLUME 30

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Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].