拟牛顿流的一种三变量域模型的有限元方法的数值分析

0．引言目前，涉及高温条件下材料蠕变性质的粘弹性流动问题已引起人们广泛的研究兴趣，不少文章讨论了如何对其进行数值求解（见［1］－－［41），首先，人们研究了较简单的仅以速度，压力两个变量来表述此现象的模型问题（如［1，2］）等．鉴于应力变量在材料性质方面的特殊重要性，最近J．Baxanzer等人在［3］中首次对应力满足幂函数规律的蠕变流研究了包含应力、速度和压力三种变量的模型问题的有限元逼近，当粘性的牛顿部分为零时（详见下述）在假定速度与应力、速度与压力有限元空间之间同时满足两种＊＊B条件以后，证明了有限元解…     

三维热传导型半导体问题的交替方向特征有限元方法

提出交替方向特征有限元方法,对电场位势方程采用混合元格式,对电子,空穴浓度方程采用交替方向特征有限元格式,对温度方程提出交替方向格式.应用向量积计算及先验估计理论和技巧,得到最佳的L2误差估计.     

常微方程组初值问题的连续有限元及守恒性

In this paper the continous finite element to solve initinal value problem for system of linear differential equations is used, and the absolute stability of the corresponding single step k-order hidden shceme is discussed. In the paper by simplified means, the superconvergence of finite element and one of it on the nodes are proved. Using the continuous finite element to solve linear Hamilton systems: Pt = Hq,qt = -Hp, the conservation of energy H(p,q) = 1/2 ap^2 bpq 1/2 cq^2 can be obtained. The computation shows that even if division is regular and the error of finite element Ph, qh is big, H(ph, qh) is almost equal H(p, q) in the range of computation accuracy.     

GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS

1. IntroductionIn the numerical simulation of the Navier-Stokes equations one encounters three seriousdifficulties in the case of large Reynolds numbers f the treatment of the incomPressibility con-dition divu = 0, the treatment of the noIilinear terms and the large time integration. For thetreatment of the incoInPressibility condition, one use the penalty method in the case of finiteelemellts [1--2l and for the treatmen of the noulinar terms and the large tfor integration, oneuse the nonlin…     

本征值有限元近似的一个抽象误差估计式

设T：LZ（fi）MLZ（fi）是自共轭全连续算子，SgCLZ（fi）是分片。次有限元空间，几：LZ、St是有限秩自共轭算子，IITh-Tllo、0（h、0）．考虑本征值问题：及其近似用（·，·）和DD·D【。·分别表示h（m中内积和范数·ID·卜F表示w认｝（m中范数，简记D卜队。为D卜卜·因为T自共轭全连续，所以它有可数无穷个本征值h，人，...人，...．其相应的本征函数（2丹构成完全标准直交系，所以VZELZ（m设几的本征值为A。l，汕。，...，汕n，相应的本征函数为如山，卜则。二1·不失一般性，可EitL。设tik一大干二＞，E是到Ah对应的本…     

二维非饱和土壤水分运动问题的半离散有限体积元模拟

本文利用基于重心对偶剖分的有限体积元法建立了二维非饱和土壤水分运动问题的数值逼近格式,讨论了离散有限体积元解的存在唯一性,并给出了最优误差估计的证明.最后给出数值算例,模拟结果表明,利用有限体积元格式来求解二维非饱和土壤水分运动问题是可靠的,且该格式具有稳定性和可实用性.     

特征值问题的自适应反迭代有限元算法

1引言设Ω∈R~2为Lipschitz单连通的有界闭区域,X为定义在Ω的Sobolev空间,a(·,·)和b(·,·)为X×X→C的有界双线性或半双线性泛函,考虑变分特征值问题:求(λ,u≠0)∈C×X使得a(u,v)=λb(u,u),(?)u∈X,其中a(·,·)满足X上的"V-强制性"条件或者连续的inf-sup条件,设M_h为Q区域上的正则三角形剖分,X_h∈X为定义在M_h有限元子空间,上述变分问题对应的有限元离散问题为:求(λ_h,u_h)∈R×X,u_h≠0使得     

二阶椭圆问题的一类广义有限元法

本文提出了求解二阶椭圆问题的一类广义有限元方法,分析了广义有限元方法的优越性,证明了二阶椭圆问题的广义有限元方法具有比标准的Galerkin有限元方法更高阶的收敛速度,根据插值算子的性质,进一步证明了有限元解的亏量迭代校正收敛到广义有限元解,并用数值例子说明广义有限元方法是有效的.     

自由边界问题的样条有限元法

本文讨论薄板弯曲自由边界问题的样条有限元法,对障碍问题和“弹塑性”弯曲问题,采用三次B样条元及二次多结点Hermite元,比较其优劣,证明离散解的收敛性.     

鱼骨形网格上二阶方程混合元的超收敛

In this paper, superconvergence of the lowest order Raviart-Thomas mixed finite element approximation for second order Neumann boundary value problem on fishbone shape meshes is analyzed. The main term of the error between the exact solution and the finite element interpolating function is determined by Bramble-Hilbert lemma on the individual finite element. A part of the main term of the error on two adjacent finite elements can be cancelled along the special direction, and thus the higher order error estimate is obtained on the whole domain by summation. Compared with the general finite element error estimate,the convergence rate can be increased from order one to order two in L2-norm by postprocessing superconvergence technique.     

THE UPWIND FINITE ELEMENT SCHEME AND MAXIMUM PRINCIPLE FOR NONLINEAR CONVECTION-DIFFUSION PROBLEM

In this paper, a kind of partial upwind finite element scheme is studied for twodimensional nonlinear convection-diffusion problem. Nonlinear convection term approximated by partial upwind finite element method considered over a mesh dual to the triangular grid, whereas the nonlinear diffusion term approximated by Galerkin method. A linearized partial upwind finite element scheme and a higher order accuracy scheme are constructed respectively. It is shown that the numerical solutions of these schemes preserve discrete maximum principle. The convergence and error estimate are also given for both schemes under some assumptions. The numerical results show that these partial upwind finite element scheme are feasible and accurate.     

OPTIMAL ERROR ESTIMATES FOR NEDELEC EDGE ELEMENTS FOR TIME-HARMONIC MAXWELL'S EQUATIONS

In this paper, we obtain optimal error estimates in both L^2-norm and H（curl）-norm for the Nedelec edge finite element approximation of the time-harmonic Maxwell＇s equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the L^2 error estimates into the L^2 estimate of a discrete divergence-free function which belongs to the edge finite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For Nedelec＇s second type elements, we present an optimal convergence estimate which improves the best results available in the literature.     

多孔介质驱动问题的混合元最小二乘特征有限元方法及其收敛分析

１．引言多孔介质二相驱动问题的数学模型是偶合的非线性偏微分方程组的初边值问题．该问题可转化为压力方程和浓度方程［１－４］．浓度方程一般是对流占优的对流扩散方程,它的对流速度依赖于比浓度方程的扩散系数大得多的Ｆａｒｃｙ速度．因此Ｄａｒｃｙ速度的求解精度直接影响着浓度的求解精度．为了提高速度的求解精度,７０年代Ｐ．Ａ．Ｒａｖｉａｔ和Ｊ．Ｍ．Ｔｈｏｍａｓ提出混合有限元方法［５］．Ｊ．ＤｏｕｇｌａｓＪｒ,Ｔ．Ｆ．Ｒｕｓｓｅｌｌ,Ｒ．Ｅ．Ｅｗｉｎｇ,Ｍ．Ｆ．Ｗｈｅｅｌｅｒ［１］－［４］,［９］,［１２］袁…     

TWO-LEVEL MULTISCALE FINITE ELEMENT METHODS FOR THE STEADY NAVIER-STOKES PROBLEM

In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.     

AN ADAPTIVE INVERSE ITERATION FEM FOR THE INHOMOGENEOUS DIELECTRIC WAVEGUIDES

We introduce an adaptive finite element method for computing electromagnetic guided waves in a closed, inhomogeneous, pillared three-dimensional waveguide at a given frequency based on the inverse iteration method. The problem is formulated as a generalized eigenvalue problems. By modifying the exact inverse iteration algorithm for the eigenvalue problem, we design a new adaptive inverse iteration finite element algorithm. Adaptive finite element methods based on a posteriori error estimate are known to be successful in resolving singularities of eigenfunctions which deteriorate the finite element convergence. We construct a posteriori error estimator for the electromagnetic guided waves problem. Numerical results are reported to illustrate the quasi-optimal performance of our adaptive inverse iteration finite element method.     

美式期权定价问题的数值方法

本文研究美式股票看跌期权定价问题的数值方法。通过将问题转化为等价的变分不等式方程，分别建立了半离散和全离散有限元逼近格式。并给出了有限元解的收敛性和稳定性分析。数值实验表明本文算法是一个高效和收敛的算法。     

用新的逼近空间求解奇性问题

用新的逼近空间求解奇性问题梁国平，孔林，何江衡（中国科学院数学研究所）ＡＮＥＷＦＩＮＩＴＥＥＬＥＭＥＮＴＳＰＡＣＥＦＯＲＳＩＮＧＵＬＡＲＩＴＹＰＲＯＢＬＥＭＳ￥ＬｉａｎｇＧｕｏ－ｐｉｎｇ；ＫｏｎｇＬｉｎ；ＨｅＪｉａｎｇ－ｈｅｎｇ（Ｉｎｓｔｉｔｕｔｅｏ...     

一类四阶抛物型积分-微分方程的混合间断时空有限元法

构造四阶抛物型积分-微分方程的混合间断时空有限元格式,利用混合有限元方法将高阶方程降阶,利用空间连续而时间允许间断的时空有限元方法离散方程,证明离散解的稳定性,存在唯一性和收敛性.     

美式债券期权定价问题的有限元方法

The aim of this paper is to investigate the finite element methods for pricing the American put option on bonds. Based on a new variational inequality equation for the option pricing problems, both semidiscrete and fully discretized finite element approximation schemes are established. It is proved that the finite element methods are stable and convergent under L2 and H^1 norms.     

CONVERGENCE ANALYSIS FOR A NONCONFORMING MEMBRANE ELEMENT ON ANISOTROPIC MESHES

Regular assumption of finite element meshes is a basic condition of most analysis offinite element approximations both for conventional conforming elements and nonconform-ing elements.The aim of this paper is to present a novel approach of dealing with theapproximation of a four-degree nonconforming finite element for the second order ellipticproblems on the anisotropic meshes.The optimal error estimates of energy norm and L~2-norm without the regular assumption or quasi-uniform assumption are obtained based onsome new special features of this element discovered herein.Numerical results are givento demonstrate validity of our theoretical analysis.