一类非线性抛物型方程的混合元方法及其误差估计

当前,混合元方法求解微分方程已广泛应用于椭圆边值问题,其理论分析也相当完善.在[1]—[2],[4]中,对线性椭圆问题的混合元方法进行了讨论,[5]对拟线性椭圆问题进行了理论分析.相对而言,关于发展型方程的混合元方法,特别是对非线性问题,其理论分析仍不够完善,缺乏统一的论述.本文对下列非线性抛物问题:     

退化椭圆问题的最小二乘混合元逼近

１．引言我们将考虑具有退化系数的椭圆问题其中 Ω为 ＩＲ２中的一个凸多边形区域,定义为这里的ｇ＞０为分片线性连续函数．从物理背景来看,问题（１．１）来源于轴对称共振结构中的电磁场研究．Ｍａｒｉｎｉ,Ｐｉｅｔｒａ（１９９５）［４］研究了问题（１．１）的混合有限元逼近,并得到了最优误差估计． 此文,我们采用一种新的混合元,即最小二乘混合元方法［５］;对退化问题（１．１）进行逼近,利用插值投影证明了近似解具有最优阶精度的收敛性．比较起经典混合元方法来,最小二乘混合元方法有两个优越性：有限元空间不必满足Ｌ…     

线性椭圆问题最低次混合有限元方法的最优最大模误差估计

本文对线性椭圆问题的最低次混合元方法提出了构造混合元空间的充分条件,并建立了新的插值算子.据此得到了混合元解,伴随向量函数及其散度的最优最大模误差估计.     

退化椭圆问题的最小二乘混合有限元方法

本文研究了电磁场中关于共振现象的一类退化的椭圆问题 ,提出了最小二乘混合有限元方法 .这一方法的好处是可以去掉传统混合元空间的LBB条件所得到的系数矩阵是对称正定的 ,使得法语解更加方便 .本文得到了最小二乘混合有限元方法的L2 和H1估计 .     

可压缩流驱动问题的一类特征混合元方法

提出数值求解可压缩流驱动问题的一个新的特征混合元 .用一个新的混合元方法解抛物型的压力方程 .新方法有两个显著的特点 :一是混合元方程具有对称正定性 ;二是压力场与流场可以分离求解 .使用特征方法解具有强对流特点的浓度方程 .在一般性条件下 ,证明了近似解具有最优阶精度的收敛性     

椭圆型对流占优扩散方程的泡函数—混合有限元方法(英文)

把泡函数有限元方法和混合有限元方法进行耦合,从而利用泡函数-混合有限元方法来求解椭圆型对流占优扩散方程,该方法不仅可以同时高精度逼近浓度(u)和浓度变化率(#u),还有效避开了传统混合有元方法中苛刻的L-BB条件,数值解的稳定性也得到了改善.     

求解stokes问题的一个新的非协调混合元格式

构造一个新的非协调混合元格式求解stokes问题,证明了该元满足B-B条件,并且有2阶收敛速度.     

高次三角形有限元的超收敛问题

关于二维区域二阶线性椭圆问题的有限元求解,[1,2]各自独立地对低次奇妙族矩形元采用单元合并技巧,获得能量的近似正交性(或称插值误差的第一弱估计),从而获得应力佳点定理.若获得更佳形式的能量正交性(或称插值误差的第二弱估计),则可获得位移佳点定理.运用以上方法,[1—8]解决了奇妙族矩形任意次元及三角形线元、二次元     

Reid-Zhi符号数值混合消元方法的一个应用

将Reid和Zhi提出的符号数值混合消元方法应用于求解多项式优化问题,将多项式优化问题转化为矩阵最小特征值求解问题,并在Maple软件中实现了算法.     

求解线性Sobolev方程的分裂型最小二乘混合元方法

本文通过引入适当的最小二乘极小化泛函,对一类线性Sobolev方程提出了两种分裂型最小二乘混合元格式,格式最大优点在于将耦合的方程组系统分裂成两个独立的子系统,进而极大降低了原问题求解的难度和规模,理论分析表明格式对原未知量及新引入的未知通量分别具有最优阶L2(Ω)模误差估计和次优阶H(div;Ω)模误差估计.数值试验很好的验证了这一点.     

Long-step primal-dual target-following algorithms for linear programming

In this paper we propose a long-step target-following methodology for linear programming. This is a general framework, that enables us to analyze various long-step primal-dual algorithms in the literature in a short and uniform way. Among these are long-step central and weighted path-following methods and algorithms to compute a central point or a weighted center. Moreover, we use it to analyze a method with the property that starting from an initial noncentral point, generates iterates that simultaneously get closer to optimality and closer to centrality.This work is completed with the support of a research grant from SHELL.The first author is supported by the Dutch Organization for Scientific Research (NWO), grant 611-304-028.The fourth author is supported by the Swiss National Foundation for Scientific Research, grant 12-34002.92.     

Trefftz,collocation, and other boundary methods—A comparison

In this article we survey the Trefftz method (TM), the collocation method (CM), and the collocation Trefftz method (CTM). We also review the coupling techniques for the interzonal conditions, which include the indirect Trefftz method, the original Trefftz method, the penalty plus hybrid Trefftz method, and the direct Trefftz method. Other boundary methods are also briefly described. Key issues in these algorithms, including the error analysis, are addressed. New numerical results are reported. Comparisons among TMs and other numerical methods are made. It is concluded that the CTM is the simplest algorithm and provides the most accurate solution with the best numerical stability. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Volume preserving RK methods for linear systems

1. IntroductionIn recent yearss there has been a great interest in constructing numerical integrationschemes for ODEs in such a way that some qualitative geometrical properties of the solutionof the ODEs are exactly preserved. R.th[ll and Feng Kang[2'31 has proposed symplectic algorithms for Hamiltollian systems, and since then st ruct ure s- preserving me t ho ds fordynamical systems have been systematically developed[4--7]. The symplectic algorithms forHamiltonian systems, the volume-pre…     

求解离散不适定问题的正则化GMERR方法

迭代极小残差方法是求解大型线性方程组的常用方法, 通常用残差范数控制迭代过程.但对于不适定问题, 即使残差范数下降, 误差范数未必下降. 对大型离散不适定问题,组合广义最小误差(GMERR)方法和截断奇异值分解(TSVD)正则化方法, 并利用广义交叉校验准则(GCV)确定正则化参数,提出了求解大型不适定问题的正则化GMERR方法.数值结果表明, 正则化GMERR方法优于正则化GMRES方法.     

A note on the biasing of Newton's direction

This note deals with the geometric interpretation of the Levenberg-Marquardt search direction when the augmented Hessian is not positive definite.     

投入产出技术在计划平衡中的理论与应用

在用投入产出技术作计划平衡时，目前一般采用最终产品法、总产品法及国民收入法等．本文从理论上研究了这些方法的可行性问题，并在此基础上提出一个较理想的综合法．最后附有实例并说明综合法的现实意义．     

Diagonalized multiplier methods and quasi-Newton methods for constrained optimization

Two approaches to quasi-Newton methods for constrained optimization problems inR ⁿ are presented. These approaches are based on a class of Lagrange multiplier approximation formulas used by the author in his previous work on Newton's method for constrained problems. The first approach is set in the framework of a diagonalized multiplier method. From this point of view, a new update rule for the Lagrange multipliers which depends on the particular quasi-Newton method employed is given. This update rule, in contrast to most other update rules, does not require exact minimization of the intermediate unconstrained problem. In fact, the optimal convergence rate is attained in the extreme case when only one step of a quasi-Newton method is taken on this intermediate problem. The second approach transforms the constrained optimization problem into an unconstrained problem of the same dimension.The author would like to thank J. Moré and M. J. D. Powell for comments related to the material in Section 13. He also thanks J. Nocedal for the computer results in Tables 1–3 and M. Wright for the results in Table 4, which were obtained via one of her general programs. Discussions with M. R. Hestenes and A. Miele regarding their contributions to this area were very helpful. Many individuals, including J. E. Dennis, made useful general comments at various stages of this paper. Finally, the author is particularly thankful to R. Byrd, M. Heath, and R. McCord for reading the paper in detail and suggesting many improvements.This work was supported by the Energy Research and Development Administration, Contract No. E-(40-1)-5046, and was performed in part while the author was visiting the Department of Operations Research, Stanford University, Stanford, California.     

s个几乎相等的素数的k次方和(Ⅰ)

假定pθ‖k,当p=2,2|k时，γ=θ 2；其它情况时，γ=θ 1。而R＝П(p-1)|kp^γ。本文在GRH（广义Riemann假设下），证明了当s=2^k 1,1≤k≤11时，任何足够大的整N≡s(modR)都可以表示为s个几乎相等的素数的k次方程。     

A comparison of some ODE solvers which require Jacobian evaluations

A variety of third-order ODE solvers which have a minimum configuration (i.e. minimum work per step) have been numerically tested and the results compared. They include implicit and explicit processes, and share the property that a Jacobian matrix must be evaluated at least once during the integration. Some of these processes have not been previously described in the literature.     

Stabilization of algebraic multilevel iteration methods; additive methods

There exist two main versions of preconditioners of algebraic multilevel type, the additive and the multiplicative methods. They correspond to preconditioners in block diagonal and block matrix factorized form, respectively. Both can be defined and analysed as recursive two-by-two block methods. Although the analytical framework for such methods is simple, for many finite element approximations it still permits the derivation of the strongest results, such as optimal, or nearly optimal, rate of convergence and optimal, or nearly optimal order of computational complexity, when proper recursive global orderings of node points have been used or when they are applied for hierarchical basis function finite element methods for elliptic self-adjoint equations and stabilized in a certain way. This holds for general elliptic problems of second order, independent of the regularity of the problem, including independence of discontinuities of coefficients between elements and of anisotropy. Important ingredients in the methods are a proper balance of the size of the coarse mesh to the finest mesh and a proper solver on the coarse mesh. This paper presents in a survey form the basic results of such methods and considers in particular additive methods. This method has excellent parallelization properties. This revised version was published online in June 2006 with corrections to the Cover Date.