相似单元的通解和有限相似单元法

在有限元方法中,将区域剖分为有限个或者无限个相似单元而构成一个组合单元,用这个方法可以计算椭圆型微分方程边值问题解的奇性和在无界区域上求解它们。在这篇文章中我们得到了问题的通解,给出了计算组合刚度矩阵的公式,这些结果适用于包括拉普拉斯方程和线性弹性力学方程组在内的一类椭圆型微分方程和方程组,而对区域的剖分除去相似性之外没有任何限制。     

各向异性外问题的Schwarz 交替法及其收敛性和误差估计

本文对于无界区域各向异性常系数椭圆型偏微分方程研究了一种基于自然边界归化的Schwarz交替法.利用极值原理证明了在连续情形最大模意义下的几何迭代收敛性,通过选取适当的共焦椭圆边界利用Fourier分析获得了不依赖各向异性程度的最优的迭代收缩因子,还在离散情形最大模意义下证明了几何收敛性,而且进一步得到了误差估计,最后,数值结果证实了迭代收缩因子和误差估计的正确性,表明了该方法在无界Ⅸ域上求解各向异性椭圆型偏微分方程的优越性.     

在具有局部内存与共享主存的并行机上并行求解线性方程组

求解线性方程组是解许多问题的核心,因此有效地求解线性方程组在科学与工程计算中是非常重要的.并行计算机的问世,使求解问题的速度和解题规模大幅度地提高.同时也使计算方法产生了变化.在传统的串行机上,LINPACK数学软件是求解线性方程组的有效软件包,然而在并行机上求解此问题,就需要设计出适合该机的并行算法.算法的优劣会对并行机的效率产生很大的影响,这里考虑的重点是并行计算一矩阵A的LU分解,亦即存在一排列矩阵P,使AP=LU.由于是在具有局部内存与共享主存的并行机上求解问题,因此算法的设计要有分布式计算的特点,又要利用共享主存的     

奇摄动四阶椭圆型偏微分方程

本文用微分不等式理论讨论了一类奇摄动四阶椭圆型偏微分方程,得到了在整个区域上一致有效的渐近展开式.     

数学物理中的总体和局部场GL方法

为了求解物理化学生物材料和金融中的微分方程,提出了一种总体(Global)和局部(Local)场方法.微分方程的求解区域可以是有限域,无限域,或具曲面边界的部分无限域.其无限域包括有限有界不均匀介质区域.其不均匀介质区域被分划为若干子区域之和.在这含非均匀介质的无限区域,将微分方程的解显式地表示为在若干非均匀介质子区域上和局部子曲面的积分的递归和.把正反算的非线性关系递归地显式化.在无限均匀区域,微分方程的解析解被称为初始总体场.微分方程解的总体场相继地被各个非均匀介质子区域的局部散射场所修正.这种修正过程是一个子域接着另个子域逐步相继地进行的.一旦所有非均匀介质子区域被散射扫描和有限步更新过程全部完成后,微分方程的解就获得了.称其为总体和局部场的方法,简称为GL方法.GL方法完全地不同于有限元及有限差方法,GL方法直接地逐子域地组装逆矩阵而获得解.GL方法无需求解大型矩阵方程,它克服了有限元大型矩阵解的困难.用有限元及有限差方法求解无限域上的微分方程时,人为边界及其上的吸收边界条件是必需的和困难的,人为边界上的吸收边界条件的不精确的反射会降低解的精确度和毁坏反算过程.GL方法又克服了有限元和有限差方法的人为边界的困难.GL方法既不需要任何人为边界又不需要任何吸收边界条件就可以子域接子域逐步精确地求解无限域上的微分方程.有限元和有限差方法都仅仅是数值的方法,GL方法将解析解和数值方法相容地结合起来.提出和证明了三角的格林函数积分方程公式.证明了当子域的直经趋于零时,波动方程的GL方法的数值解收敛于精确解.GL方法解波动方程的误差估计也获得了.求解椭圆型,抛物线型,双曲线型方程的GL模拟计算结果显示出我们的GL方法具有准确,快速,稳定的许多优点.GL方法可以是有网,无网和半网算法.GL方法可广泛应用在三维电磁场,三维弹塑性力学场,地震波场,声波场,流场,量子场等方面.上述三维电磁场等应用领域的GL方法的软件已经由作者研制和发展了。     

二维椭圆型偏微分方程的反势问题

将Radon变换及其反投影变换原理应用于二维椭圆型偏微分方程反势问题的求解，从另一个角度解决了小扰动情况下椭圆型偏微分方程的反势问题．     

解单障碍问题的非重叠区域分解法

1．引言在实际中的许多物理问题、工程问题以及各类经济平衡问题都可用变分不等式来描述．本文考虑这类问题的数值解．众所周知，区域分解法的思想可朔源到19世纪70年代提出的Schwarz交替法，但直到本世纪中期才用于数值计算．而真正获得发展还是在近十几年．由于并行机与并行算法的发展，使得Schwarz算法的优良并行性能得以开发利用，从而使得这种区域分解新技术不仅应用于偏微分方程数值解，而且广泛应用于其它各类科学与工程计算问题．近几年来，重叠型区域分解已经被成功地应用于求解椭圆型变分不等式，早期的结果见［6］．我们还可从…     

六边形Fourier谱方法

首先,建立了晶格Fourier分析的一般理论,并具体研究了六边形区域上周期函数的数值逼近.在此基础上,提出了六边形区域上的椭圆型偏微分方程的周期问题求解的六边形Fourier谱方法,设计了相应谱格式快速实现算法,建立了Fourier谱方法的稳定性与收敛性理论.同方形区域上的经典Fourier谱方法一样,六边形Fourier谱方法可以充分利用快速Fourier变换,并具备了"无穷阶"的谱收敛速度.     

多维区域上椭圆型方程Dirichlet问题的勒让德多小波多层扩充算法

用小波伽辽金方法求解多维区域上椭圆型方程齐次Dirichlet问题,构造了近似解空间的两个等价的勒让德多小波基,使得快速求解离散后的线性方程组的多层扩充算法得以实现.数值算例表明该算法是有效的.     

半导体问题的交替方向有限元方法

该文用交替方向有限元方法求解半导体问题的Energy Trans port (ET)模型。对模型中椭圆型的电子位势方程采用交替方向迭代法，对流占优扩散的电子浓度和空穴浓度方程采用特征交替方向有限元方法，热传导方程利用Patch逼近采用交替方向有限元方法求解。利用微分方程的先验估计理论和技巧，分别得到了椭圆型方程和抛物型方程的最优H+1和L+2误差估计。     

Non-Conforming Domain Decomposition with Hybrid Method

We present a non-conforming domain decomposition technique for solving elliptic problems with the finite element method. Functions in the finite element space associated with this method may be discontinuous on the boundary of subdomains. The sizes of the finite meshes, the kinds of elements and the kinds of interpolation functions may be different in different subdomains. So, this method is more convenient and more efficient than the conforming domain decomposition method. We prove that the solution obtained by this method has the same convergence rate as by the conforming method, and both the condition number and the order of the capacitance matrix are much lower than those in the conforming case.     

A priori error analysis of the mortar finite element method for variational inequalities

The mortar finite element method is a special domain decomposition method, which can handle the situation where meshes on different subdomains need not align across the interface. In this article, we will apply the mortar element method to general variational inequalities of free boundary type, such as free seepage flow, which may show different behaviors in different regions. We prove that if the solution of the original variational inequality belongs to H²(D), then the mortar element solution can achieve the same order error estimate as the conforming P₁ finite element solution. Application of the mortar element method to a free surface seepage problem and an obstacle problem verifies not only its convergence property but also its great computational efficiency. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

非协调区域分解的Lagrangian乘子法

§1.引言 近年来随着并行计算机的迅速发展,求解椭圆型方程的区域分解法愈来愈引起人们的兴趣和重视.但是,目前能够见到的有限元区域分解法几乎都要求有限元空间在跨过子区域的边界时是协调的,必然限制有限元区域分解算法的优越性. [3]提出了一种非协凋区域分解法——非协调区域分解的杂交法.采用简化杂交法处理各子区域交界处的非协调性,这种方法在子区域的内部和边界采用两套不同的变量,允许内部变量在跨过各子区域的边界时不连续.但是这种方法有它的局限性,即要求边界变量在各子区域的顶点处必须保持连续性,这对推广到三维空间的情形带来很大的困难.本文提出一种非协调区域分解的Lagrangian乘子法,引进Lagrangian乘子来处理各子区域交界处的非协调性.这种方法也在子区域内部和边界采用两套不同的变量,它不仅允许内部变量在越过各子区域边界时的非协调性,并且还允许边界变量在各子区域的顶点处可以不连续,这就弥补了[3]的不足.同时,这种算法具有[3]的优点,即在不     

关于解椭圆型问题的两个子区域不重叠区域分解算法

关于解椭圆型问题的两个子区域不重叠区域分解算法顾金生，胡显承（清华大学）ＯＮＴＨＥＤＯＭＡＩＮＤＥＣＯＭＰＯＳＩＴＩＯＮＭＥＴＨＯＤＳＦＯＲＥＬＬＩＰＴＩＣＰＲＯＢＬＥＭＳＷＩＴＨＴＷＯＳＵＢＳＴＲＵＣＴＵＲＥＳ￥ＧｕＪｉｎ－ｓｈｅｎｇ；ＨｕＸｉａｎ...     

A NON-OVERLAPPING DOMAIN DECOMPOSITION ALGORITHM BASED ON THE NATURAL BOUNDARY REDUCTION FOR WAVE EQUATIONS IN AN UNBOUNDED DOMAIN

In this paper, a new domain decomposition method based on the natural boundary reduction, which solves wave problems over an unbounded domain, is suggestted. An circular artificial boundary is introduced. The original unbounded domain is divided into two subdomains, an internal bounded region and external unbounded region outside the artificial boundary. A Dirichlet-Neumann(D-N) alternating iteration algorithm is constructed. We prove that the algorithm is equavilent to preconditional Richardson iteration method. Numerical studies are performed by finite element method. The numerical results show that the convergence rate of the discrete D-N iteration is independent of the finite element mesh size.     

Two-level Schwarz method for unilateral variational inequalities

The numerical solution of variational inequalities of obstacletype associated with second-order elliptic operators is considered.Iterative methods based on the domain decomposition approachare proposed for discrete obstacle problems arising from thecontinuous, piecewise linear finite element approximation ofthe differential problem. A new variant of the Schwarz methodology,called the two-level Schwarz method, is developed offering thepossibility of making use of fast linear solvers (e.g., linearmultigrid and fictitious domain methods) for the genuinely nonlinearobstacle problems. Namely, by using particular monotonicityresults, the computational domain can be partitioned into (mesh)subdomains with linear and nonlinear (obstacle-type) subproblems.By taking advantage of this domain decomposition and fast linearsolvers, efficient implementation algorithms for large-scalediscrete obstacle problems can be developed. The last part ofthe paper is devoted to illustrate numerical experiments.     

SUBSTRUCTURE PRECONDITIONERS FOR NONCONFORMING PLATE ELEMENTS

1.IntroductionInthispaper,wegeneralizetheBPSalgorithm[1]tononconformingelementfproximationsofthebiharmonicequation.WeconstructapreconditionerforMor:elementbysubstructuringonthebasisofafunctiondecompositionfordiscretebibmonicfunctions.Thefunctiondecomposit…     

Studies of an interface relaxation domain decomposition technique using finite elements on a parallel computer

Studies are presented for an interface relaxation domain decomposition technique using finite elements on an iPSC/2 D5 Hypercube Concurrent computer. The general type of problem to be solved is one governed by a partial differential equation. The application of the approach, however, will be extended to a free boundary value problem by appropriate modification of the numerical scheme. Using the domain decomposition technique, the computation domain is subdivided into several subdomains. In addition, on the interfaces between two adjacent subdomains are imposed a continuity condition on one side and an equilibrium condition on the other side. Successive overrelaxation iterative processes are then carried out in all subdomains with a relaxation process imposed on the interfaces. With this domain decomposition technique, the problem can be solved parallelly until convergence is reached both in the interiors and on the interfaces of all subdomains. Moreover, the formulation includes a simple domain decomposer that automatically divides a finite element mesh into a list of subdomains to guarantee load balancing. Furthermore, it is shown, through numerical experiments performed on an example problem of free surface seepage through a porous dam, how the values of the relaxation parameters, the choice of imposed boundary conditions, and the number of subdomains (i.e., the number of processors used) affect the solution convergence in this parallel computing environment. © 1993 John Wiley & Sons, Inc.     

The Mortar finite element method with Lagrange multipliers

Summary. The present paper deals with a variant of a non conforming domain decomposition technique: the mortar finite element method. In the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices (and the edges in 3D) of subdomains. It is shown that, written under primal hybrid formulation, the approximation problem, issued from a discretization of a second order elliptic equation in 2D, is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms. Finally the parallelization advantages consequence of this variant are also addressed. Received December 1, 1996 / Revised version received November 23, 1998 / Published online September 24, 1999     

Convergence of a substructuring method with Lagrange multipliers

Summary. We analyze the convergence of a substructuring iterative method with Lagrange multipliers, proposed recently by Farhat and Roux. The method decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains plus a coarse problem for the subdomain nullspace components. For linear conforming elements and preconditioning by the Dirichlet problems on the subdomains, we prove the asymptotic bound on the condition number , or ,where is the characteristic element size and subdomain size. Received January 3, 1995