Lagrangian乘子区域分解法的一类预条件子

1．引言非重叠区域分解的Lagrangian乘子法已被许多作者讨论［1今它是一类非协调区域分解法（与通常的非协调元区域分解不同），特别适合于非匹配网格的情形（即相邻子域在公共边或公共面上的结点不重合，参见14］［6］）．这种方法的一个最大优点是不要求界面变量在内交点（或内交边）上的连续性，从而界面方程易于建立，程序易于实现，而又正因为这个特点，使得界面矩阵的预条件子不能按通常的方法构造，故目前还未见到理想的预条件子（或者条件数差，或者应用上不方便）．本文在很大程度上解决了这一问题．1）工作单位：湘潭大学数学系…     

抛物问题的拉格朗日乘子区域分解法

１．引言 近年来,一类新的非重叠区域分解方法一非匹配网格区域分解法,日益引起人们的广泛兴趣,并已成为当今区域分解方法研究的热门课题。这类区域分解方法的特点是：相邻子区域在公共边（或面）上的结点可以不重合,从而能解决许多传统区域分解方法不便解决的问题（如变动网格问题）．目前主要有两类方法来处理这种区域分解的强非协调性：Ｍｏｒｔａｒ无法（见［１－２］和［９－１０］）和拉格朗日乘子法（见［５］,［８］,［１１］和［１２］）．拉格朗日乘子法比Ｍｏｒｔａｒ无法有明显的优点：（１）界面变量（即拉格朗日乘子）…     

一类次椭圆方程弱解的正则性

在区域Ω上考虑一类由退化向量场形成的Schrodinger方程： ∑i,j=1^mXi^＊（aij（x）Xju）-vu=0 其中X1,…,Xm为R^n（n≥）3上满足Hormander条件的实C^∞向量场,Xi^＊为Xi的形式共轭,v属于Kato类的某一类比Kη^loc（Ω）.并得到以下结果：若u为以上方程的弱解,则｜Xu｜^2w=∑i=1^m｜Xiu｜^2w∈Kη^loc（Ω）.     

几类一阶微分方程的统一方程和统一变换

在高等数学的微分方程一章中,一阶变量可分离方程是最基本的方程,分离变量后积分,即得通解（通积分）：式中x（s）羊0（否则y一C）,C为积分常数（下同）．对于一阶非齐次线性方程由于其对应的齐次方程将常数U变易为函数U什）,即作变量代换方程（3）就变为关于函数X（X）和变量X的变量可分离方程积分后代入（5）式,即得非齐次线性方程（3）的通解对,右端为零次齐次。数。（于卜齐次方程当机y缸）学VX（否则,变量可分离）,作变换可得变量可分离方程UU‘＋U一J（U）,于是（7）的通解对于伯努利（Bernoulli）方程（n羊0,豆,否…     

一阶非齐次线性微分方程的齐次解法

现行高等数学教材对于一阶非齐次线性微分方程均采用常数变易法求通解，而本文是运用变量替换法将非齐次方程化为齐次方程求出通解。对于一阶非齐次线性微分方程做变换两边关于x求导数即有（1）式整理得令y的系数和常数项均为零这时变换后的微分方程为Z′＝0，解得Z一C（C为任意常数），于是将A（X）、B（X）和Z代入（2）式得原方程通解为从以上解法我们可以得到两方面启示：～方面可见变量替换法在解微分方程中同常数变易法都不失为行之有效的方法。另一方面常数变易法的解法思路是齐次方程～非齐次方程，而变量替换法的解法思路是非齐…     

具有多变时滞中立型微分方程的振动性

考虑具有多变时滞中立型微分方程[x(t)-∑i=1^lpi(t)x(t-τi(t))]′ ∑j=1^mqj(t)x(t-σj（t））=0,获得了该方程所有解振动的几族充分条件．其中定理3的条件是“Sharp”条件，即当Pi(t)，τi(t)，qj(t)，σj(t)(i=1，2，…，l，j=1，2，…，m)为常数时，条件是充分必要的．     

求解界面方程的Chebyshev加速算法

非重叠区域分解算法在于建立和求解相关的界面方程．建立界面方程在理论上虽。然容易推导，例如某些问题可用Gauss块消去法，但在实际计算时并不可行，所以界面方程在一些算法中是陷式的．而求解界面方程一般要进行预处理，本提出一种区域分解算法，可得出界面方程的显式表达．算法是完全并行的，所得出的界面方程的系数矩阵的条件数已与网参数无关，事实上就是(Sh^（1）)^-1Sh，进而可直接用收敛速度较快的Chebyshev加速算法求解该界面方程，在充分应用并行计算方法的条件下，本算法与[4]中的算法相比计算效率提高．     

基于几何非协调分解的Lagrange乘子区域分解方法

本文考虑将Lagrange乘子区域分解方法应用于几何非协调分解的情况来求解二阶椭圆问题.由于采用几何非协调区域分解,每个局部乘子空间关联到多个界面,我们按照一定的规则选取合适的乘子面来定义乘子空间.利用局部正则化技巧,可以消去内部变量,得到关于Lagrange乘子的界面方程.采用一种经济的预条件迭代方法求解界面方程,且相关的预条件子是可扩展的.     

抛物型方程有限元解的长时间渐近性态

1引言设Ω为R2中有界凸多边形区域，f(x)∈L2（Ω），λ为非负常数，面为LaPlace算子，考虑下面定解问题：我们在空间区域采用有限元剖分，在时间轴上采用［1］提出的一类差分格式，本文证明（1．1）的有限元解当L、co时收敛到下面椭圆方程（1．2）的有限元解：从而可应用［2］提出的具有并行本性的差分格式得到椭圆问题（1．幻的有限元解．2．预备知识设J为o上的拟一致三角剖分，vnCHI（n）为相应于J的线性协调元空间，7T：H‘（m、K为插值算子，使对J的任意节点P，7T。（P）＝。（P）·我们用下述格式求解（1．1）：求。h＋IEVh…     

一个包含欧拉函数的方程

设n为任意正整数,如果n〉1,设n=p1^α1p2^α2…pk^αk是n的标准分解式,函数Ω（n）定义为Ω（1）=0,Ω（n）=∑i=1^kαi,φ（n）为Euler函数,本文的主要目的是利用初等方法研究方程φ（φ（n））=2Ω（n）的可解性,并获得该方程的所有正整数解,从而彻底解决了前学者提出的一个问题．     

Preconditioned iterative methods on sparse subspaces

When some rows of the system matrix and a preconditioner coincide, preconditioned iterations can be reduced to a sparse subspace. Taking advantage of this property can lead to considerable memory and computational savings. This is particularly useful with the GMRES method. We consider the iterative solution of a discretized partial differential equation on this sparse subspace. With a domain decomposition method and a fictitious domain method the subspace corresponds a small neighborhood of an interface. As numerical examples we solve the Helmholtz equation using a fictitious domain method and an elliptic equation with a jump in the diffusion coefficient using a separable preconditioner.     

A preconditioner for a kind of coupled FEM-BEM variational inequality

In this paper we are concerned with a kind of nonlinear transmission problem with Signorini contact conditions. This problem can be described by a coupled FEM-BEM variational inequality. We first develop a preconditioning gradient projection method for solving the variational inequality. Then we construct an effective domain decomposition preconditioner for the discrete system. The preconditioner makes the coupled inequality problem be decomposed into an equation problem and a “small” inequality problem, which can be solved in parallel. We give a complete analysis to the convergence speed of this iterative method.     

A two level domain decomposition preconditioner based on local Dirichlet-to-Neumann maps

Coarse grid correction is a key ingredient in order to have scalable domain decomposition methods. In this Note we construct the coarse grid space using the low frequency modes of the subdomain DtN (Dirichlet–Neumann) maps, and apply the obtained two-level preconditioner to the linear system arising from an overlapping domain decomposition. Our method is suitable for the parallel implementation and its efficiency is demonstrated by numerical examples on problems with high heterogeneities.     

A domain decomposition preconditioner for hermite collocation problems

We propose a preconditioning method for linear systems of equations arising from piecewise Hermite bicubic collocation applied to two‐dimensional elliptic PDEs with mixed boundary conditions. We construct an efficient, parallel preconditioner for the GMRES method. The main contribution of the article is a novel interface preconditioner derived in the framework of substructuring and employing a local Hermite collocation discretization for the interface subproblems based on a hybrid fine‐coarse mesh. Interface equations based on this mesh depend only weakly on unknowns associated with subdomains. The effectiveness of the proposed method is highlighted by numerical experiments that cover a variety of problems. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 135–151, 2003     

Large scale petroleum reservoir simulation and parallel preconditioning algorithms research

Solving large scale linear systems efficiently plays an important role in a petroleum reservoir simulator, and the key part is how to choose an effective parallel preconditioner. Properly choosing a good preconditioner has been beyond the pure algebraic field. An integrated preconditioner should include such components as physical background, characteristics of PDE mathematical model, nonlinear solving method, linear solving algorithm, domain decomposition and parallel computation. We first discuss some parallel preconditioning techniques, and then construct an integrated preconditioner, which is based on large scale distributed parallel processing, and reservoir simulation-oriented. The infrastructure of this preconditioner contains such famous preconditioning construction techniques as coarse grid correction, constraint residual correction and subspace projection correction. We essentially use multi-step means to integrate totally eight types of preconditioning components in order to give out the final preconditioner. Million-grid cell scale industrial reservoir data were tested on native high performance computers. Numerical statistics and analyses show that this preconditioner achieves satisfying parallel efficiency and acceleration effect.     

Large scale petroleum reservoir simulation and parallel preconditioning algorithms research

Solving large scale linear systems efficiently plays an important role in a petroleum reservoir simulator, and the key part is how to choose an effective parallel preconditioner. Properly choosing a good preconditioner has been beyond the pure algebraic field. An integrated preconditioner should include such components as physical background, characteristics of PDE mathematical model, nonlinear solving method, linear     

Bilus: A block version of ilus factorization

ILUS factorization has many desirable properties such as its amenability to the skyline format, the ease with which stability may be monitored, and the possibility of constructing a preconditioner with symmetric structure. In this paper we introduce a new preconditioning technique for general sparse linear systems based on the ILUS factorization strategy. The resulting preconditioner has the same properties as the ILUS preconditioner. Some theoretical properties of the new preconditioner are discussed and numerical experiments on test matrices from the Harwell-Boeing collection are tested. Our results indicate that the new preconditioner is cheaper to construct than the ILUS preconditioner.     

Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations

We consider an abstract parameter dependent saddle-point problem and present a general framework for analyzing robust Schur complement preconditioners. The abstract analysis is applied to a generalized Stokes problem, which yields robustness of the Cahouet-Chabard preconditioner. Motivated by models for two-phase incompressible flows we consider a generalized Stokes interface problem. Application of the general theory results in a new Schur complement preconditioner for this class of problems. The robustness of this preconditioner with respect to several parameters is treated. Results of numerical experiments are given that illustrate robustness properties of the preconditioner.