SINE TRANSFORM PRECONDITIONERS FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.     

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.     

Navier—Stokes方程区域分解法的收敛性

０引言区域分解方法是近年来迅速发展的偏微分方程数值方法．区域分解方法及其收敛性的研究大多是在线性偏微分方程下得到的,对于非线性问题,经典的技巧在收敛性证明时遇到了困难．流体计算是一个较为复杂的非线性问题,数值模拟过程中因节点多．网格复杂,所以计算量很大．由于区域分解方法不但可以缩小求解规模,进行并行计算,而且可以在不同区域选取不同离散方法和模型,因此对Ｎ－Ｓ方程区域分解方法的研究会有较高的实用价值,也可以对其它非线性问题数值方法研究提供新的途径．本文首先给出了Ｎ－Ｓ方程的最优控制方法以及一些重要…     

A PARALLEL ITERATIVE DOMAIN DECOMPOSITION ALGORITHM FOR ELLIPTIC PROBLEMS

1.IntroductionNolloverlappillgdomaindecolllpositionnletllodshavereceivedalotofattentionlenlsilllldallowefficielltparallelisnl.F'Orarecentdevelopmelltofthesemethods,werefertot…     

A NONOVERLAPPING DOMAIN DECOMPOSITION METHOD FOR EXTERIOR 3-D PROBLEM

1. IntroductionIn recent y6ars, the elliptic boUndaly value problems ill unbounded domains have dlawnmore and more attention. TO solve an equation in an unbounded domain numerically, a basicidea is to licit the computation to a bounded domain by introducing an artWial boundary.Based on this idea, many numerical methods, such as the coupling of BEM and FEM, the FEMwith boundary conditions at atilicial boundary) the coupled finite-~ie elemellt ndhodthe DDM(domain decomposition method)(cf.,…     

THE CONVERGENCE OF A DOMAIN DECOMPOSITION ALGORITHM FOR PARABOLIC PROBLEMS

At recent, Hourgat et gave a domain decomposition algorithm for elliptic problems which can be implemented in parallel. Many numerical experiments have illustrated its efficiency. In the present paper, we apply this algorithm to solve the discrete parabolic problems, analyse its convergence and show that its convergence rale is about (1 - 2p + σp2 ) which is nearly optimal and independent of the parameter τ, where σ τ O((1 +H )(1 + ln(H / h))2 ). 0 < p < 1 / σ,τ,h,H are the time step size, finite element parameter and subdomain diameter, respectively.     

A PARALLEL NONOVERLAPPING DOMAIN DECOMPOSITION METHOD FOR STOKES PROBLEMS

A nonoverlapping domain decomposition iterative procedure is developed and analyzed for generalized Stokes problems and their finite element approximate problems in R^N（N=2,3）. The method is based on a mixed-type consistency condition with two parameters as a transmission condition together with a derivative-free transmission data updating technique on the artificial interfaces. The method can be applied to a general multi-subdomain decomposition and implemented on parallel machines with local simple communications naturally.     

DOMAIN DECOMPOSITION METHODS WITH NONMATCHING GRIDS FOR THE UNILATERAL PROBLEM

1. IntroductionDomain decomposition methods (DDMs) with nonmatching grids, which have been deveLoped in recent years3 are a quite new class of nonconforming DDMs. As this kind of DDMscan be aPplied to solving many practical problems which can't be handled…     

结构三角网上抛物方程的有限差分区域分解算法

1引言对于大型科学与工程计算问题,并行计算是必需的．构造高效率的数值并行方法一直是人们关心的问题,并且已有了大量的研究．在三层交替计算方法的研究中出现了许多既具有明显并行性又绝对稳定的差分格式(见[1]-[5])．在只涉及两个时间层的算法研究中,Dawson等人(见[6])首先发展了求解一维热传导方程的区域分解算法,并将其推广到     

一类积分方程的稳定性与吸引域

本文通过建立非线性积分不等式,讨论了一类具有时滞积分方程的渐近稳定性．获得了简捷而实用的充分准则,并给出了确定吸引域的方法．     

CONSTRUCTION OF A PRECONDITIONER FOR DOMAIN DECOMPOSITION METHODS WITH POLYNOMIAL LAGRANGIAN MULTIPLIERSS

1. IntroductionIn recent years the non-overlapping domain decomposition methods (DDMs) with nonmatching grids have attracted particular attention of computional eXPerts and engineers (see[1]--[9]). This kind of DDM allows non--coincidence of nodal points at common edges (or commonfaces) of two neighbouring subdomains. Thus it can be applied to solving the problems ofchanging meshes (for example, the multi-body contact problems in solid mechanics an'd therelative motion problems in oil explo…     

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）…     

Capacitance matrix method using strips with alternating neumann and dirichlet boundary conditions

A method for solving numerically second-order elliptic equations in rectangular regions is discussed. The rectangle is divided into strips. The original problem is replaced by subproblems with alternating Neumann and Dirichlet boundary conditions, and an auxiliary problem. The latter, called the capacitance matrix system, is solved by an efficient numerical procedure using the conjugate gradient method.     

非线性双曲型方程的混合元方法

1　引　　言考虑下述非线性双曲型方程的混合问题:c(x,u)utt-.(a(x,u)u)=f(x,u,t),　　x∈Ω,t∈J,(1.1)u(x,0)=u0(x),　　x∈Ω,(1.2)ut(x,0)=u1(x),　　x∈Ω,(1.3)u(x,t)=-g(x,t),　　(x,t)∈Ω×J,(1.4)其中ΩR2是一具有Lipschitz边界Ω的有界区域,J=[0,T],0相似文献   

一类特殊矩阵方程的并行预处理变形共轭梯度算法

研究了求解一类矩阵方程AXB=C,提出了一种并行预处理变形共轭梯度法．该方法给出一种迭代法的预处理模式．首先给出的预处理矩阵是严格对角占优矩阵,构造并行迭代求解预处理矩阵方程的迭代格式,进而使用变形共轭梯度法并行求解．通过数值试验,预处理变形共轭梯度法与直接使用变形共轭梯度法相比较,该算法不仅有效提高了收敛速度,而且具有很高的并行性．     

ON BLOCK PRECONDITIONERS FOR PDE-CONSTRAINED OPTIMIZATION PROBLEMS

Recently, Bal proposed a block-counter-diagonal and a block-counter-triangular precon- ditioning matrices to precondition the GMRES method for solving the structured system of linear equations arising from the Galerkin finite-element discretizations of the distributed control problems in （Computing 91 （2011） 379-395）. He analyzed the spectral properties and derived explicit expressions of the eigenvalues and eigenvectors of the preconditioned matrices. By applying the special structures and properties of the eigenvector matrices of the preconditioned matrices, we derive upper bounds for the 2-norm condition numbers of the eigenvector matrices and give asymptotic convergence factors of the preconditioned GMRES methods with the block-counter-diagonal and the block-counter-triangular pre- conditioners. Experimental results show that the convergence analyses match well with the numerical results.     

THE RESTRICTIVELY PRECONDITIONED CONJUGATE GRADIENT METHODS ON NORMAL RESIDUAL FOR BLOCK TWO-BY-TWO LINEAR SYSTEMS

The restrictively preconditioned conjugate gradient （RPCG） method is further developed to solve large sparse system of linear equations of a block two-by-two structure. The basic idea of this new approach is that we apply the RPCG method to the normal-residual equation of the block two-by-two linear system and construct each required approximate matrix by making use of the incomplete orthogonal factorization of the involved matrix blocks. Numerical experiments show that the new method, called the restrictively preconditioned conjugate gradient on normal residual （RPCGNR）, is more robust and effective than either the known RPCG method or the standard conjugate gradient on normal residual （CGNR） method when being used for solving the large sparse saddle point problems.     

ERROR ANALYSIS OF THE SECOND-ORDER SERENDIPITY VIRTUAL ELEMENT METHOD FOR SEMILINEAR PSEUDO-PARABOLIC EQUATIONS ON CURVED DOMAINS

The second-order serendipity virtual element method is studied for the semilinear pseudo-parabolic equations on curved domains in this paper.Nonhomogeneous Dirichlet boundary conditions are taken into account,the existence and uniqueness are investigated for the weak solution of the nonhomogeneous initial-boundary value problem.The Nitsche-based projection method is adopted to impose the boundary conditions in a weak way.The interpolation operator is used to deal with the nonlinear term.The Crank-Nicolson scheme is employed to discretize the temporal variable.There are two main features of the proposed scheme:(ⅰ)the internal degrees of freedom are avoided no matter what type of mesh is utilized,and(ⅱ)the Jacobian is simple to calculate when Newton's iteration method is applied to solve the fully discrete scheme.The error estimates are established for the discrete schemes and the theoretical results are illustrated through some numerical examples.