Generalized Schwartz algorithm with variable parameters-猫眼课题宝

DOI：

10.1515/rnam.1990.5.1.1

发表时间：

1990

期刊：

影响因子：

通讯作者：

中科院分区：

其他

文献类型：

作者：

V. Agoshkov;V. Lebedev

本文介绍了一类特殊的算子——粒子输运理论中的反射算子和椭圆边值问题中的庞加莱算子。利用这些算子的性质和迭代过程的一般理论，提出了构造和研究广义Schwartz方法（域分解方法）的各种算法的技术。证明了一些特定算法的收敛速度估计。schwartz迭代算法以解决拉普拉斯方程[15]的Dirichlet问题而闻名，其中域Ω被分解为两个具有非零相交的子域ΩΙ和Ω2。研究这种方法在子域ΩΙ和Ω2只有一个公共边界的情况下的推广是很有趣的。最近，研究这种广义Schwartz算法（或域分解方法）应用于各种数学物理问题的可行性引起了人们的兴趣[1,9,10,12-14,17]。在[1,9,10,12-14]中考虑平稳迭代过程；在[17]中，参数是通过利用二次泛函的一步最小化来选择的。本文研究了变参数非平稳迭代方法的有效性，该方法推广了一些已知的迭代方法，提出了新的迭代方案，并给出了最优参数集。本文还探讨了b[4]中提出的块切比雪夫法的可行性。为了解决公式化问题，有必要研究在边界的公共部分上定义的一些算子的频谱的位置；这些算子被称为庞加莱-斯特克洛夫算子。作者认为有责任指出，他们对本文讨论的问题的兴趣是由[1,13,14]中得到的结果激发的，他们对A. Poincare和V. A. Steklov的论文的注意力是由V. P. Mikhailov引起的。求解双循环矩阵系统的块CHEBYSHEV方法让我们回顾一下块CHEBYSHEV方法[4]的描述，因为它的迭代格式非常适合研究广义Schwartz算法。设DV D2， C^和C2是线性算子将巴拿赫空间Β映射到自身，算子D^和D2是可逆的，v^ v2， gl和n2是b的元素，考虑方程系统ll = 12 + #！> 22 =会+£·i·)假设/ / = D2 ~ C2D1 ~ C1, Sp (/ /) e (ra, Af), 1 ^)。（1.6）这里，cu>为t序列'[14]。在算子//对称的情况下，该方法的收敛性在一定范数下由公式Remark 1.2估计。方法（1.2）可以推广为与k相关的算子D.、Ct和g.的方法。这些算子和右边的g.可以从等价于系统（1.1）的方程组U1 = U2 + *ljc ‘ 2Jk2 = 2/2 + fyc i’中找到。1.2的话。使用c^ Φ 1的方法（1.2）的其他版本的优化在[4]中构造。2、广义Schwartz算法输运方程两个半空间的情况。让我们考虑在整个空间中给定的输运方程μ do c a dx 21。设/(*,/*)是x的有限函数，对于jt >, c = cx，对于* 0，c = c^。用下面的方法构造迭代序列ν*(μ)。使用给定的函数ν£（μ），求出方程（2.1）的解φ^(χ,μ)，它满足条件φ^Ο，μ) =ν*(μ)对于μ > 0。设置和ν*)。然后，对* 0求方程（2.1）的φ ι χ， μ)解，对// >求set，/i) v2(/i))解。对于任意函数ν(/^)€Ζ^，对于* >存在一个唯一解φ^χ，μ)对问题Λ -7^ + ^ι = ν dx 2 -1（2 2）因此，我们在L^ 1上定义了一个线性算子51，使每个函数v（/^）与制裁^1（//）€l2对应，根据规则w^/ι) = n (0,-/i)：“i = i·同样，对于j: < 0，存在一个唯一解φ2（χ,μ）

The paper introduces special classes of operators-reflection operators in the particle transport theory and Poincare Steklov operators in elliptic boundary value problems. The properties of these operators and the general theory of iterative processes are used to propose techniques for construction and investigation of various algorithms of the generalized Schwartz method (the domain decomposition method). Estimates of the convergence rate for a number of specific algorithms are proved. The Scwartz iterative algorithm is widely known for solving the Dirichlet problem for the Laplace equation [15] in which the domain Ω is decomposed into two subdomains ΩΙ and Ω2 which have a nonzero intersection. It is interesting to investigate the generalization of this method to the case where the subdomains ΩΙ and Ω2 have only a common boundary. An interest has recently arisen in investigating the feasibilities of such generalized Schwartz algorithm (or of the domain decomposition method) as applied to various problems of mathematical physics [1,9,10,12-14,17]. Stationary iterative processes were considered in [1,9,10,12-14]; in [17], the parameters were chosen by making use of the one-step minimization of a quadratic functional. This paper investigates the efficiency of non-stationary iterative methods with variable parameters, which generalize some methods known before, suggests new iterative schemes and presents optimal sets of parameters. The paper also investigates the feasibilities of the block Chebyshev method suggested in [4]. To solve the formulated problems, it is necessary to investigate the location of the spectrum of some operators defined on the common part of the boundary; these operators are called Poincare-Steklov operators. The authors consider it their duty to note that their interest in the problems discussed in the paper was stimulated by the results obtained in [1,13,14], and their attention to the papers by A. Poincare and V. A. Steklov was drawn by V. P. Mikhailov. 1. BLOCK CHEBYSHEV METHOD FOR SOLVING SYSTEMS WITH TWO-CYCLIC MATRICES Let us recall the description of the block Chebyshev method [4] as its iterative scheme is well suited for investigating the generalized Schwartz algorithm. Let DV D2, C^ and C2 be linear operators mapping the Banach space Β into itself, the operators D^ and D2 be invertible, v^ v2, gl and ̂ be elements of B. Consider the system of equation ll = 12 + #!> 22 = ll +£· i·) Assume that // = D2~C2D1~C1, Sp(//) e [ra,Af], where 1 ^). (1.6) Here, cu>is a T-sequence' [14]. In the case of the symmetricity of the operator //, the convergence of the method is estimated in a certain norm by the formula Remark 1.2. Method (1.2) can be generalized into the method with the operators D., Ct and g. dependent on k. These operators and the right-hand sides g. can be found from the system of equations U1 = U2 + *ljc ' 2Jk2 = 2/2 + fyc i') equivalent to system (1.1). Remark 1.2. Other versions of optimization of method (1.2) with c^ Φ 1 are constructed in [4]. , Generalized Schwartz algorithm 2. TRANSPORT EQUATION 2.1. Case of two half -spaces. Let us consider the transport equation μ do c a dx 2 1 given in the entire space. Let /(*,/*) be a function finitary in x, c = cx for jt > 0 and c = c^ for * 0. Construct the iterative sequence ν*(μ) in the following way. Using a given function ν£(μ), find for jc > 0 the solution φ^(χ,μ) to equation (2.1), which satisfies the condition φ^Ο,μ) =ν*(μ) for μ > 0. Set and ν*) . Then, find the solution φ ι χ , μ ) to equation (2.1) for * 0 and set ,/i) v2(/i)) for // > 0. For any function ν(/^)€Ζ^ there exists for * > 0 a unique solution φ^χ,μ) to the problem Λ -7^ + ^ι = ν dx 2 -1 (2 2) We have thus defined on L^ a linear operator 51 putting each function v(/^) into correspondence with the Sanction ^1(//)€l2 by the rule w^/ι) = ̂ (0,-/i): "i = i· Likewise, for j: < 0 there exists a unique solution φ2(χ,μ) to the problem

Generalized Schwartz algorithm with variable parameters

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