一种高度并行的Schwarz型混乱松弛法及其收敛性证明

文1—2]把混乱松弛思想引入到Schwarz交替法中,构造了一种Schwarz型混乱松弛法。但这个方法在进行第n+1步迭代时,在拟边界上必须要用到第n步迭代的值,从而影响了算法的并行性,得不到相应的同步或异步MIMD并行算法、为此,本文给出一种高度并行的Schwarz型混乱松弛法,这个方法包括了Schwarz交替法及其相应的同步和异步MIMD并行算法。对于二阶线性与非线性微分方程Dirichlet问题,本文应用微分方程极值原理证明了该方法的收敛性。

A Highly Parallel Schwarz-Chaotic Relaxation Method and Its Convergence Proof

The papers 1-2] introduced the chaotic relaxation idea into the Schwarz alternating procedure and give a Schwarz-chaotic relaxation method. The method must use the n -step iterative result on pesudo-boundary when it calculates the ( n + 1)-step iterative result. So the parallelism of this method vanishes,and neither synchronous nor asynchronous MIMD parallel algorithm is obtained. In this paper, we give a highly parallel Schwarz-chaotic relaxation method. The method involves the Schwarz alternating procedure and the relevant (synchronous and asynchronous) MIMD parallel algorithms. Using the maxima principle of differential equation,we prove the convergence of this method for the Dirichlet problem of the second order linear and nonlinear differential equation.