Parallel Numerical processing of Unstructured Grid

本文研究了由非结构网格离散化偏微分方程得到的线性方程的解，重点研究了并行处理。提出了Bi-CGSTAB方法的多网格预处理方法，并将该方法应用于非结构网格上离散的对流扩散方程。结果表明，该算法比MILU预处理具有更好的收敛性，适用于这类问题的广泛领域。2)基于内部点消除的区域分解方法提出了一种新的区域分解方法，该方法的边界由双阵列点组成。虽然该方案需要的边界点数量增加了一倍，但所得电容矩阵的结构变得简单，从而可以并行应用基于对角块的预处理。我们还实现了并行化，其中一个域由多个PU处理。一般正对称矩阵的预条件法非结构化网格离散化后的不规则矩阵的一般预条件法。针对ICCG方法难以并行化的特点，提出了一种块Jacobi预处理的变体，并在并行机上实现。这种预条件易于并行化，并且具有比Jacobi点更好的收敛性。预处理。

In this research, the solution of linear equations which come out from the discretization of partial differenctial equations by an unstructured grid is studied with special emphasis on parallel processing.1) Preconditoning of the Bi-CGSTAB methodWe proposed a Multi-Grid preconditioning of the Bi-CGSTAB method and applied this MGBI-CGSTAB method to a convection-diffusion equation discretized on an unstructured grid. We showed that this algorithm gives better convergence behavior than the MILU preconditioning and is applicable to wide area of problems of this type.2) Domain Decomposiotion Method based on interior point eliminationWe proposed a new domain decomposition method where the boundary consists of double array of points. Although this scheme requires twice larger number of boudary points, the structure of the resulting capacitance matrix becomes simple so that the preconditioning based on diagonal block can be applied parallelly. We also implemented a parallelization where one domain is handled by multiple PU's.3) Precoditioning method for general positive symmetric matrixA general preconditoning method for an irregular matrix which comes from the discretization on an unstructured grid. Since the ICCG method is hard to parallelize, we proposed a variant of block Jacobi preconditioning and implemented it on a parallel machine. This pre-conditioning is easily parallelizable and gives better convergence properties than the point Jacobi. preconditioning.