Parallel Numerical processing of Unstructured Grid

本文研究了用非结构网格离散偏微分方程组所产生的线性方程组的求解问题，重点讨论了并行处理的问题。1)提出了一种多网格预条件的BiCGSTAB方法，并将该方法应用于离散在非结构网格上的对流扩散方程。2)基于内点消除的区域分解方法我们提出了一种新的区域分解方法，该方法的边界由两个点组成。虽然该方案需要两倍多的边界点，但所得到的电容矩阵的结构变得简单，使得基于对角线块的预条件可以并行应用。我们还实现了一个区域由多个PU处理的并行化。3)一般正对称矩阵的预条件方法非结构网格上的离散化产生的不规则矩阵的一般预条件方法。针对ICCG算法难以并行化的问题，提出了一种改进的块Jacobi预条件算法，并在并行机上实现了该算法。这种预条件易于并行化，并且比点Jacobi具有更好的收敛特性。预适应。

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.