Parallel Numerical processing of Unstructured Grid

本文主要研究了非结构网格下偏微分方程离散后的线性方程组的并行求解问题。1）Bi-CGSTAB方法的预处理我们提出了Bi-CGSTAB方法的多重网格预处理方法，并将该方法应用于非结构网格下的对流扩散方程。结果表明，该算法比MILU预处理算法具有更好的收敛性，适用范围更广。2）基于内点消去的区域分解方法提出了一种新的边界由双点阵组成的区域分解方法。该方案虽然需要两倍多的边界点，但所得到的电容矩阵结构变得简单，从而可以直接应用基于对角块的预处理。3）一般正对称矩阵的预编码方法非结构化网格离散化后的非规则矩阵的预编码方法。由于ICCG方法难以并行化，我们提出了一种块Jacobi预处理的变体，并在并行机上实现了它。这种预处理是很容易并行化，并给出更好的收敛性能比点雅可比。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.