Abstract We consider a method due to P. Vassilevski and Yu. A. Kuznetsov [4, 10] for solving linear systems with matrices of low Kronecker rank such that all factors in Kronecker products are banded. Most important examples of such matrices arise from discretized div K grad operator with diffusion term k1(x)k2(y)k3(z). Several practical issues are addressed: an MPI implementation with distribution of data along processor grid inheriting Cartesian 3D structure of discretized problem; implicit deflation of the known nullspace of the system matrix; links with two-grid framework of multigrid algorithm which allow one to remove the requirement of Kronecker structure in one or two of axes. Numerical experiments show the efficiency of 3D data distribution having the scalability analogous to (structured) HYPRE solvers yet the absolute timings being an order of magnitude lower, on the range from 10 to 104 cores.

中文翻译：

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关于离散椭圆问题的快速直接法的注记

摘要 我们考虑了 P. Vassilevski 和 Yu 的方法。A. Kuznetsov [4, 10] 用于求解具有低 Kronecker 秩矩阵的线性系统，使得 Kronecker 乘积中的所有因子都是带状的。此类矩阵的最重要示例来自具有扩散项 k1(x)k2(y)k3(z) 的离散化 div K grad 算子。解决了几个实际问题：沿处理器网格分布数据的 MPI 实现，继承了离散化问题的笛卡尔 3D 结构；系统矩阵的已知零空间的隐式紧缩；与多网格算法的双网格框架链接，允许一个或两个轴去除克罗内克结构的要求。