Hierarchical node method for solving large-scale sparse linear equations in parallel.

In this paper, the hierarchical node method (HNM) is proposed for solving large-scale sparse linear equations in the context of elastic mechanics. HNM decomposes complex problems into simple subproblems using superposition and reduces computational costs using locality. In HNM, the coarse node stiffness matrix and subdomain stiffness matrix remain unchanged during the iteration process, which makes HNM suitable for solving multiple right-hand side problems. The mapping relationship between coarse nodes and fine nodes is obtained by solving local finite element problems rather than using the interpolation method, which enables HNM to solve complex topology problems rapidly. To accelerate the convergence rate of HNM, an optimal coarse node stiffness matrix is proposed, and a calculation method is provided based on the regularized least squares method. Large-scale numerical experiments show that HNM exhibits excellent scalability and a near-linear speedup. Compared to mainstream algorithms such as multigrid, HNM achieves faster convergence speeds when solving multiple right-hand side problems. [ABSTRACT FROM AUTHOR]