Investigation on an optimal aggregation level for a parallel meshless multigrid method based on domain decomposition method

We developed three parallel algorithms for a meshless geometric multigrid (GMG) method proposed for the finite element discretization of elliptic partial differential equation. These methods for parallel multigrid (PMG) are based on the message passing interface (MPI) for domain decomposition and coarse matrix aggregation (CMA) algorithm for coarser levels. Using coarse matrices obtained by a parallel Galerkin condition for the present meshless GMG, we proposed a parameter by which an optimal aggregation level is determined. This parameter is defined as the ratio of total number of external interface nodes from all the subdomains before aggregation to the number of non-zero entries of gathered matrix after aggregation. Three methods —M1, M2, and M3— are classified depending on how the coarsest matrix is solved and the number of coarser levels for which CMA is applied. M1 (M2) solves the coarsest matrix via an iterative (direct) solver applying CMA only for the coarsest level, whereas M3 determines the multigrid levels with CMA based on the parameter and employs a direct solver for the coarsest matrix. We found that M3 is more efficient than the others and much more efficient in the case of complicated geometry because communication overhead is reduced compared to the other methods. Furthermore, the present PMG could achieve super-linear scalability owing to the cache effect for a large problem.