GPA: Intrinsic Parallel Solver for the Discrete PDE Eigen-Problem

A class of geometric asynchronous parallel algorithms for solving large-scale discrete PDE eigenvalues has been studied by the author (Sun in Sci China Math 41(8): 701–725, 2011; Sun in Math Numer Sin 34(1): 1–24, 2012; Sun in J Numer Methods Comput Appl 42(2): 104–125, 2021; Sun in Math Numer Sin 44(4): 433–465, 2022; Sun in Sci China Math 53(6): 859–894, 2023; Sun et al. in Chin Ann Math Ser B 44(5): 735–752, 2023). Different from traditional preconditioning algorithm with the discrete matrix directly, our geometric pre-processing algorithm (GPA) algorithm is based on so-called intrinsic geometric invariance, i.e., commutativity between the stiff matrix A and the grid mesh matrix \(G{:}\,AG=GA\) $G:AG=GA$ . Thus, the large-scale system solvers can be replaced with a much smaller block-solver as a pretreatment. In this paper, we study a sole PDE and assume G satisfies a periodic condition \(G^m=I, m<< \textrm{dim}(G)\) $G^m=I,m<<\text{dim}(G)$ . Four special cases have been studied in this paper: two-point ODE eigen-problem, Laplace eigen-problems over L-shaped region, square ring, and 3D hexahedron. Two conclusions that “the parallelism of geometric mesh pre-transformation is mainly proportional to the number of faces of polyhedron” and “commutativity of grid mesh matrix and mass matrix is the essential condition for the GPA algorithm” have been obtained.