Construction of intrinsic schemes for eigen-computation based on the polyhedron grid matrix in 3-D
Jiachang Sun · Scientia Sinica Mathematica · 2023
A geometric asynchronous parallel algorithm for solving large-scale discrete mathematical-physical systems in the 2-D case was recently presented by the author in 2022. Different from the traditional preconditioning, we applied the intrinsic geometric invariance to get the grid matrix G, and required the discrete PDE (partial differential equation) stiff matrix A, and the mass matrices B and G satisfy the reciprocal relations A G = G A and B G = G B, where G satisfies G^m=I, m łl N=dim(G), i.e., large scale system solvers can be transformed to a smaller block-solver as a pre-treatment in real or complex domain. In this paper, we expand our geometry pre-processing asynchronous algorithm (GPA) to the 2-D irregular mesh and the 3-D mathematical-physical discrete eigenvalue problems over more wide polyhedron grids such as hexahedrons, tetrahedrons and dodecahedrons. We give the theory and numerical examples of an efficient asynchronous parallel order reduction algorithm. We obtain the conclusion that “the parallelism of 3-D geometric mesh pre-transformation is mainly proportional to the number of the faces of the polyhedron”, and further find that “the reciprocity of the grid mesh matrix and the stiff matrix is an important basis for the feasibility and reliability of the GPA algorithm”.