Commutation of Geometry-Grids and Fast Discrete PDE Eigen-Solver GPA

Abstract

A geometric intrinsic pre-processing algorithm(GPA for short) for solving large-scale discrete mathematical-physical PDE in 2-D and 3-D case has been presented by Sun (in 2022–2023). Different from traditional preconditioning, the authors apply the intrinsic geometric invariance, the Grid matrix G and the discrete PDE mass matrix B, stiff matrix A satisfies commutative operator BG = GB and AG = GA, where G satisfies G ^m = I, m ≪ dim(G). A large scale system solvers can be replaced to a more smaller block-solver as a pretreatment in real or complex domain.

In this paper, the authors expand their research to 2-D and 3-D mathematical physical equations over more wide polyhedron grids such as triangle, square, tetrahedron, cube, and so on. They give the general form of pre-processing matrix, theory and numerical test of GPA. The conclusion that “the parallelism of geometric mesh pre-transformation is mainly proportional to the number of faces of polyhedron” is obtained through research, and it is further found that “commutative of grid mesh matrix and mass matrix is an important basis for the feasibility and reliability of GPA algorithm”.

Keywords

Mathematical-physical discrete eigenvalue problems / Commutative operator / Geometric pre-processing algorithm / Eigen-polynomial factorization

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