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

作     者：Jiachang SUN Jianwen CAO Ya ZHANG Haitao ZHAO Jiachang SUN;Jianwen CAO;Ya ZHANG;Haitao ZHAO

作者机构：Laboratory of Parallel Software and Computational ScienceInstitute of SoftwareChinese Academy ofSciencesBeijing 100190China 

出 版 物：《Chinese Annals of Mathematics,Series B》 (数学年刊（B辑英文版）)

年 卷 期：2023年第44卷第5期

页      面：735-752页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported by the Basic Research Plan on High Performance Computing of Institute of Software(No.ISCAS-PYFX-202302) the National Key R&D Program of China(No.2020YFB1709502) the Advanced Space Propulsion Laboratory of BICE and Beijing Engineering Research Center of Efficient and Green Aerospace Propulsion Technology(No.Lab ASP-2019-03)。 

主　　题：Mathematical-physical discrete eigenvalue problems Commutative operator Geometric pre-processing algorithm Eigen-polynomial factorization 

摘      要：A geometric intrinsic pre-processing algorithm(GPA for short)for solving largescale 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,mdim(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 polyhedronis 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.