Parallel Iteration Solution Of Poisson Equation Based On The PVM

The finite element method is one of the important numerical methods to solve the partial differential equation in the modern engineering design and analyzing and is an effective method to research structure analyze problem nowadays. More and more larger-scale scientific calculation questions have been put forward in the extensive fields of science and project at present. And the traditional finite element algorithm analysis that regards serial computer as material base can't satisfy the demands of the scientific research and the development of the engineering and technology. The appearance of high-performance parallel computing system undoubtedly brings the opportunity to larger-scale calculation of the science and engineering. The challenge how to fully make use of the computing ability of the system, especially to numerical parallel algorithm, is seriously put forward .The network parallel system made of a few computers is a comparatively popular parallel processing way at present .The research of the parallel finite element starts from the end of the seventies of the 20th century. The past research is mainly based on the parallel machines and vector ones. The traditional finite element seeks to vectorization and parallelization, besides, various strategies and technologies to increase parallel degree are explored in every level of finite element analysis and designing process. However, the research to realize finite element parallel computation under the distributed environment in which is based on the news transmitting, just began in recent years. Parallel strategy of area partition method or EBE method is adopted in most literature.The discrete process of finite element and the solution of the finite element system of equations are analyzed in the paper. In the condition that matrix and vector use row partitioning memory, parallel computation and parallel assembly of total stiffness matrix of the finite element are realized by the row element unit contribution method, and JOR iteration algorithm is adopted to parallel solve the finite element equation set. The numerical experimentation is carried out on the PVM network parallel computation platform which is composed of 1-4 microcomputers and is based on news transmitting. The result of the experiments shows that the algorithm has high parallel degree to finite element solution of Poisson equation, and is easily realized and has higher acceleration ratio and machine efficiency.In the paper the character of overall stiffness matrix is analyzed after Poisson equation is discrete, and the selection of the relaxation factor towards JOR iteration algorithm is discussed, and the relaxation factor selection theorem and optimum relaxation factor selection theorem which adapt the positive definite system of equations are given, and the proof of the theorems is also given. Forefathers have already made a large number of works to the selection of JOR method relaxation factor. In the article , let coefficient matrix A be the H matrix, and letρbe the spectral radius of matrix |D|^-1|B| then the convergent area of JOR method is (0,2/(1 +ρ))(ρ< 1). In the paper , letλ_max andλ_min, respectively, be the maximal and minimal eigenvalues of positive definite symmetric coefficient matrix A , then the relaxation factorωof JOR method belongs to (0,2/λ_max) and the optimum relaxation factorωis 2/(λ_max +λmin), when the iteration scheme of JOR method is x_{k +1} =ω(b - A_xk )+ x_k. In this paper the selection of relaxation factor is discussed under the condition that JOR iteration scheme...