A comparison of vectorized methods for solving the two-dimensional diffusion equation: multigrid versus polynomial preconditioned conjugate gradient

摘要

This paper presents the result of a study to develop efficient vectorized multigrid (MG) methods for solving the two-dimensional diffusion equation on a rectangular domain with Dirichlet-Neumann boundary conditions for particular application to groundwater flow problems. Three different algorithms have been implemented on the CDC CYBER 205, the “best” one being selected on the basis of the degree of continuity and isotropy of the coefficients of the particular problem at hand. To demonstrate the superiority of our algorithms, we compare performances for several sample problems with those resulting from application of the polynomial preconditioned conjugate-gradient (PPCG) method. Except for “small” problems, MG solution rates are found to be substantially greater than those obtained using PPCG, and the ratio of the rates increases dramatically with problem size.