Solution of elliptic PDEs by fast poisson solvers using a local relaxation factor

Abstract

Using a local discrete Fourier analysis, one- and two-step iterative procedures are developed to solve a large class of 2-D and 3-D nonseparable elliptic partial differential equations (PDEs). The one-step procedure is a modified D'Yakanov-Gunn iterative procedure in which the relaxation factor is grid point dependent. The two-step procedure is designed to accelerate the one-step procedure. Both are easy to implement, and applicable to a variety of boundary conditions. They are also computationally efficient as indicated by the results of numerical comparisons with other established methods. Furthermore, these new algorithms possess two important properties which the traditional iterative methods lack, i.e., (i) the convergence rate is relatively insensitive to grid cell size and aspect ratio, and (ii) the convergence rate can be easily estimated using the coefficient of the PDE being solved. For a set of constant coefficient model problems, it is show theoretically that the two-step vs. one-step computational efficiency ratio ranges between 1 and 2. The higher ratio is realized whenever the need to accelerate the one-step procedure is greater, i.e., whenever its convergence rate is lower. It is also shown numerically that the two-step procedure can substantially outperform the one-step procedure in the numerical solution of many PDEs with variable coefficients.