Convergence rates of iterative treatments of partial differential equations

Abstract

The development of high-speed digital computers has made feasible the numerical solution by iterative methods of some partial differential equations. The convergence rates of several such iterative methods are estimated here. It is found that with the familiar elementary iterative methods some quite simple problems require prohibitive computational labor. The iterative methods here considered are related to the various forms of the “relaxation method” in that they involve successively applied local corrections to improve an approximate solution. However, these iterative methods are routinized in conformity with the requirements of automatic computers while the relaxation method is flexible and depends in an essential way on the skill of its practitioners.