Higher order multigrid methods

Abstract

This paper is concerned with the treatment of higher order multi-grid techniques for obtaining accurate finite difference approximations to partial differential equations. The three basic techniques considered are a multi-grid process involving smoothing via higher order difference approximations, iterated defect corrections with multi-grid used as an inner loop equation solver, and tau-extrapolation. Efficient versions of each of these three basic schemes are developed and analyzed by local mode analysis and numerical experiments. The numerical tests focus on fourth and sixth order discretizations of Poisson’s equations and demonstrate that the three methods performed similarly yet substantially better than the usual multi-grid method, even when the right-hand side lacked sufficient smoothness.