Higher Order Multi - Grid Methods

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. Introduction. The goal of the numerical solution of partial differential equations is to obtain the highest accuracy possible within the constraints imposed by limitations in computer time and storage. Using higher order discretizations of the differential equation provides a means of obtaining this accuracy without requiring large amounts of storage. Higher order approximations, however, are more expensive to obtain due to the added complexity of the resulting discretized equations. Recent advances in fast solution techniques have made it more feasible to attempt these super accurate approximations. The Multi-Grid method is one such fast solver which is easily adapted to accommodate higher order processes. This paper treats three higher order multi-grid methods applied to finite difference discretizations of a linear partial differential equation. This paper is primarily concerned with comparing the numerical performance of the three higher order multi-grid solution processes. The model problem used for the experiments is given by (1) LU = F in the interior of ß, U = G on the boundary of ß(8ß), where L is the Laplace operator, ß is the unit square in R2, F is a continuous function defined on the interior of ß and G is a continuous function defined on 9ß. Several discretizations of (1), using second, fourth and sixth order differences, are considered. A number of algorithms based on these higher order multi-grid methods are tested in numerical experiments for their efficiency of solution (i.e., the work required to obtain a given accuracy). The algorithms are defined by various choices of discretizations and multi-grid processes, grid transfers, types of relaxation, etc. Received June 21, 1982; revised November 2, 1983. 1980 Mathematics Subject Classification. Primary 65N20, 65F10. * This work was supported by the National Science Foundation under grant no. MCS78-03847 and the Air Force Office of Scientific Research under grant no. F33615-79-3223. ©1984 American Mathematical Society 0025-5718/84 $1.00 + $.25 per page 89 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use