Comparison of S and V cycles in multigrid method for linear elliptic equations with variable coefficients

Abstract

We implement and compare the Sawtooth (S) and Vcycles in multigrid method (mgd) for the Poisson equation and for the fourth-order 13-point methods derived by Krishnaiah et al. Defect correction in the mgd context is also studied. Computations reveal that the S cycle is much cheaper compared to the V cycle in most of the situations. Defect correction is much cheaper than the direct application of the 13-point formula, giving almost the fourth-order accurate results.

Sawtooth cycle revisited

“…Multigrid methods move the hierarchy recursively, and one recursive procedure is referred to as a cycle. Although many methods have been proposed to accelerate multigrid methods (see, e.g., the work of McAdams et al), most of these methods conventionally employ the V‐cycle multigrid methods, and little attention has been paid to other cycles such as the W‐cycle, the F‐cycle, and the sawtooth cycle . As pointed out by Satteluri et al, the sawtooth cycle has potential for being an alternative to the V‐cycle due to the low computation cost, but the sawtooth cycle has not been used in the computer graphics field so far.…”