Multigrid is a powerful solver for large-scale linear systems that arise from discretized partial differential equations. The cycle index, denoted by \(\gamma \) , of standard multigrid schemes is a fixed positive integer over all coarse levels. In particular, \(\gamma =1\) corresponds to the V-cycle and \(\gamma =2\) to the W-cycle. A common wisdom in the field of multigrid is that a multigrid method with \(\gamma \ge 2\) can carry over the uniform convergence of two-grid ones. In practice, one may use variable indices in order to combine the advantages of different cycles. The resulting methods can be seen as multigrid schemes with fractional cycle indices. In this paper, we present a convergence theory for VW-cycle and WV-cycle multigrid methods, which are formed by performing the V- and W-cycles alternately (i.e., \(\gamma \approx 1.5\) ). Our convergence theory shows that, if exact two-grid convergence factors are uniformly bounded by \(\sigma <1-1/\sqrt{2}\) , then the corresponding VW-cycle (resp., WV-cycle) multigrid convergence factor can be uniformly bounded by \(1/(1-\sigma )^{2}-1\) (resp., \(\sigma ^{2}/(1-\sigma )^{2}+\sigma /(1-\sigma )^{3}\) ).
