Analysis of Aggregation-Based Multigrid

Aggregation-based multigrid with standard piecewise constant like prolongation is investigated. Unknowns are aggregated either by pairs or by quadruplets; in the latter case the grouping may be either linewise or boxwise. A Fourier analysis is developed for a model two-dimensional anisotropic problem. Most of the results are stated for an arbitrary smoother (which fits with the Fourier analysis framework). It turns out that the convergence factor of two-grid schemes can be bounded independently of the grid size. With a sensible choice of the (linewise or boxwise) coarsening, the bound is also uniform with respect to the anisotropy ratio, without requiring a specialized smoother. The bound is too large to guarantee optimal convergence properties with the V-cycle or the standard W-cycle, but a W-cycle scheme accelerated by the recursive use of the conjugate gradient method exhibits near grid independent convergence.