 Geometric multigrid V-cycle

A geometric multigrid V-cycle pre-smooths an approximate solution, restricts its residual, recursively solves the coarse error equation once from zero, prolongs and adds that error correction, and post-smooths. The recursion ends in a direct solve on a small coarsest grid. With fixed sweep counts and bounded stencil work, coarsening in $d$ dimensions gives total work $O(N){\sum }_{j\ge 0}{2}^{-dj}=O(N)$ per cycle. A mesh-independent convergence factor also requires compatible coarse approximation and effective smoothing; linear work alone does not prove convergence.