Interpolation or prolongation – interpolating a correction computed on a coarser grid into a finer grid. Correction – Adding prolongated coarser grid solution onto the finer grid. There are many choices of multigrid methods with varying trade-offs between speed of solving a single iteration and the rate of convergence with said iteration. The 3 main types are V-Cycle, F-Cycle, and W-Cycle. For a discrete 2D problem, F-Cycle takes 83% more time to compute than a V-Cycle iteration while a W-Cycle iteration takes 125% more. If the problem is set up in a 3D domain, then a F-Cycle iteration and a W-Cycle iteration take about 64% and 75% more time respectively than a V-Cycle iteration ignoring overheads. Typically, W-Cycle produces similar convergence to F-Cycle. However, in cases of convection-diffusion problems with high Péclet numbers, W-Cycle can show superiority in its rate of convergence per iteration over F-Cycle. The choice of smoothing operators are extremely diverse as they include Krylov subspace methods and can be preconditioned. Any geometric multigrid cycle iteration is performed on a hierarchy of grids and hence it can be coded using recursion. Since the function calls itself with smaller sized (coarser) parameters, the coarsest grid is where the recursion stops. In cases where the system has a high condition number, the correction procedure is modified such that only a fraction of the prolongated coarser grid solution is added onto the finer grid. Computational cost This approach has the advantage over other methods that it often scales linearly with the number of discrete nodes used. In other words, it can solve these problems to a given accuracy in a number of operations that is proportional to the number of unknowns. Assume that one has a differential equation which can be solved approximately (with a given accuracy) on a grid with a given grid point density . Assume furthermore that a solution on any grid may be obtained with a given effort from a solution on a coarser grid . Here, is the ratio of grid points on "neighboring" grids and is assumed to be constant throughout the grid hierarchy, and is some constant modeling the effort of computing the result for one grid point. The following recurrence relation is then obtained for the effort of obtaining the solution on grid : And in particular, we find for the finest grid that Combining these two expressions (and using ) gives