Multigrid method: Difference between revisions

Revision as of 11:47, 12 March 2024

Revision as of 07:04, 15 April 2024

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* '''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. These differ in which and how many coarse-grain cycles are performed per fine iteration. The V-Cycle algorithm executes one coarse-grain V-Cycle. F-Cycle does a coarse-grain V-Cycle followed by a coursecoarse-grain F-Cycle, while each W-Cycle performs two coarse-grain W-Cycles per iteration. For a [[Discrete Poisson equation#On a two-dimensional rectangular grid|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 [[Overhead (computing)|overheads]]. Typically, W-Cycle produces similar convergence to F-Cycle. However, in cases of [[Convection–diffusion equation|convection-diffusion]] problems with high [[Péclet number]]s, 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 [[Preconditioner|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.