Multigrid method: Difference between revisions

Revision as of 08:38, 7 August 2022

Revision as of 20:40, 26 August 2022

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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. 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 setupset 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.