Multigridded

Multigridded is a term used in computational mathematics and numerical analysis to describe a class of algorithms designed to solve partial differential equations (PDEs) or other large systems of linear equations. The core idea behind multigridded methods is to accelerate the convergence of iterative solvers by employing a hierarchy of grids. Instead of solving the problem on a single, fine grid, which can be computationally expensive, multigridded methods operate on a sequence of grids of decreasing resolution.

These methods typically start with a coarse grid, where the problem is approximated and solved relatively quickly. The solution obtained on the coarse grid is then projected onto a finer grid and used as an initial guess for a solver operating on that finer grid. This process is repeated, moving from coarser to finer grids, with the information from the coarser levels helping to smooth out the error on the finer levels. Conversely, the finer grids can provide more accurate residual information that is then smoothed on the coarser grids. This "coarse-to-fine" and "fine-to-coarse" transfer of information is fundamental to the effectiveness of multigridded techniques.

The efficiency of multigridded algorithms stems from their ability to address different error components at different scales. On fine grids, local, high-frequency errors are effectively smoothed by standard iterative methods like Jacobi or Gauss-Seidel. However, these standard methods converge slowly for low-frequency errors, which represent global trends or smoother variations in the solution. Multigridded methods tackle these low-frequency errors by transferring them to coarser grids, where they become high-frequency errors and can then be efficiently smoothed. This multi-scale approach allows multigridded solvers to achieve convergence rates that are often independent of the grid size, making them highly scalable for large problems.