Multigrid and Multilevel Methods in Science and Engineering

The fundamental idea behind all multigrid methods is to compute using diierent scales in order to obliterate error components of diierent scales. Error components can be thought of as waves. A long wave, which has low frequency, on one grid is a short wave, or high frequency, on a coarse enough related grid. A number of iterative methods (e.g., relaxation and alternating direction implicit methods) are very eecient at damping short wave error components, but not not as eecient at damping long wave error components. A particularly well written motivation for multigrid methods can be found in Brigg's book 1]. Closely related to multigrid are multilevel methods. For problems arising from partial diierential equations on grids, there is no real diierence between these methods. For problems in which the grid is not used in forming the coarser approximations or for problems in which there never was a grid, there is a methodology that is quite similar to multigrid. A sequence 1