Project 21 returns to the large sparse linear systems considered in project 20, but now develops a new iterative scheme, called the multigrid method.
In project 20, you saw that many of the iterative methods converged quickly when the problem size was small, but that convergence slowed down as N increased. In some ways, this is due to the fact that a change to a single variable affects the entire solution; in an iterative method, the change in any variable only spreads slowly. Essentially, each iteration spreads the updated value only to the "neighbors" of the variable, that is, those variables which share an equation with the given variable. So in a big problem, there's a long delay as the improvements to the solution work their way around.
The multigrid method actually constructs a family of grids of varying sizes, and a corresponding family of solutions; on the small grids, communication is fast but approximation is poor. The idea is that you want to work on the fine grid, but you use the coarser grids to quickly spread the improved solution values around.
This sounds like a great deal of effort, but it can be done in an orderly way, and the improvement in convergence rate can be impressive.
In the case study, we start out on a simple 1D version of the Poisson equation, where the multigrid idea is easy to implement. We then move on to consider the same kind of 2D problems we looked at in projects 13, 15, 16 and 20.
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