It is intuitively obvious that whenever iterative techniques are used to solve a finite element or finite difference problem it is useful to start from a coarse mesh solution and then to use this coarse mesh solution as a starting point for iteration in a finer mesh. This process repeated on many meshes has been used frequently and obviously accelerates the total convergence rate. This acceleration is particularly important when a hierarchical formulation of the problem is used. We have indeed discussed such hierarchical formulations in Chapter 4 of The Finite Element Method: Its Basis and Fundamentals (Zienkiewicz, Taylor and Zhu) and the advantages are pointed out there.

The simple process which we have just described involves going from coarser meshes to finer ones. However, it is not useful if no return to the coarser mesh is done. In hierarchical solutions such returning is possible as the coarser mesh matrix is embedded in the finer one with the same variables and indeed the iteration process can be described entirely in terms of the fine mesh solution. The same idea is applied to the multigrid form of iteration in which the coarse and fine mesh solution are suitably linked and use is made of the fact that the fine mesh iteration converges very rapidly in eliminating the higher frequencies of error while the coarse mesh solution is important in eliminating the low frequencies.

To describe the process let us consider the problem of

which we discretize incorporating the boundary conditions...