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Multigrid Methods

Simple iterative methods (such as the Jacobi method) tend to damp out high frequency components of the error fastest (see §). This has led people to develop methods based on the following heuristic:

1. Perform some steps of a basic method in order to smooth out the error.
2. Restrict the current state of the problem to a subset of the grid points, the so-called ``coarse grid'', and solve the resulting projected problem.
3. Interpolate the coarse grid solution back to the original grid, and perform a number of steps of the basic method again.

The method outlined above is said to be a ``V-cycle'' method, since it descends through a sequence of subsequently coarser grids, and then ascends this sequence in reverse order. A ``W-cycle'' method results from visiting the coarse grid twice, with possibly some smoothing steps in between.

An analysis of multigrid methods is relatively straightforward in the case of simple differential operators such as the Poisson operator on tensor product grids. In that case, each next coarse grid is taken to have the double grid spacing of the previous grid. In two dimensions, a coarse grid will have one quarter of the number of points of the corresponding fine grid. Since the coarse grid is again a tensor product grid, a Fourier analysis (see for instance Briggs [41]) can be used. For the more general case of self-adjoint elliptic operators on arbitrary domains a more sophisticated analysis is needed (see Hackbusch [116], McCormick [145]). Many multigrid methods can be shown to have an (almost) optimal number of operations, that is, the work involved is proportional to the number of variables.

From the above description it is clear that iterative methods play a role in multigrid theory as smoothers (see Kettler [133]). Conversely, multigrid-like methods can be used as preconditioners in iterative methods. The basic idea here is to partition the matrix on a given grid to a structure with the variables in the second block row corresponding to the coarse grid nodes. The matrix on the next grid is then an incomplete version of the Schur complement The coarse grid is typically formed based on a red-black or cyclic reduction ordering; see for instance Rodrigue and Wolitzer [175], and Elman [89].

Some multigrid preconditioners try to obtain optimality results similar to those for the full multigrid method. Here we will merely supply some pointers to the literature: Axelsson and Eijkhout [16], Axelsson and Vassilevski [21][22], Braess [34], Maitre and Musy [142], McCormick and Thomas [146], Yserentant [213] and Wesseling [210].