The smoother is the central component of a multigrid algorithm. It is usually a linear iteration, like the Gauß-Seidel method. This is the smoother used in the example of the multigrid workbench.

Alternative smoothers include

- Richardson iteration
- Damped Gauß-Jacobi iteration
- Symmetric Gauß-Seidel
- Alternating direction Gauß-Seidel
- Red-Black Gauß-Seidel
- Four color Gauß-Seidel
- Zebra relaxation (consisting of exact solvers along lines of unknowns, also in alternating direction variants)
- Incomplete factorization
- (suitably chosen) successive over relaxation (SOR)
- ... and many more ...

To demonstrate the features of relaxation methods, we apply the Gauß-Seidel iteration to the example of the multigrid workbench. The true solution and the initial guess are

respectively.

The results after 1, 10, 100, 1000 sweeps of Gauß-Seidel are

Clearly, the solution is
approximating the true solution correctly only after 1000 iterations.
This confirms the theoretical result that on a grid with
n(=31)
gridlines
O(n^{2})
relaxations are necessary to obtain satisfactory results.

Though over-relaxation methods (SOR) could accelerate this to O(n) iterations, multigrid methods are superior, since they need only O(1) iterations.

Ulrich Ruede , Thu Feb 2 21:05:47 MEZ 1995

Updated by Craig C. Douglas