The Multigrid Workbench: Smoother

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

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(n2) 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