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On the finest level (with 33 33 grid points) the result of presmoothing and the interpolated coarse grid correction are combined. The result is

Then a single sweep of the Gauß-Seidel method is applied to smooth the result.

This is the result of the V-Cycle. Qualitatively it looks almost like the true solution, however, it is still about 10% wrong. In contrast to smoothing alone, all Fourier components of the error have been reduced.

However, one single
V-cycle
does not produce the correct result.
The V-cycle must be iterated, each application improving the result by
approximately one digit.
The
convergence rate,
however, is independent of the problem size, so that multigrid
is an *asymptotically* optimal solver.

In practice, multigrid should be used combined with
nested iteration
(also called *full multigrid*).

Ulrich Ruede , Thu Feb 2 21:04:55 MEZ 1995

Updated by Craig C. Douglas