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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).
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Ulrich Ruede , Thu Feb 2 21:04:55 MEZ 1995
Updated by Craig C. Douglas