10-08-2012, 02:01 PM
Multigrid Tutorial
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First observation toward
multigrid
• Many relaxation schemes have the smoothing
property, where oscillatory modes of the error
are eliminated effectively, but smooth modes
are damped very slowly.
• This might seem like a limitation, but by using
coarse grids we can use the smoothing property to
good advantage.
Reason #1 for using coarse
grids: Nested Iteration
• Coarse grids can be used to compute an improved
initial guess for the fine-grid relaxation. This is
advantageous because:
– Relaxation on the coarse-grid is much cheaper (1/2 as
many points in 1D, 1/4 in 2D, 1/8 in 3D)
– Relaxation on the coarse grid has a marginally better
convergence rate, for example
Second observation toward
multigrid
• Recall the residual correction idea: Let v be an
approximation to the solution of Au=f, where the
residual r=f-Av. The the error e=u-v satisfies
Ae=r.
• After relaxing on Au=f on the fine grid, the error
will be smooth. On the coarse grid, however, this
error appears more oscillatory, and relaxation will
be more effective.
• Therefore we go to a coarse grid and relax on the
residual equation Ae=r, with an initial guess of e=0.