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Suppose we are dealing with a nonlinear problem, say $$ A u := L u + G(u) = f $$ the nonlinearity of the operator $A$ is the polynomial type, ie, $L$ is a linear operator, and $G(u) = u^k$, or more generally we have a Fréchet derivative bound like: $$ \big\|A(u) - A(v) -DA(u)(u-v) \big\|\leq C \|A(u) - A(v)\| $$ where $D$ is the Fréchet derivative for the operator $A$.

We would like to use some numerical method together with an iterative solver to get a finite dimensional approximation to this problem. For a two-grid cycles multigrid, before we compute the residual at a finer level of mesh size $h$(vs coarse level mesh size $2h$), according to the book written by Saad chapter 13, we would like to apply a presmoothing operator $\mathcal{S}$ to the initial guess $u^h_0$ at the finer level:

  1. Presmooth: $u^h := \mathcal{S}(A_h,u^h_0,f^h)$
  2. Compute the residual: $r^h = f^h - A_h u^h$

Then in next few steps we could go back the coarse mesh $2h$ to solve and correct $u^h$ at the finer level. Here $A_h$ is a finite dimensional linearization matrix of the operator $A$.

Now my questions are: What are the criteria for choosing a "good" presmooth operator for nonlinear multigrid method? and how to construct this presmooth operator? Is there a routine to follow? Appreciate if you could share me some references having examples similar to this type of problem.

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  • $\begingroup$ Do you intend that statement about $D$ to hold for all $u$ and $v$? And do you intend for $D$ to be the Fréchet differential operator or the derivative? $\endgroup$ – Jed Brown Feb 24 '12 at 23:16
  • $\begingroup$ @JedBrown Thanks for the heads up Jed, I meant to say $D$ being the Fréchet differential operator, and I just corrected my typos. $\endgroup$ – Shuhao Cao Feb 24 '12 at 23:29
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Brandt's book Multigrid techniques has a good discussion of the Full Approximation Scheme, the 1982 Guide to Multigrid Development also has some discussion. As with linear problems, the requirement is for the smoothers to remove high-frequency components of the residual. The formal spectral analysis only applies to linearized problems, but FAS theory only needs local linearization. Trottenberg, Oosterlee, and Schueller's 2001 book on multigrid is also a good reference and contains local Fourier analysis for nonlinear problems.

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