# Question about the smoothing operators in multigrid methods for nonlinear PDEs

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.

• 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? – Jed Brown Feb 24 '12 at 23:16
• @JedBrown Thanks for the heads up Jed, I meant to say $D$ being the Fréchet differential operator, and I just corrected my typos. – Shuhao Cao Feb 24 '12 at 23:29