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:
- Presmooth: $u^h := \mathcal{S}(A_h,u^h_0,f^h)$
- 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.