I am trying to construct transfer operators $I^H_h, \, I^h_H$ for multigrid where $H \ne 2h$. I have gone through Briggs' tutorial, Hemker's paper, Hackbush's book, Trottenberg's book, but the details there are extremely scarce on the derivation of the presented schemes. In Hemker's paper ("On the order of prolongations and restrictions in multigrid procedures") there is the following statement:
If an interpolation rule leaves all polynomials of degree k – 1
invariant, then both the low and the high frequency order are equal to
k.
Later on in a proof the meaning of what leaving all polynomials invariant is made more precise:
If the restriction stencil $[c_j]$ leaves all polynomials $P_k$ of
degree $\leq k$ invariant, then
$$P_k(x_k) = \sum_j c_j P_k(x_{k-j}),$$
And additionally:
If the interpolation stencil $[c_j]$ leaves all polynomials $P_k$ of degree
$\leq k$ invariant, then
$$P_k(x_k) = \sum_j c_{2j-p} P_k(x_{k+p-2j}),$$
where $p \in \{0,1\}^n$. As I understand it $x_k = kh$ (the notation $x_k$ is unfortunately not defined in the paper). So I am assuming that from the above the weights $c_j$ can somehow be derived. There is also a note that "the theory can be made more general" (in the sense of not requiring $H=2h$), but I could not find the cited report anywhere.
On the other hand in Trottenberg's book, there is a remark that the full weighting scheme can be derived from:
$$\int_{[x-h,x+h]\times[y-h,y+h]}w(x,y)\,dxdy = \int_{[x-h,x+h]\times[y-h,y+h]}(I^{2h}_hw)(x,y)\,dxdy,$$
by approximating the rhs with the midpoint rule and approximating the lhs with the trapezoidal rule.
Is there some reference going into more details on how the transfer operators are actually derived? Ideally with a generalization to $H\ne 2h$. Notably, for the case $H=2h$, the full weighting scheme should be derivable from linear interpolation, and higher order schemes from cubic, quintic, etc. interpolation. In Hackbush's book tensor products are used to construct higher-dimensional analogues even for the stencil resulting from cubic interpolation. Does the same idea work for $H\ne 2h$? And how is this related to the above conditions of Hemker of polynomial invariance under the restriction/interpolation stencils?
Best Answer
I have found an answer to this. Basically one writes the interpolant for the coarse data points and then resamples (regardless of the ratio of $H$ to $h$). Some details can be found in my cross-post at the scicomp stack: Prolongation and restriction operators in multigrid for high order PDEs
More details regarding the optimality wrt grids can be found in: Multigrid methods: grid transfer operators and subdivision schemes.