coarsening coefficient matrixes (A2h, A4h...) for geometric multigrid method in 2-D/3-D

Question

I am learning about multigrid methods from the textbook section 6.3 Multigrid Methods, which shows a geometric multigrid algorithm for 1-D examples in detail, including how to build restriction/interpolation matrices (i.e., eqs.1, 3) and how to get a coarse coefficient matrix (eq. 6).
I realize that these matrices and way of coarsening the coefficient matrix are important for multigrid methods.
But for the 2-D case, only the ways of getting restriction and interpolation matrices are shown in the book, not the coefficient matrix for coarse grid (i.e., $A_{2h}$, $A_{4h}$, etc.).
This missing link is barrier for me to go further to learn multigird methods.
Can anybody help me?

rchilton1980 · Answer

For geometric multigrid, the $mathbf A_{2h}$ and $mathbf A_{4h}$ (etc) matrices are just discretizations of the same PDE on coarser grids.
For instance, if your original $mathbf A_{h}$ was a finite difference approximation of the laplacian on a 64 point grid, then $mathbf A_{2h}$ would be a finite difference approximation of the laplacian on a 32 point grid, etc. You should also see some sort of relationship where $mathbf A_{2h} = mathbf R mathbf A_h mathbf R^T$ (maybe to within a constant? I forget).
In algebraic multigrid, only the latter relationship really applies. There is no "fine grid" or "coarse grid", just the matrices themselves, and the restriction operator itself uniquely defines the coarsened problem(s).

coarsening coefficient matrixes (A2h, A4h...) for geometric multigrid method in 2-D/3-D

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