Multigrid methods for structured grids with odd no. of rows and columns
 

During the coarsening step of a multigrid method, we need to create a smaller grid out of the bigger one.
This gets easier to do when the no. of rows and columns are even (and most preferably a power of 2). Just dividing by 2 gets us the dimensions of the new coarsened grid.


What to do when the no. of rows and columns are odd?


I got the idea of just taking the middle row and column of the fine grid and pretend they're actually made of two rows and columns. So, the total no. of rows and columns become even, and we can simply copy over the middle column from the fine grid to the coarse grid. I expect that might create some issues in the preservation of the conserved variables.

There is no requirement in the geometric multigrid that the reduction must be by a power of 2 everywhere all the time. What matters is that you have a hierarchy of coarser and coarser meshes.

You do not have to do a reduction by 2 everywhere all the time. You can do mostly a reduction of 2 and a reduction by 3 when needed. A 1D example, a 5x1 matrix can be merged into a 2x1 and 3x1 matrix. A 5x5 can be merged into a 2x2, 3x2, 2x3, and 3x3. You can also turn your brain completely off and merge the entire 5x1 into a single cell. If you can follow that much, it comes down to constructing the correct restriction and prolongation operators at each level.


It's tempting to merge by powers of 2 when you give an example of a grid with an even x even number of rows and columns, all with uniform spacings... But keep in mind that's not necessarily desirable if you have the same even by even grid but the spacing is non-uniform...