Error criteria on coarsest level in Multigrid method
 

Hi, I am studying multigrid method.

I refered to numerical recipes, A multigrid tutorial etc.

In reference, they said "on coarsest grid, solve exactly Au=f"

I solved simple Laplace’s equation with various error criteria ( e-01 ~ e-09) and utilized residual norm on coarsest grid.

But when I used error criteria e-01, it shows fastest elapsed time.

I tested Laplace's equation on 1D, 2D and 3D. And used structured mesh. every case showed same trends (error criteria e-01 case is fastest).

1. Why do they say "on coarsest grid, solve exactly Au=f" ?
it shows good performance when we solved the equation roughly rather than exactly.
both results are the same, but with difference on elapsed time.
does this mean my code is incorrect? And why does the textbook say " solve Au=f exactly"?

In multi grid, the details of the 'coarsest level solve' actually do not matter much, both with respect to performance (in 3D, each level has eight times less nodes than the next finer one, so with 8 levels the system you solve at the coarsest has two million times less nodes than the finest level) an with respect to accuracy, because normally you are using multiple V-cycles through the whole hierarchy until the errors residual drops below your prescribed accuracy threshold.

However, considering the fact that the coarsest level solve can be million times cheaper per degree of freedom than the full problem solve, it does make sense to invest some more iterations there for a truly exact solution at the coarsest level, just for reasons of efficiency.