# What does it take to prove that a multigrid algorithm scales linearly with system size?

It is my understanding that the multigrid solution techniques are generally the preferable method to solve large Poisson problems. Now assume I have written a multigrid solver that is tailored to my special needs that actually works quite well, i.e. the residual norm decreases by a factor ~10 with every successive V-cycle.

How would I actually show/prove/test for textbook efficiency? Can optimal scalability only be shown for a fixed continuous problem that is discretized onto increasingly finer grids? If so, are there any special properties such a problem should have?

• What do you mean by "system size"? The size of the problem or the number of computational units that compute the solution in parallel? – Jakub Klinkovský May 13 '17 at 9:14
• The size of the problem. I.e. the number of unknowns. – user20867 May 15 '17 at 6:11