# What kind of problem or matrices are suitable for multigrid method?

For Poisson or Convection-diffusion equation as follows: $$-\Delta u=f,\qquad u|_\Omega = g.$$ or $$-\Delta u +\vec{w}.\nabla u=f,\qquad u|_\Omega = g.$$ using FDM or FEM discretization, we can get $$Ax=b.$$ I have known that Multigrid method is a very efficient method for these equations and I have done some numerical examples using AMG code bulit-in some known software, e.g., ifiss. Indeed, I do not know why AMG works so efficient (because the principles seem so complex that I cannot understand, then I just use it), from which I indeed can get an optimal reslut, i.e., iteration step independent on mesh size $$h$$, and CPU time is almost linear with the unknowns of the system of linear equations.

My questions is as follows:

1. why amg works so efficient for Poisson and C-D equations? (can you give me some simple and clear explations without detailed principles?)
2. What other else matrices are suitable for amg method? so far, I have just known that matrices from Poisson and C-D equations are best fit for AMG? anyone else?
• These are very broad questions. Have you read some literature about the geometric multigrid method, and how it relates to the algebraic multigrid method? – Wolfgang Bangerth Dec 2 '19 at 16:46
• Thanks Prof. Bangerth, I tried to read some but it is too complicated for me to understand it. Then my teacher told us we can just use the existing software to precondition a linear system and do not need to understand the inner principle. So I just know that for Poisson equations, it is suitable for AMG, and I just can use the software to solve the Ax=b using AMG preconditioner but I do not know the reasons. Thanks Prof. Can you tell me what kind of matrices are fit for AMG, simply? – sunshine Dec 3 '19 at 2:37
• Well, the problem is that if I explain what matrices are suitable, the explanation will not be much simpler than the one you find in the literature. It really is a complicated subject. But the short answer is that multigrid works particularly well for matrices that result from elliptic operators and that are, as a consequence, symmetric, positive definite, and sparse. – Wolfgang Bangerth Dec 4 '19 at 4:53