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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?
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  • $\begingroup$ These are very broad questions. Have you read some literature about the geometric multigrid method, and how it relates to the algebraic multigrid method? $\endgroup$ – Wolfgang Bangerth Dec 2 at 16:46
  • $\begingroup$ 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? $\endgroup$ – sunshine Dec 3 at 2:37
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    $\begingroup$ 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. $\endgroup$ – Wolfgang Bangerth Dec 4 at 4:53

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