algebraic multigrid vs geometric multigrid
 

Hi,

What is the advantages and disadvantages of Algebraic Multigrid compared with Geometric Multigrid? From my experience, algebraic multigrid is much slower than geometric multigrid. Is that always true? Why?

Hi Anna,
Algebraic multigrid is supposed to work better with complex geometries and also with highly anisotropic problems. It would be nice if you can get somewhere the book "Multigrid" from Trottemberg, there you can find a really good explanation about it, it has an appendix about algebraic multigrid.

Could you give a basic conclusion about what are the advantages of geometric multigrid method compared with algebraic multigrid method? Is it always faster?

To clarify, geometric multigrid actually discretizes the problem on a series of grids whereas algebraic multigrid simply works with the linear system. When working with unstructured grids it is really difficult to define a series of meshes for geometric multigrid so algebraic multigrid is usually used instead. It's not really a matter of speed, more so one of convenience.

The main fact about the geometric multigrid is that, by discretizing the problem on each grid level, the non-linearity is not lost, say, for Navier-Stokes equations. I don't know if for linear problems there is yet some advantage in the geometric approach.

Geometric multigrid can be used for both implicit and explicit solvers. Geometric multigrid construction is quite difficult for unstructured meshes. There are many ways to construct coarser grid for GMG. I have coded agglomeration technique for GMG which is quite generic. In case of anisotropy coarse grid construction plays an important role. There is no issue of linearity and non-linearity in case of geometric multigrid, as only the mesh changes but the system is still non-linear. There may be loss in restriction and prolongation operators (not sure of this).

Algebraic multigrid can be applied only for implicit methods. Algebraic method works on the matrix with the spirit of geometric multigrid concept. Since matrix operation is linear, AMG is basically linear. AMG is easy to implement doesn't need complex geometry coarsening. Anisotropy can be controlled by selecting the appropriate coarse level matrix.