# SMG¶

SMG is a parallel semicoarsening multigrid solver for the linear systems arising from finite difference, finite volume, or finite element discretizations of the diffusion equation,

$\nabla \cdot ( D \nabla u ) + \sigma u = f$

on logically rectangular grids. The code solves both 2D and 3D problems with discretization stencils of up to 9-point in 2D and up to 27-point in 3D. See [Scha1998], [BrFJ2000], [FaJo2000] for details on the algorithm and its parallel implementation/performance.

SMG is a particularly robust method. The algorithm semicoarsens in the z-direction and uses plane smoothing. The xy plane-solves are effected by one V-cycle of the 2D SMG algorithm, which semicoarsens in the y-direction and uses line smoothing.

# PFMG¶

PFMG is a parallel semicoarsening multigrid solver similar to SMG. See [AsFa1996], [FaJo2000] for details on the algorithm and its parallel implementation/performance.

The main difference between the two methods is in the smoother: PFMG uses simple pointwise smoothing. As a result, PFMG is not as robust as SMG, but is much more efficient per V-cycle.

# SysPFMG¶

SysPFMG is a parallel semicoarsening multigrid solver for systems of elliptic PDEs. It is a generalization of PFMG, with the interpolation defined only within the same variable. The relaxation is of nodal type- all variables at a given point location are simultaneously solved for in the relaxation.

Although SysPFMG is implemented through the SStruct interface, it can be used only for problems with one grid part. Ideally, the solver should handle any of the seven variable types (cell-, node-, xface-, yface-, zface-, xedge-, yedge-, and zedge-based). However, it has been completed only for cell-based variables.