Smoothed aggregation is a well-known algebraic multigrid method that is based on minimizing the energy of grid transfer basis functions. It has been used on a wide variety of problems and has had particular success in elasticity.
An overview of the parallel smoothed aggregation multigrid method is first given. Then, several generalizations of the smoothed aggregation idea are discussed. These generalizations are intended to improve method robustness for anisotropic problems and highly variable coefficient problems as well as extend the applicability of smoothed aggregation to other problem domains such as the eddy current equations and the Navier-Stokes equations.
The central theme of this talk will be generalizations of the prolongator smoothing step which maintain the exact interpolation of null space components. Two particular ideas will be emphasized. The first concerns adapting the prolongator smoother to irregular aggregates. This involves shifting support between basis functions in a way that maintains the low energy nature of the basis functions while improving the sparsity of the resulting multigrid operators. The second idea considers generalizations of the notion of "energy" and the use of local damping parameters. This allows smoothed aggregation to be used on nonsymmetric systems and often improves the smoothing of individual basis functions (leading to faster rates of convergence).
We present numerical experiments that compare the new methods to traditional smoothed aggregation on a variety of problems.