On Generalizing the AMG Framework

Robert D. Falgout

Lawrence Livermore National Laboratory
P.O. Box 808, L-561
Livermore, CA 94551

Panayot S. Vassilevski


In recent years, much work has been done to increase the robustness of Algebraic Multigrid (AMG) methods. The classical AMG method of Ruge and Stüben [5] used heuristics based on properties of M-matrices. Although this algorithm works remarkably well for a wide variety of problems [3], the M-matrix assumption still limits its applicability. To address this, a new class of algorithms was developed based on multigrid theory: AMGe [1], element-free AMGe [4], and spectral AMGe [2]. All of these algorithms assume a basic framework in their construction. Namely, they assume that relaxation is a simple pointwise method, then they build the coarse-grid correction step to eliminate the so-called algebraically smooth error left over by the relaxation process. In the AMGe methods above, this is done with the help of a measure (or weak-approximation property) that defines the approximation property that interpolation must satisfy in order to achieve uniform convergence assuming a pointwise relaxation.

In this talk, we introduce a new measure for constructing algebraic multigrid methods. The purpose of this new measure is to generalize the AMG framework to include problems such as Maxwell's Equations, which has a particularly large null-space when discretized using the common Nédélec finite elements. In the old framework, it is necessary to take all O(N) of these null-space components to the coarse grid, which results in a non-optimal method. This problem can be overcome by using something other than pointwise relaxation to damp a subspace of these near-null-space components on the fine grid. Examples include overlapping block relaxation or the so-called Hiptmair smoother (Brandt's distributive relaxation). The measure proposed here takes into account the smoothing process and changes the AMGe heuristic in a subtle but important way. The hope is that this new measure will allow us to develop AMG methods that can handle difficult problems such as Maxwell's equations or Helmholtz.

This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.


[1] M. Brezina, A. J. Cleary, R. D. Falgout, V. E. Henson, J. E. Jones, T. A. Manteuffel, S. F. McCormick, and J. W. Ruge, Algebraic multigrid based on element interpolation (AMGe), SIAM J. Sci. Comput., 22 (2000), pp. 1570 -1592.
[2] T. Chartier, R. D. Falgout, V. E. Henson, J. E. Jones, T. A. Manteuffel, S. F. McCormick, J. W. Ruge, and P. S. Vassilevski, Spectral AMGe, SIAM J. Sci. Comput. To appear.
[3] A. J. Cleary, R. D. Falgout, V. E. Henson, J. E. Jones, T. A. Manteuffel, S. F. McCormick, G. N. Miranda, and J. W. Ruge, Robustness and scalability of algebraic multigrid, SIAM J. Sci. Comput., 21 (2000), pp. 1886-1908.
[4] V. E. Henson and P. S. Vassilevski, Element-free AMGe: general algorithms for computing interpolation weights in AMG, SIAM J. Sci. Comput., 23 (2001), pp. 629-650.
[5] J. W. Ruge and K. Stüben, Algebraic multigrid (AMG), in Multigrid Methods, S. F. McCormick, ed., vol. 3 of Frontiers in Applied Mathematics, SIAM, Philadelphia, PA, 1987, pp. 73-130.