Approximation and Coupling Estimators for Algebraic Multigrid

Jan Mandel

Center for Computational Mathematics
Department of Mathematics
University of Colorado at Denver
P.O. Box 173364, Campus Box 170
Denver, CO 80217-3364


A-priori estimates of approximation of fine grid functions by coarse grid functions are important for the design of robust Algebraic Multigrid methods. A number of coarsening schemes will work well on an easy problem, such as a the Laplace equation discretized by linear elements on a reasonable unstructured grid. Methods that incorporate rigid body modes, such as [1], work also very well for elasticity. Realistic problems, however, typically include elements violating shape limits, large jumps of coefficients, and special kinds of elements that destroy the numerical relevance of the underlying differential equations, such as side constraints on the values of degrees of freedom, enforced by large penalties ("stiff spring" or "contact" elements in engineering parlance), or even arbitrary additional equations that are eliminated before the matrix is passed to the solver ("multiple point constraints"). Such problems are hard to solve by Algebraic Multigrid even if they are symmetric and positive semidefinite. Without a-priori numerical estimates of the rate of convergence, with a rigorous foundation, an Algebraic Multigrid algorithm is essentially based just on the hope that the problem will not have anything unexpected and things will work out in the end.

One common estimate that can be computed a-priori is the weak approximation property, which bounds the error of the best approximation in Euclidean norm of a fine grid vector by the prolongation of a coarse grid vector in terms of the energy norm of the fine grid vector. The weak approximation property is known to imply a bound on the two-level convergence factor, albeit a fairly pessimistic one. In the smoothed aggregation method [1], two-level as well as multilevel convergence bounds can be obtained [2] from the weak approximation property for the so-called tentative prolongator, which is simply the transpose of the matrix of a weighted aggregation of degrees of freedom. The actual prolongation used in the multigrid algorithm is then obtained by smoothing the tentative prolongation.

In the absence of multiple point constraints, the constant in the weak approximation property can be bounded rigorously from the solution of eigenvalue problems based on local element matrices. In [3], it was proposed to select the columns of the tentative prolongator as the eigenvectors of the local problems and to control the convergence of algebraic multigrid by choosing the number of the eigenvectors and by selecting the amount of smoothing of the prolongation.

In practice, the problem to be solved is most conveniently given in terms of a single global stiffness matrix with all constraints incorporated. Then the information contained in the local stiffness matrices is lost, and to bound the constant in the weak approximation property rigorously by the solution of local eigenvalue problems, one would need to decompose the global matrix into the sum of positive semidefinite local matrices. We show that in general, such decomposition does not exists if the dimension of the coarse space is more than two. To estimate the contribution of a single aggregate (or, equivalently, of a block of coarse basis functions) to the constant in the weak approximation property, we decompose the global matrix into local matrices corresponding to the decomposition of the set of all nodes into the given aggregate and its complement. We also present further approximation techniques to reduce the cost of the estimation. The resulting estimates are not rigorous but they are still practically useful.

We show that bad constants in the weak approximation condition will sometimes arize when the coarsening does not follow strong couplings of the nodes. An analysis of the mechanism how this happens shows when coarsening along weak couplings can still be tolerated. We observe that while strong couplings can be usually quite well determined from the local stiffness matrices, the computation of strong couplings from the global matrix is not reliable. We propose a new method to estimate the strength of the coupling between two nodes from an approximate local matrix corresponding to the pair of nodes.

[1] P. Vanek, J. Mandel, and M. Brezina, Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems, Computing 56 (1996) 179-196
[2] P. Vanek, M. Brezina, and J. Mandel, Convergence of Algebraic Multigrid Based on Smoothed Aggregation, UCD/CCM Report 126, February 1998. Revised February 2000. To appear in Numerische Mathematik.
[3] M. Brezina, C. Heberton, J. Mandel, and P. Vanek, An Iterative Method with Convergence Rate Chosen a Priori, UCD/CCM Report 140, April 1999.