Aggregation-Based Algebraic Multilevel Preconditioning

We propose a preconditioning technique that is applicable in a “black box” fashion to linear systems arising from second order scalar elliptic PDEs discretized by finite differences or finite elements with nodal basis functions. This technique is based on an algebraic multilevel scheme with coarsening by aggregation. We introduce a new aggregation method which, for the targeted class of applications, produces semicoarsening effects whenever desirable, while the number of nodes is decreased by a factor of about 4 at each level, regardless of the problem at hand. Moreover, the number of nonzero entries per row in the successive coarse grid matrices remains approximately constant, ensuring small set up cost and modest memory requirements. This aggregation technique can be used in an algebraic multigrid (AMG)-like framework, but better results are obtained with an algebraic multilevel scheme based on a block approximate factorization of the matrix. In this scheme, the block pivot corresponding to fine grid nodes is approximated by a modified incomplete LU (MILU) factorization. To enhance robustness and avoid any potential breakdown, the coarsening process is refined by recasting as “coarse” fine grid nodes for which the corresponding pivot in this MILU factorization would be negative or too small. Numerical results display the efficiency, the scalability, and the robustness of the resulting preconditioner on a wide set of discrete scalar PDE problems, ranging from the two-dimensional Poisson equation to three-dimensional convection-diffusion problems with high Reynolds number and strongly varying convection.