Substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice. These systems often arise from discretizing partial differential equations over unstructured meshes, and the particular parameters or geometry of the physical problem being discretized may be unavailable to the solver. Algebraic multilevel methods have been of particular interest in this context because of their promises of optimal performance without the need for explicit knowledge of the problem geometry. These methods construct a hierarchy of coarse problems based on the linear system itself and on certain assumptions about the low-energy components of the error.
For smoothed aggregation methods applied to discretizations of elliptic problems, these assumptions typically consist of knowledge of the near-nullspace of the weak form. We describe a recently developed extension of the smoothed aggregation method in which good convergence properties are achieved in situations where explicit knowledge of the near-nullspace components may be unavailable. This extension is accomplished by using the method itself to determine near-nullspace components when none are provided. The coarsening process is modified to use and improve the computed components.
Numerical experiments include test problems arising in the LANL's hydrocode SAGE, featuring cell-centered discretization, automatic mesh refinement and coefficient discontinuities.