Generic iterative methods can be robust in the sense that they converge for a large class of matrix equations, but they typically do not obtain full efficiency without taking the origin of the problem into account. Multigrid and other methods that do take the problem origin into account can obtain full efficiency, but they are usually tailored to specific aspects of the target application. Algebraic multigrid (AMG) is a method that attempts to take the middle ground by providing a matrix equation solver that is removed from problem details, but that attains efficiency comparable to conventional multigrid. In essence, AMG abstracts multigrid principles to the matrix level by developing 'coarser' matrices and interlevel transfer operators based solely on the original matrix entries.
AMGe is a special version of AMG designed to exploit the special nature of matrices that arise in finite element discretization of partial differential equations. Assuming access to the element stiffness matrices, AMGe uses a measure derived from multigrid theory to determine local representations of algebraically "smooth" error components. We believe that this measure and the resulting representations provide the basis for effective design of coarse grids, coarse grid matrices, and interlevel transfers. Our aim is to use this measure for defining multigrid components in order to create a more robust AMG scheme. This talk will discuss this measure and recent results in AMGe on difficult problems that arise in linear elasticity.