The Algebraic Multilevel Method

The multilevel method introduced here is algebraic in the sense that it is defined in terms of the coefficient matrix only. In particular, the coarse level is just a subset of unknowns with no geometric interpretation, hence the name “multilevel” rather than “multigrid.” For diagonally dominant SPD problems, we derive an a posteriori upper-bound for the condition number of the V(0,0)-cycle. For diffusion problems, this upper-bound indicates that the condition number grows only polynomially with the number of levels, independent of the meshsize and the possible discontinuities in the diffusion coefficient. This indicates that the nearly singular eigenvectors of A are handled well by the coarse-level correction, so the V(1,1)-cycle that uses relaxation to handle the rest of the error modes as well should converge rapidly.