Optimal V-cycle algebraic multilevel preconditioning (Q2713565)

This paper deals with the iterative solution of large sparse symmetric positive (non-negative) definite linear systems arising from the discretization of second order elliptic partial differential equations. The author considers algebraic multilevel preconditioning methods based on the recursive use of a \(2\times 2\) block incomplete factorization procedure in which the Schur complement is approximated by a coarse grid matrix. The combination of these techniques with a smoothing procedure which is much the same as the one used in standard multigrid algorithms, except that smoothing is not required on the finest grid, is analyzed. NEWLINENEWLINENEWLINEThe theoretical results prove optimal convergence properties for the V-cycle under an assumption similar to the `approximation property' of the classical multigrid convergence theory. On the other hand, the presented numerical experiments made on both 2D and 3D problems show that the condition number is close to that of the two-level method. Further, the method appears to be robust in the presence of discontinuity and anisotropy, even when the material interfaces are not aligned with the coarse grid.