Grid-independent Convergence Based on Preconditioning Techniques

Today numerical calculations are no longer restricted to a class of simple problems, but cope with complicated simulations and complex geometries. In many situations the accuracy of the numerical solution is determined by the limited amount of computer power and memory. Therefore much attention has been given to the development of numerical methods for solving the large sparse system of equations Ax = b obtained by discretising some partial differential equation. Since direct methods require much computer storage and CPU-time, a large variety of iterative methods has been derived. In this paper we will focus on iterative methods like MICCG and algebraic multigrid. Gustafsson [1] has shown that for several problems the CPU-time using MICCG is O(N 5/4) in 2 dimensions and O(N 7/6) for 3D-problems, where N is the total number of unknowns. Multigrid methods perform even better and for a large class of problems they have an optimal order of convergence: the amount of work and storage is proportial to the number of unknowns N. However, due to the required proper smoothers and the restriction and prolongation operators at each level, the implementation of multigrid for practical problems is much more complicated than that of MICCG. Here we look for a combination of these properties: an incomplete LU-decomposition such that the preconditioned system can be solved with the optimal computational complexity O(N) by a conjugate gradient-like method. The basic idea behind this preconditioning technique is the same as in multigrid methods. In Section 2 a preconditioning technique is described which uses a partition of the unknowns based on the sequence of grids in multigrid.