Multilevel preconditioning methods for finite element matrices for the approximation of second-order elliptic problems are considered. Using perturbations of the local finite element matrices by zero-order terms it is shown that one can control the smallest eigenvalues. In this way in a multilevel method one can reach a final coarse mesh, where the remaining problem to be solved has a condition number independent of the total degrees of freedom, much earlier than if no perturbations were used. Hence, there is no need in a method of optimal computational complexity to carry out the recursion in the multilevel method to a coarse mesh with a fixed number of degrees of freedom. 1
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