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This paper is the second part of a work on stabilizing the classical hierarchical basis HB by using wavelet-like basis functions. Implementation techniques are of major concern for the multilevel preconditioners proposed by the authors in the first part of the work, which deals with algorithms and their mathematical theory. Numerical results are presented to confirm the theory established there. A comparison of the performance of a number of multilevel methods is conducted for elliptic problems of three space variables.