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A type of incomplete decomposition preconditioner based on local block factorization is considered, for the matrices derived from discreting 2-D or 3-D elliptic partial differential equations. We prove that the condition numbers of the preconditioned matrices are small, which means that the constructed preconditioners are effective. Further we consider an efficient parallel version of the preconditioner which depends only on a single integer argument. When its value is small, the iterations needed on multiple processors to converge is much more than on a single processor But with the increase of this value, the difference decreases step by step. Finally, we have many experiments on a cluster of 6 PCs with main frequencies of 1.8GHz. The results show that the local block factorization constructed are efficient in serial implementation, if compared to some well-known effective preconditioners, and the parallel versions are also efficient.