J. Comp. Math., 9 (1991), pp. 17-27. Multigerid Multi-Level Domain Decomposition Sheng Zhang 1, Hong-ci Huang 11 Computing Center, Academia Sinica, Beijing, China Received 1990-3-10 Revised Online 2006-12-8 Abstract The domain decomposition method in this paper is based on PGG (Preconditioned Conjugate Gradient method). If N is the number of subdomains, the number of sub-problems solved parallelly in a PGG step is $\frac{4}{3}(1-\frac{4^{\log N+1}}{1})N$. The condition number of the preconditioned system does not exceed $O(1+\log N)^3$. It is completely independent of the mesh size. The number of iterations required, to decrease the energy norm of the error by a fixed factor, is proporitional to $O(1+\log N)^{\frac{3}{2}}$ . Key words: