Multigrid Multi-Level Domain Decomposition

Abstract

The domain decomposition method in this paper is based on PCG (Preconditioned Conjugate Gradient method). If $N$ is the number of subdomains, the number of sub-problems solved parallelly in a PCG step is $\frac{4}{3}(1-\frac{1}{4^{\log N+1}})N$. 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 proportional to $O(1+\log N)^{\frac{3}{2}}$ .