Domain decomposition preconditioners for the conjugate gradient method (Q1122935)

The paper is devoted to the comparison of different DD (domain decomposition) preconditioners M for the conjugate gradient method applied to the solution of the linear system \(Ax=b\) of algebraic equations arising from the standard five-point finite difference approximation to a self-adjoint linear 2nd-order elliptic PDE in a rectangle with Dirichlet, or Neumann boundary conditions (model problem). The DD preconditioners M are derived by various approximate (incomplete) factorizations obtained from the well-known exact DD factorization. A complexity analysis is made with respect to the arithmetical operations per iteration step and to the degree of parallelism of the methods, but no estimate of the relative condition number \(\kappa (M^{-1}A)\) is given. The numerical experiments presented aim to show the sensitivity of the methods to the number of subdomains \((=number\) of processors) for a constant discretization parameter h. To clarify the dependence of the iteration numbers on h, numerical tests for varying h would be useful.