Optimal Alternating Direction Implicit Preconditioners for Conjugate Gradient methods

In this work the (Extrapolated) Alternating Direction Implicit ((E)ADI) Preconditioners for the Conjugate Gradient (CG) Method are applied for the solution of the second order elliptic partial differential equation (PDE) in a rectangle under Dirichlet boundary conditions. It is shown that the spectral condition number for the discretized model problem, when the ADI-CG Method is used, is an order of magnitude smaller than that of the CG Method. The PDE is approximated by uniform 5- and 9-point stencils and it is shown that the determination of the optimal condition number is equivalent to that of the determination of the optimal EADI scheme. Analytic expressions for the optimal acceleration and extrapolation parameters are obtained in both cases of the 5- and 9-point schemes; those for the 5-point complete known ones while those for the 9-point are novel. Finally, numerical examples show that the EADI preconditioners proposed make the CG methods compete well against most of the fastest well-known methods.