Stationary biparametric ADI preconditioners for conjugate gradient methods

In the present article we determine optimal stationary biparametric ADI preconditioners for the conjugate gradient methods when applied for the solution of a model problem second order elliptic PDE. The PDE is approximated by 5- and 9-point stencils. As was proved in Hadjidimos and M. Lapidakis [Optimal alternating direction implicit preconditioners for conjugate gradient methods, 〈 http://www.math.uoc.gr/ ∼ hadjidim/hadlap05.ps 〉 ] the problem of determining the best ADI preconditioner is equivalent to that of determining the optimal extrapolated (E) ADI method. So, analytic expressions are found for the optimal acceleration and extrapolation parameters for both discretizations where those for the 9-point stencil ones are new. Finally, numerical examples run using other well-known preconditioners show that the ADI ones we propose are very competitive.