Optimal Alternating Direction Implicit Preconditioners for Conjugate Gradient methods

The (Extrapolated) Alternating Direction Implicit Preconditioners for the class of Conjugate Gradient Methods are applied for the solution of the second order elliptic equation in a rectangle under Dirichlet boundary conditions. The PDE is approximated by uniform meshes of 5− and 9−point difference schemes and analytic expressions for the optimal acceleration and extrapolation parameters are obtained in both cases. The ones for the 5−point schemes complete others already known while those for the 9−point schemes are new. Numerical examples are presented to show the superiority of the preconditioners proposed. Part of the work of this author ([email protected]) was funded by the Program Pythagoras of the Greek Ministry of Education The work of this author ([email protected]) was done under a scholarship from the Greek State Scholarships Foundation