On the Convergence of the Preconditioned Group Rotated Iterative Methods In The Solution of Elliptic PDEs

The convergence rates of the explicit group methods derived from the standard and skewed (rotated) finite difference operators depend on the spectral properties of the coefficient matrices resulted from these group discretization formulas. By applying appropriate preconditioner, we may transform the resulting linear system into another equivalent system that has the same solution, but has a better spectral property than its unpreconditioned form. In Saeed and Ali [11], the application of a specific splittingtype block preconditioner to the Explicit Decoupled Group Successive Over-Relaxation (EDG SOR) method was presented where the preconditioned scheme was shown to have a better rate of convergence compared to its unpreconditioned counterpart. In this paper, some new Fundamental theorems and lemmas related to the convergence properties of this preconditioned scheme will be established and presented.