Iterative methods for Robbins problems

Preconditioned iterative methods are described for the solution of an elliptic partial differential equation over an unit square region with Robbins boundary conditions. Transform based preconditioners are constructed and analyzed. The motivation is to exploit the fast inversion of transform based systems via the fast transform. We prove that transform based preconditioners can be chosen so that the condition number of the preconditioned system can be reduced from O(n2) to O(1). Numerical results are reported to illustrate the effectiveness of the preconditioners.