Eigendecomposition of Domain Decomposition Interface Operators for Constant Coefficient Elliptic Problems

In this paper the authors derive the exact eigendecomposition of the interface operators arising in domain decomposition methods for general five-point discretizations to constant coefficient elliptic equations on rectangular domains. The special case of convection-diffusion problems is studied in some detail, including both central and upwind differencing for the convection term and flows normal and tangential to the interface. It is shown that preconditioners based on the diffusion operator alone may give very slow convergence when the cell Reynolds number is of order $O(1)$.