Support operator method on waveguide problems

The finite difference method has been successfully used to solve waveguide problems on rectangular grids. We introduce the support operator method (SOM) (Hyman, J.M. and Shashkov, M., Comp. Math. Appl., vol.33, no.4, p.81-104, 1997; Appl. Num. Math., vol.25, p.413-42, 1997) with unstructured quadrilateral cells to solve these problems. In SOM, a discrete approximation is defined for a first order differential operator that satisfies the appropriate integral identity. This initial discrete operator, called the natural operator, then supports the construction of other discrete operators, using discrete formulations of the identities for differential operators. The SOM method is based on fundamental mathematical principles that correspond to basic physical principles, and this method provides accurate, robust and stable approximations to differential operators on a nonuniform, nonsmooth, and unstructured grid. We also give a complexity analysis which shows that this algorithm is very efficient.