Many problems involving internal interfaces can be formulated as partial differential equations with singular source terms. Numerical approximation to such problems on a regular grid necessitates suitable regularizations of delta functions. We study the convergence properties of such discretizations for constant coefficient elliptic problems using the Immersed Boundary (IB) method, which is both a mathematical formulation and a numerical scheme widely used to solve fluid-structure interaction problems, as an example. IB schemes use a uniform Cartesian mesh for the fluid, a Lagrangian curvilinear mesh for the immersed structure, and discrete delta functions for communication between these two grids. We show how the order of the differential operator, order of the finite difference discretization, and properties of the discrete delta function all influence the local convergence behavior. In particular, we show how a recently introduced property of discrete delta functions - the smoothing order - is important in the determination of local convergence rates. We apply our theories to stationary Stokes flow problem and obtain both local and L^P convergence results. We examine the predicted results with numerical simulations.