In this talk we will present accurate two and three dimensional methods for interpolating singular or smoothed force fields. The methods are meant to be used in particle mesh or particle-particle particle-mesh calculations so that the resulting schemes conserve momentum. The interpolation weights use a discretization of the differential equation the interaction potential satisfies, and the methods are most accurate when the force satisfies a homogeneous elliptic differential equation or system of equations away from the sources. We will show why the precise accuracy of the interpolation formulas depends on the accuracy of certain corresponding quadrature formulas, and give results of numerical experiments.