We 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, which have previously been used by Anderson and Colella to spread charge from particles to the mesh (but not to interpolate the force from the mesh to the particles) use discretizations of the differential equations

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... forces satisfy. The methods are most accurate when the forces satisfy homogeneous elliptic differential equations or systems of equations, and the precise accuracy levels of the interpolation formulas depend on the accuracy of certain corresponding quadrature formulas. We describe the methods and give results of numerical experiments which demonstrate their effectiveness. 2000 Mathematics Subject Classification. Primary: 65D05; Secondary: 65Z05.