Fast computation of elastic and hydrodynamic potentials using approximate approximations

We propose fast cubature formulas for the elastic and hydrodynamic potentials based on the approximate approximation of the densities with Gaussian and related functions. For densities with separated representation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures. We obtain high order approximations up to a small saturation error, which is negligible in computations. Results of numerical experiments which show approximation order $$\mathscr {O}(h^{2M})$$ O ( h 2 M ) , $$M=1,2,3,4$$ M = 1 , 2 , 3 , 4 , are provided.