Ricerc@Sapienza

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 O(h2M) , M= 1 , 2 , 3 , 4 , are provided.

We propose a fast method for high order approximations of the solution of the Cauchy problem for the linear non-stationary Stokes system in R^3 in the unknown velocity u and kinematic pressure P. The density f(x,t) and the divergence vector-free initial value g(x) are smooth and rapidly decreasing as |x| tends to infinity. We construct the vector u in the form u=u1+u2 where u1 solves a system of homogeneous heat equations and u2 solves a system of non-homogeneous heat equations with right-hand side f-graf P. Moreover, P=-L( div f)) where L denotes the harmonic potential.