Justification of the fast multipole method for the stokes system in the case of the interior diriclet problem

Justification of the fast multipole method for the stokes system. Part II. Exterior domain problems

“…Therefore, we will need several FMM cycles to calculate one matrix-vector product in our case. As it was in the case of the interior Dirichlet problem presented in [27], this method essentially reduces the computational costs, which will be demonstrated by numerical test computations.…”

“…Hence, if we find for the matrices (10), (13) and (19) the representation of the form (21), then we will be able to use FMM by the calculation of hydrodynamical potentials and assentially reduce the computational costs by the numerical solution of our exterior problems (6). We have already done it in the case of the interior Dirichlet problem for the Stokes equations in [27], where we presented a suitable representation of the velocity part of the hydrodynamical double layer tensor (19) and derived the corresponding multipole and Taylor expansion, as well as its shifting and converting operators. Therefore, we will only discuss in the following the hydrodynamical single layer tensor (10) and its normal stress tensor (13).…”

“…A complex representation of the double layer tensor (19) is similar to the representation (30) of the normal stress tensor and has already been derived and discussed in [27].…”

“…Statements, allowing us to manipulate the multipole expansion from Theorems 4.2 and 4.3 in a manner required by the fast multipole algorithm, are exactly the same as in [27] and [7]. Thus, we only present them here without giving the corresponding proofs.…”