In this article we discuss the application of fictitious domain methods to the numerical simulation of incompressible viscous flow with suspended moving particles. The model coupling the Navier-Stokes equations from fluid dynamics with the Newton equations for the particle motion has been extensively studied in the literature. Among the problems for its practical application are fluidized beds, sedimention, a blood flow around artificial heart valve, etc.

The solution method discussed here combines finite element discretizations in space, time discretization by a projection scheme and the method of characteristics for the treatment of the convection term. The key points of our method are locally refined locally adapted grids for space discretization and efficient iterative solvers based on fictitious domain methods. The methodologies we follow in this paper were proposed and studied by G.Astrakhatsev (1978), G.Marchuk and Yu.Kuznetsov (1986), and R.Glowinski and Yu.Kuznetrov (1998). We shall show in Section 4, that the concrete choice of the optimal domain embedding is strongly governed by the computational domain topology. Therefore, we focus in our research on simulations with a few solid particles to investigate in details the behavior of iterative solvers for the case of particle collisions.