An accurate method for the motion of particles in a Stokes fluid

The work has two independent parts. The first part concerns the convergence toward equilibrium of discrete gradient flows or of some discretizations of autonomous systems which admit a Lyapunov function. The study is performed assuming sufficient conditions for the solutions of the continuous problem to converge toward a stationary state as time goes to infinity. Under mild hypotheses, the discrete system has the same property. This leads to new results on the large time asymptotic behavior of some known non-linear schemes. The second part concerns the numerical simulation of the motion of particles suspended in a viscous fluid. It is shown that the most widely used methods for computing the hydrodynamic interactions between particles lose their accuracy in the presence of large non-hydrodynamic forces and when at least two particles are close from each other. This case arises in the context of medical engineering for the design of nano-robots that can swim. This loss of accuracy is due to the singular character of the Stokes flow in areas of almost contact. A new method is introduced here. Numerical experiments are realized to illustrate its better accuracy.