Convergence of a vector penalty projection scheme for the Navier Stokes equations with moving body

In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to treat the displacement of a moving body in incompressible viscous flows in the case where the interaction of the fluid on the body can be neglected. The presence of the obstacle inside the computational domain is treated with a penalization method introducing a parameter η to enforce the velocity on the solid boundary. The incompressibility constraint is approached using a Vector Projection method which introduces a relaxation parameter ε. We show the stability of the scheme and that the pressure and velocity converge towards a limit when the relaxation parameter ε and the time step δt tend to zero with a proportionality constraint ε = λδt. Finally, when η goes to 0, we show that the problem admits a weak limit which is a weak solution of the Navier-Stokes equations with no-slip condition on the solid boundary.