  In this article, we develop a least--squares/fictitious domain method for
direct simulation of fluid particle motion with Navier slip boundary condition
at the fluid--particle interface. Let $\Omega$ and $B$ be two bounded domains
of $\mathbb{R}^{d}$ such that $\overline{B} \subset \Omega$. The motion of
solid particle $B$ is governed by Newton's equations. Our goal here is to
develop a fictitious domain method where one solves a variant of the original
problem on the full $\Omega$, followed by a well--chosen correction over $B$
and corrections related to translation velocity and angular velocity of the
particle. This method is of the virtual control type and relies on a
least--squares formulation making the problem solvable by a conjugate gradient
algorithm operating in a well chosen control space. Since the fully explicit
scheme to update the particle motion using Newton's equation is unstable, we
propose and implement an explicit--implicit scheme in which, at each time step,
the position of the particle is updated explicitly, and the solution of
Navier-Stokes equations and particle velocities are solved by the the
least--squares/fictitious domain method implicitly. Numerical results are given
to verify our numerical method.
