  The volume penalization (VP) or the Brinkman penalization (BP) method is a
diffuse interface method for simulating multiphase fluid-structure interaction
(FSI) problems in ocean engineering and/or phase change problems in thermal
sciences. The method relies on a penalty factor (which is inversely related to
body's permeability $\kappa$) that must be large to enforce rigid body velocity
in the solid domain. When the penalty factor is large, the discrete system of
equations becomes stiff and difficult to solve numerically. In this paper, we
propose a projection method-based preconditioning strategy for solving volume
penalized (VP) incompressible and low-Mach Navier-Stokes equations. The
projection preconditioner enables the monolithic solution of the coupled
velocity-pressure system in both single phase and multiphase flow settings. In
this approach, the penalty force is treated implicitly, which is allowed to
take arbitrary large values without affecting the solver's convergence rate or
causing numerical stiffness/instability. It is made possible by including the
penalty term in the pressure Poisson equation. Solver scalability under grid
refinement is demonstrated. A manufactured solution in a single phase setting
is used to determine the spatial accuracy of the penalized solution.
Second-order pointwise accuracy is achieved for both velocity and pressure
solutions. Two multiphase fluid-structure interaction (FSI) problems from the
ocean engineering literature are also simulated to evaluate the solver's
robustness and performance. The proposed solver allows us to investigate the
effect of $\kappa$ on the motion of the contact line over the surface of the
immersed body. It also allows us to investigate the dynamics of the free
surface of a solidifying metal
