Fluid-solid interaction has been a challenging subject due to their strongnonlinearity and multidisciplinary nature. Many of the numerical methods forsolving FSI problems have struggled with non-convergence and numericalinstability. In spite of comprehensive studies, it has been still a challengeto develop a method that guarantees both convergence and stability. Our discussion in this work is restricted to the interaction of viscousincompressible fluid flow and a rigid body. We take the monolithic approach byGibou and Min that results in an extended Hodge projection. The projectionupdates not only the fluid vector field but also the solid velocities. Wederive the equivalence of the extended Hodge projection to the Poisson equationwith non-local Robin boundary condition. We prove the existence, uniqueness,and regularity for the weak solution of the Poisson equation, through which theHodge projection is shown to be unique and orthogonal. Also, we show thestability of the projection in a sense that the projection does not increasethe total kinetic energy of fluid and solid. Also, we discusse a numericalmethod as a discrete analogue to the Hodge projection, then we show that theunique decomposition and orthogonality also hold in the discrete setting. Asone of our main results, we prove that the numerical solution is convergentwith at least the first order accuracy. We carry out numerical experiments intwo and three dimensions, which validate our analysis and arguments.
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