We consider a domain decomposition method for a fluid-structure interaction problem. The fluid-structure interaction problem involves two mathematical models, each posed on a different domain, so that domain decomposition occurs naturally. Our approach to a domain decomposition method is based on a strategy in which unknown data at the interface is determined through an optimization process. We prove that the solution of the optimization problem exists. And we show that the Lagrange multiplier rule may be used to transform the constrained optimization problem into an unconstrained one and that rule is applied to derive an optimality system from which optimal solutions may be obtained. We then study a gradient method for solving optimization problem.