A Lagrange multiplier formulation for the finite element discretization of FSI
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We provide a general variational framework for the finite element discretization of fluid-structure interaction problems. In this talk we consider the case of an elastic body immersed in an incompressible fluid, it can be either a thick or a thin body. Our approach is based on the fictitious domain method and makes use of a distributed Lagrange multiplier. Our technique allows us to use computational grids for fluid and solid completely independent one from each other, at the price to evaluate some coupling terms which involve basis functions associated to both the meshes. We present a recent result on the existence and uniqueness of the solution in a linearized case, see~\cite{existence}. Moreover, we discuss the finite element discretization, with particular attention to the choice of the finite element spaces ensuring the solvability of the problem at each time step, and to some computational aspects related with the evaluation of the coupling terms. With the aid of the simpler interface problem presenting the same features as the saddle point problem resulting from the time discretization of the FSI system, we analyze new choices for the finite element spaces and obtain error estimates.
