Numerical analysis of a linearised fluid-structure interaction problem

This paper describes the numerical analysis of a time dependent linearised fluid structure interaction problems involving a very viscous fluid and an elastic shell in small displacements. For simplicity, all changes of geometry are neglected. A single variational formulation is proposed for the whole problem and generic discretisation strategies are introduced independently on the fluid and on the structure. More precisely, the space approximation of the fluid problem is realized by standard mixed finite elements, the shell is approximated by DKT finite elements, and time derivatives are approximated either by midpoint rules or by backward difference formula. Using fundamental energy estimates on the continuous problem written in a proper functional space, on its discrete equivalent, and on an associated error evolution equation, we can prove that the proposed variational problem is well posed, and that its approximation in space and time converges with optimal order to the continuous solution.