  We consider a fictitious domain formulation for fluid-structure interaction
problems based on a distributed Lagrange multiplier to couple the fluid and
solid behaviors. How to deal with the coupling term is crucial since the
construction of the associated finite element matrix requires the integration
of functions defined over non-matching grids: the exact computation can be
performed by intersecting the involved meshes, whereas an approximate coupling
matrix can be evaluated on the original meshes by introducing a quadrature
error. The purpose of this paper is twofold: we prove that the discrete problem
is well-posed also when the coupling term is constructed in approximate way and
we discuss quadrature error estimates over non-matching grids.
