In this study the coupled overlapping domain (COD) technique, previously developed within our group, is extended to make it suitable for fluid-structure interaction (FSI) computations with flexible elastic solids. A standard FSI benchmark from literature is implemented to verify the computations of the COD technique. However, in this study the incompressible neo-Hookean material model is applied instead of the compressible St. Venant-Kirchhoff model. The results of the COD technique are compared to the solutions obtained with a full finite element method. The performed computations demonstrate the COD technique accurately computes the solids displace- ments and the forces on the elastic solid. The results of the COD technique also show a good resemblance with the results obtained by other researchers, even with the use of a different material model. Additionally, the COD technique is applied to compute a rotating elastic beam in a square box filled with fluid. The computations show that the COD technique is capable of computing large deformations and translations, which makes this technique suitable for a wide range of applications without the need for computationally expensive remeshing.