Components of a general method for computational fluid dynamics with moving boundaries are developed, implemented and tested. The method adapts the boundary fitted mesh to the changing spatial domain by deforming it and, when necessary due to grid quality requirements, regenerating it. The software TRITET, used for regeneration, is tested and an interface between it and the flow solver OpenFOAM has been developed. Different algorithms for the deformation are evaluated and compared. The study recommends the use of an elliptic equation with variable coefficient for the mesh deformation. Finally, the full method has been implemented and applied for the solution of the three-dimensional Navier-Stokes equations for the flow around a rotating ellipsoid.