In this talk, we first introduce an unconditionally energy stable scheme to solve the inextensible interface problem with bending arising from vesicle dynamics in fluid flows.  The fundamental problem is formulated by the immersed boundary method where the non-stationary Stokes equations are considered, with the elastic tension and bending forces expressed in terms of Dirac delta function along the interface. The elastic tension is one of the solution variables and plays the role of Lagrange multiplier to enforce the inextensibility of the interface. The scheme uses a semi-implicit discretization for the non-stationary Stokes part and it is time-lagged for the interface position so that the whole scheme becomes linear.  Meanwhile, this time-lagging technique can be used to prove that the proposed immersed boundary scheme is unconditionally energy stable.  Furthermore, an efficient immersed boundary projection method based on the scheme is developed so that the whole numerical algorithm takes only a linearithmic complexity by using preconditioned GMRES and FFT-based solvers.