Unconditionally energy stable schemes for the inextensible interface problem with bending

In this paper, we develop unconditionally energy stable schemes to solve the inextensible interface problem with bending. The fundamental problem is formulated by the immersed boundary method where the nonstationary 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 which plays the role of Lagrange multiplier to enforce the inextensibility of the interface. Both the backward Euler and Crank-Nicolson methods are introduced and it can be proved that the total energy, i.e., kinetic energy and bending energy, is discretely bounded. The numerical results show that both schemes are unconditionally energy stable without any time-step restriction. The backward Euler scheme is also applied to study the dynamics of vesicles suspended in a shear flow.