In this paper, we consider numerical approximations of the volume-conserved phase-field elastic bending energy model for lipid vesicles where a nonlocal term is added to the model such that the total volume can be conserved precisely. We further develop two linear and unconditionally energy stable schemes by combining the recently developed IEQ and SAV approaches with the stabilization technique, where several extra stabilization terms are added to enhance the stability and keep the required accuracy while using large time steps. Through the comparisons with some other type numerical schemes for simulating numerous benchmarking numerical examples in 2D and 3D, we demonstrate the robustness of the new nonlocal volume-conserved model, as well as the stability and the accuracy of the developed schemes, numerically.

在本文中，我们考虑脂质囊泡的体积守恒相场弹性弯曲能量模型的数值近似，其中将非局部项添加到模型中，以便可以精确守恒总体积。通过将最近开发的IEQ和SAV方法与稳定技术相结合，我们进一步开发了两种线性和无条件的能量稳定方案，其中添加了几个额外的稳定项以增强稳定性并在使用较大时间步长的同时保持所需的精度。通过与其他类型的数值方案进行比较以模拟2D和3D中的大量基准数值示例，我们从数值上证明了新的非局部体积守恒模型的鲁棒性以及所开发方案的稳定性和准确性。