Abstract: Surfaces with constant mean curvature always arise as interfaces in many physical problems, and are the mathematical abstraction of physical soap films and soap bubbles.The flexibility and high quality of subdivision surfaces make them to be a powerful tool for designing surfaces.In this paper, we construct the constant mean curvature subdivision surfaces with given boundaries using a forth-order geometric partial differential equation deduced from a second-order energy functional and a second-order geometric partial differential equation.These equations are solved by a finite element method.We adopt the limit functions of the extended Loop's subdivision scheme as the finite element space because this scheme can treat surfaces with boundaries.The constant mean curvature subdivision surfaces can be approximately constructed with any topology of the control mesh and any shaped boundaries.