An adaptive high-order surface finite element method for the self-consistent field theory on general curved surfaces

In this paper, we develop an adaptive high-order surface finite element method (FEM) to solve self-consistent field equations of polymers on general curved surfaces. It is an improvement of the existing algorithm of [J. Comp. Phys. 387: 230-244 (2019)] in which a linear surface FEM was presented to address this problem. The high-order surface FEM is obtained by the high-order surface geometrical approximation and high-order function space approximation. In order to describe the sharp interface in the strong segregation system more accurately, an adaptive FEM equipped with a novel Log marking strategy is proposed. Compared with the traditional strategy, this new marking strategy can not only label the elements that need to be refined or coarsened, but also give the refined or coarsened times, which can make full use of the information of a posterior error estimator and improve the efficiency of the adaptive algorithm. To demonstrate the power of our approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces. Numerical results illustrate the efficiency of the proposed method, especially for strong segregation systems.