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.

中文翻译：

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一般曲面自洽场论的自适应高阶曲面有限元方法

在本文中，我们开发了一种自适应高阶曲面有限元方法 (FEM) 来求解一般曲面上聚合物的自洽场方程。它是对现有算法的改进 [J. 比较 物理。387: 230-244 (2019)]，其中提出了线性表面有限元法来解决这个问题。高阶曲面有限元法是通过高阶曲面几何逼近和高阶函数空间逼近得到的。为了更准确地描述强分离系统中的尖锐界面，提出了一种配备新颖Log标记策略的自适应FEM。与传统策略相比，这种新的标记策略不仅可以标记需要细化或粗化的元素，还可以给出细化或粗化的次数，可以充分利用后验误差估计量的信息，提高自适应算法的效率。为了证明我们的方法的威力，我们研究了几个不同曲面上的二嵌段共聚物的自组装模式。数值结果说明了所提出方法的效率，特别是对于强分离系统。