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

TL;DR

This paper introduces an adaptive high-order surface finite element method with spectral deferred correction for solving polymeric self-consistent field equations on curved surfaces, improving accuracy and efficiency especially for strongly segregated systems.

Contribution

The paper develops a novel adaptive high-order surface FEM with a Log marking strategy, enhancing accuracy and computational efficiency for complex surface-based polymer modeling.

Findings

Numerical results confirm the method's high accuracy and convergence.

The adaptive strategy effectively captures sharp interfaces in strongly segregated systems.

The approach demonstrates efficiency on various curved surfaces.

Abstract

In this paper, we develop an adaptive high-order surface finite element method (FEM) incorporating the spectral deferred correction method for chain contour discretization to solve polymeric self-consistent field equations on general curved surfaces. The high-order surface FEM is obtained by the high-order surface geometrical approximation and the high-order function space approximation. Numerical results demonstrate that the precision order of these methods is consistent with the theoretical prediction. In order to describe the sharp interface in the strongly segregated system more accurately, an adaptive FEM equipped with a new Log marking strategy is proposed. Compared with the traditional strategy, the Log 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 ecciency 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 ecciency of the proposed method, especially for strongly segregated systems with economical discretization nodes.