  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.
