An Adaptive High-Order Surface Finite Element Method for Polymeric Self-Consistent Field Theory on General Curved Surfaces (Advances in Applied Mathematics and Mechanics, 2024)
In this paper, we develop an adaptive high-order surface finite elementmethod (FEM) incorporating spectral deferred correction method for chain contourdiscretization to solve polymeric self-consistent field equations on general curved surfaces. The high-order surface FEM is obtained by the high-order surface geometricalapproximation and the high-order function space approximation. Numerical resultsdemonstrate that the precision order of these methods is consistent with theoreticalprediction. In order to describe the sharp interface in the strongly segregated system more accurately, an adaptive FEM equipped with a new $Log$ marking strategy isproposed. The $Log$ marking strategy can not only label the elements that need to berefined or coarsened, but also give the refined or coarsened times, which can makefull use of the information of a posterior error estimator and improve the efficiency ofthe adaptive algorithm. To demonstrate the power of our approach, we investigate theself-assembled patterns of diblock copolymers on several distinct curved surfaces. Numerical results illustrate the efficiency of the proposed method, especially for stronglysegregated systems with economical discretization nodes.