Fully discrete subdivision-based IGA scheme with decoupled structure and unconditional energy stability for the phase-field crystal model on surfaces

In this work, we aim to numerically solve the phase-field crystal (PFC) model to simulate atomic growth on manifolds. The geometric complexity, pronounced curvature variations, and nonlinearities inherent in the physical model pose significant challenges, necessitating the development of efficient and robust numerical schemes that can handle strong coupling and nonlinear terms while accurately accounting for curved geometries. To address these challenges, we first adopt a subdivision-based isogeometric analysis (IGA) for spatial discretization. This approach effectively resolves geometric complexities by offering hierarchical refinability, geometric exactness, and adaptability to arbitrary topologies, while eliminating geometric errors commonly encountered in traditional finite element methods. For temporal discretization, the highly nonlinear terms in the model are addressed using the Invariant Energy Quadratization (IEQ) method, which linearizes the nonlinear terms and guarantees strict unconditional energy stability. However, the introduction of auxiliary variables in the IEQ method results in a linearly coupled system. To overcome this limitation and further enhance computational efficiency, we incorporate the Zero-Energy-Coupling (ZEC) approach, ultimately constructing a scheme that achieves second-order accuracy, linearity, unconditional energy stability, and a fully decoupled structure. We rigorously prove the energy stability and solvability of the proposed scheme and validate its accuracy and robustness through extensive numerical experiments conducted on manifolds, demonstrating its capability to handle intricate geometric structures and nonlinear dynamics effectively.