Time-space adaptive time-filtered finite element method for the Cahn–Hilliard–Navier–Stokes model
Hao Wang, Yaoyao Chen · Journal of Computational and Applied Mathematics · 2026
The adaptive finite element method (AFEM) is essential for long-term numerical simulations of multiscale problems, particularly for complex systems like the coupled Cahn–Hilliard–Navier–Stokes (CHNS) equations. This work presents a time-space AFEM for long-time CHNS simulations, incorporating a time-filtered stepping scheme that achieves second-order accuracy while maintaining unconditional stability. The proposed algorithm integrates interface tracking with adaptive mesh refinement through a recovery-type a posteriori error estimator, employing superconvergent patch recovery (SPR), which uses linear elements for the phase field and quadratic elements for velocity field. Combined with an adaptive time-stepping strategy driven by phase-field derivatives, the method ensures geometric consistency and computational efficiency while maintaining mesh-interface conformity. Numerical experiments demonstrate the method’s robustness in handling the intricate coupling of phase-field and hydrodynamic interactions, while maintaining computational stability over extended time scales. The results highlight the algorithm’s dual capability of achieving high-resolution simulations and preserving interface-mesh conformity throughout the time evolution process.