Unconditionally Stable, Variable Step DLN Methods for the Allen-Cahn Active Fluid Model: A Divergence-free Preserving Approach

This paper addresses the divergence-free mixed finite element method (FEM) for nonlinear fourth-order Allen-Cahn phase field coupled active fluid equations. By introducing an auxiliary variable $w = \Delta u$, the original fourth-order problem is converted into a system of second-order equations, thereby easing the regularity constraints imposed on standard $H^2$-comforming finite element spaces. To further refine the formulation, an additional auxiliary variable $\xi$, analogous to the pressure, is introduced, resulting in a mixed finite element scheme that preserves the divergence-free condition in $which = \Delta u$ inherited from the model. A fully discrete scheme is then established by combining the spatial approximation by the divergence-free mixed finite element method with the variable-step Dahlquist-Liniger-Nevanlinna (DLN) time integrator. The boundedness of the scheme is rigorously derived under suitable regularity assumptions. Additionally, an adaptive time-stepping strategy based on the minimum dissipation criterion is carried out to enhance computational efficiency. Several numerical experiments validate the theoretical findings and demonstrate the method's effectiveness and accuracy in simulating complex active fluid dynamics.