A Divergence-free Preserving Mixed Finite Element Method for Thermally Driven Active Fluid Model

In this report, we propose a divergence-free preserving mixed finite element method (FEM) for the system of nonlinear fourth-order thermally driven active fluid equations. By introducing two auxiliary variables, we lower the complexity of the model and enhance the robustness of the algorithm. The auxiliary variable $w = \Delta u$ is used to convert the original fourth-order system to an equivalent system of second-order equations, thereby easing the regularity constraints imposed on standard $H^2$-conforming finite element space. The second variable $\eta$, analogous to the pressure, helps the scheme preserve the divergence-free condition arising from the model. The two-step Dahlquist-Liniger-Nevanlinna (DLN) time integrator, unconditionally non-linear stable and second-order accurate under non-uniform time grids, is combined with the mixed FEM for fully discrete approximation. Due to the fine properties of the DLN scheme, we prove the boundedness of model energy and the associated error estimates under suitable regularity assumptions and mild time restrictions. Additionally, an adaptive time-stepping strategy based on a minimum-dissipation criterion is to balance computational costs and time efficiency. Several numerical experiments validate the theoretical findings and demonstrate the method's effectiveness and accuracy in simulating complex active fluid dynamics.