This paper develops a mixed finite element method (FEM) for solving nonlinear fourth-order active fluid equations. By introducing an auxiliary variable \(\displaystyle w = -\varDelta u \) , we reduce the fourth-order problem to a second-order system which avoids the need for \(\displaystyle H^2\) -conforming elements. Since w is inherently divergence-free, we further introduce a pressure-like variable \(\displaystyle \phi \) and propose an extended variational formulation that enforces weakly divergence-free constraints on both u and w. The fully discrete schemes are constructed by coupling the spatial mixed FEM with the variable-step Dahlquist–Liniger–Nevanlinna (DLN) time integrator. The boundedness and error estimates of the proposed schemes are rigorously established under suitable regularity assumptions. To improve computational efficiency, an adaptive time-stepping strategy based on a minimum-dissipation criterion is developed. Numerical experiments validate the theoretical results and demonstrate the accuracy and efficiency of the proposed numerical methods.
