2025/07/28 by Zheng, Nan, Guo, Xu, Pei, Wenlong +1
#FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2507.21392
This paper is concerned with mixed finite element method (FEM) for solving the two-dimensional, nonlinear fourth-order active fluid equations. By introducing an auxiliary variable w=-Δu, the original fourth problem is transformed into a system of second-order equations, which relaxes the regularity requirements of standard H2-conforming finite spaces. To further enhance the robustness and efficiency of the algorithm, an additional auxiliary variable ϕ, treated analogously to the pressure, is introduced, leading to a divergence-free preserving mixed finite element scheme. A fully discrete scheme is then constructed by coupling the spatial mixed FEM with the variable-step Dahlquist-Liniger-Nevanlinna (DLN) time integrator. The boundedness of the scheme and corresponding error estimates can be rigorously proven under appropriate assumptions due to unconditional non-linear stability and second-order accuracy of the DLN method. To enhance computational efficiency in practice, we develop an adaptive time-stepping strategy based on a minimum-dissipation criterion. Several numerical experiments are displayed to fully validate the theoretical results and demonstrate the accuracy and efficiency of the scheme for complex active fluid simulations.