Divergence-free Preserving Mix Finite Element Methods for Fourth-order Active Fluid Model

TL;DR

This paper develops a divergence-free mixed finite element method for two-dimensional fourth-order active fluid equations, introducing auxiliary variables and adaptive time-stepping to improve robustness, efficiency, and accuracy, validated through numerical experiments.

Contribution

It introduces a novel divergence-free preserving mixed FEM with auxiliary variables and adaptive time-stepping for active fluid models, improving stability and efficiency.

Findings

Proved unconditional nonlinear stability and second-order accuracy.

Validated the scheme's effectiveness through numerical experiments.

Demonstrated improved computational efficiency with adaptive time-stepping.

Abstract

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 $w=-\Delta u$, the original fourth problem is transformed into a system of second-order equations, which relaxes the regularity requirements of standard $H^2$-conforming finite spaces. To further enhance the robustness and efficiency of the algorithm, an additional auxiliary variable $\phi$, 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.