An Unconditionally Stable, Energy-Preserving Finite Element Method with Adaptive Filtering for Geotactic Bioconvection

Geotactic bioconvection describes the macroscopic convective patterns that arise in dense suspensions of free-swimming microorganisms exhibiting negative geotaxis, a preferential upward swimming against the direction of gravity. As the population accumulates near the upper surface, the resulting top-heavy system becomes unstable. Fingers of concentrated organisms start to descend into the bulk fluid, and the collective dynamics generates large-scale flow patterns that persist over long times. This process shares important features with Rayleigh-B\'{e}nard thermal convection. Both feature adverse density gradients that drive buoyancy-induced instabilities, although the underlying physical origins differ. One is the thermal expansion and one is the cell movement.

Bioconvection has extensive practical applications, including microbial enhanced oil recovery, large-scale biofuel production, wastewater treatment, and the design of microfluidic lab-on-a-chip systems. In many of these settings, understanding the long-time asymptotic behavior and large-scale flow structures is essential, which demands accurate simulations over a long period of time on sufficiently fine meshes. Moreover, in the Rayleigh-Bénard flow, it is well known that the flow transitions to weakly turbulent, chaotic dynamics when the Rayleigh number exceeds a critical threshold. We observe analogous behavior in the bioconvection system. When the Rayleigh number, defined appropriately for the bioconvection setting, is sufficiently large, the flow exhibits chaotic dynamics that require even finer spatial resolution to resolve.

However, direct numerical simulation of the coupled Navier-Stokes and concentration transport system on such fine meshes is prohibitively expensive, particularly for the long integration times of practical interest. To address this, we develop a spatially adaptive filtering approach inspired by the Leray-$\alpha$ regularization model. The proposed method preserves the energy structure of the original system while enabling accurate long-time simulations at significantly reduced computational cost.