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Convergence analysis and adaptive computation of a Banach-space mixed finite element method for generalized bioconvective flows

Eligio Colmenares, Ricardo Ruiz-Baier, Dalidet Sanhueza

TL;DR

This work addresses stationary generalized bioconvective flows in which concentration-dependent viscosity couples Navier–Stokes dynamics with microorganism transport. It develops a fully mixed finite element method formulated in Banach spaces, introducing auxiliary variables to obtain a robust first-order system and recasting the problem as a single fixed-point operator. The authors establish well-posedness and optimal a priori error estimates for the discrete scheme, and develop a residual-based a posteriori error estimator with proven reliability and efficiency to enable adaptive refinement. Numerical experiments in two and three dimensions confirm the theoretical rates, demonstrate the effectiveness of adaptivity for singular solutions and complex geometries, and illustrate robustness under an Einstein–Batchelor–type viscosity law. The resulting framework yields physically meaningful postprocessed quantities, preserves skew-symmetry in convective terms, and provides a rigorous foundation for efficient simulation of complex bioconvective phenomena.

Abstract

We develop and analyse an adaptive fully mixed finite element method for stationary generalized bioconvective flows, where the Navier--Stokes equations with concentration-dependent viscosity are coupled with a conservation law for swimming microorganisms. The formulation introduces auxiliary variables including the trace-free velocity gradient, a symmetric pseudo-stress tensor, the concentration gradient, and a semi-advective microorganism flux, which also allows for a consistent treatment of Robin-type boundary condition. The variational problem is posed within a Banach space framework and reformulated as a fixed-point operator. Existence of solutions follows from Schauder's theorem, while uniqueness is obtained under suitable data assumptions. The discrete problem is constructed using Raviart--Thomas finite element spaces together with piecewise polynomial approximations on macroelement-structured meshes, and existence of discrete solutions is established via Brouwer's theorem. An a priori error analysis yields optimal convergence rates. We further derive a residual-based a posteriori error estimator and prove its reliability using global inf-sup conditions, Helmholtz decompositions, and suitable projection operators, while efficiency is ensured through localization techniques and bubble functions. Numerical experiments in two and three dimensions confirm the theoretical results, demonstrate the effectiveness of adaptive refinement for singular solutions and complex geometries with inclusions, and illustrate the robustness of the method for a bioconvective benchmark with plume formation governed by an Einstein--Batchelor-type viscosity law.

Convergence analysis and adaptive computation of a Banach-space mixed finite element method for generalized bioconvective flows

TL;DR

This work addresses stationary generalized bioconvective flows in which concentration-dependent viscosity couples Navier–Stokes dynamics with microorganism transport. It develops a fully mixed finite element method formulated in Banach spaces, introducing auxiliary variables to obtain a robust first-order system and recasting the problem as a single fixed-point operator. The authors establish well-posedness and optimal a priori error estimates for the discrete scheme, and develop a residual-based a posteriori error estimator with proven reliability and efficiency to enable adaptive refinement. Numerical experiments in two and three dimensions confirm the theoretical rates, demonstrate the effectiveness of adaptivity for singular solutions and complex geometries, and illustrate robustness under an Einstein–Batchelor–type viscosity law. The resulting framework yields physically meaningful postprocessed quantities, preserves skew-symmetry in convective terms, and provides a rigorous foundation for efficient simulation of complex bioconvective phenomena.

Abstract

We develop and analyse an adaptive fully mixed finite element method for stationary generalized bioconvective flows, where the Navier--Stokes equations with concentration-dependent viscosity are coupled with a conservation law for swimming microorganisms. The formulation introduces auxiliary variables including the trace-free velocity gradient, a symmetric pseudo-stress tensor, the concentration gradient, and a semi-advective microorganism flux, which also allows for a consistent treatment of Robin-type boundary condition. The variational problem is posed within a Banach space framework and reformulated as a fixed-point operator. Existence of solutions follows from Schauder's theorem, while uniqueness is obtained under suitable data assumptions. The discrete problem is constructed using Raviart--Thomas finite element spaces together with piecewise polynomial approximations on macroelement-structured meshes, and existence of discrete solutions is established via Brouwer's theorem. An a priori error analysis yields optimal convergence rates. We further derive a residual-based a posteriori error estimator and prove its reliability using global inf-sup conditions, Helmholtz decompositions, and suitable projection operators, while efficiency is ensured through localization techniques and bubble functions. Numerical experiments in two and three dimensions confirm the theoretical results, demonstrate the effectiveness of adaptive refinement for singular solutions and complex geometries with inclusions, and illustrate the robustness of the method for a bioconvective benchmark with plume formation governed by an Einstein--Batchelor-type viscosity law.
Paper Structure (14 sections, 29 theorems, 231 equations, 6 figures, 3 tables)

This paper contains 14 sections, 29 theorems, 231 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The fully-mixed formulation
  3. The equivalent first-order system
  4. The fully mixed formulation
  5. Well-posedness of the continuous problem
  6. The Galerkin scheme
  7. Discretization and Finite Element Spaces
  8. Well-posedness of the discrete problem
  9. A priori error analysis
  10. A posteriori error analysis
  11. Preliminary results for the a posteriori error analysis
  12. Residual-based a posteriori error estimator reliability
  13. Residual-based a posteriori error estimator efficiency
  14. Numerical results

Key Result

Lemma 2.1

The forms ${\mathscr{B}}:\mathbf{H}\times\mathbb{H}_{0}({\mathbf{div}}_{4/3};\Omega) \rightarrow \mathrm{R}$ and $\widetilde{{\mathscr{B}}}:\widetilde{\mathbf{H}}\times\mathbf{H}_{\Gamma}({\mathrm{div}}_{4/3};\Omega)\rightarrow \mathrm{R}$ defined in B^S and B^C, possess the following properties

Figures (6)

  • Figure 5.1: Example 1. Approximate solutions for the 2D manufactured problem with $\ell=2$: velocity line integral contours, strain rate magnitude, pseudostress magnitude, concentration profile, diffusive flux and total flux magnitudes, and postprocessed pressure.
  • Figure 5.2: Example 1. Approximate solutions for the 3D manufactured problem with $\ell=2$: velocity streamlines, strain rate magnitude, pseudostress magnitude, slices of concentration profile, diffusive flux and total flux streamlines, and slices of postprocessed pressure.
  • Figure 5.3: Example 2. Convergence for each field variable against non-smooth solutions on an L-shaped domain using uniform and adaptive mesh refinement with barycentrically refined meshes. The bottom right plot shows effectivity indexes for all cases. For reference we show triangles indicating linear and cubic slopes.
  • Figure 5.4: Example 2. Approximate solutions for the 2D manufactured problem on an L-shaped domain with $\ell=2$ and rendered on a coarse mesh: velocity line integral contours, strain rate magnitude, pseudostress magnitude, slices of concentration profile, diffusive flux and total flux line integral contours, and postprocessed pressure. The bottom row shows four adaptively refined meshes.
  • Figure 5.5: Example 3. Bioconvection of micro-organisms on a cavity with square inclusions and mixed boundary conditions. Velocity line integral contours, concentration profile, and total flux line integral contours for the case $\alpha = 0.3$. The bottom row shows three samples of adaptively refined meshes.
  • ...and 1 more figures

Theorems & Definitions (37)

  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Lemma 2.3
  • Remark 2.2
  • Lemma 2.4
  • Theorem 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 27 more