Speaker
Description
In this talk, we consider a coupled Chemotaxis--(Navier--)Stokes system on a bounded domain Ω ⊂ ℝ². The system consists of two parabolic partial differential equations coupled with an incompressible (Navier--)Stokes equation, describing the functions (z, c, 𝐮), where z = z(𝐱, t) denotes the cell density, c = c(𝐱, t) the substance concentration, 𝐮 = 𝐮(𝐱, t) with 𝐮 = (u₁, u₂)ᵀ the velocity field of the fluid at position 𝐱 ∈ Ω at time t ∈ (0, T], and p = p(𝐱, t) is the pressure.
The Chemotaxis--(Navier--)Stokes system describes the interaction between cells (e.g., bacteria) and a chemical signal or substance in liquid environments. This phenomenon plays an important role in biological applications. It is well known that physical quantities, such as density, should remain positive. Therefore, it is of great importance to develop numerical methods that ensure the approximate solutions also remain positive.
The fully discrete scheme, based on the linear conforming finite element method for the discretization of the spatial variable and the backward Euler method for the temporal variable, fails to preserve the positivity of the approximate solution. To ensure positivity preservation, we propose a stabilized scheme using the Algebraic Flux Correction method. The resulting fully discrete scheme is nonlinear, and a fixed-point argument is employed alongside a superconvergence argument to demonstrate its existence and uniqueness. Furthermore, we prove that its solutions remain uniformly bounded, provided the solution is sufficiently regular and the time and mesh steps are appropriately chosen, i.e., k = 𝒪(h¹⁺^ε), 0 < ε < 1/3.
In addition, we derive error estimates in L^∞ (0, T; L²), L²(0, T; H¹) for the cell density, and L^∞ (0, T; H¹) for the substance concentration as well as for the velocity field. We also present numerical experiments validating our theoretical results.
