Francisco Guillén-González

Francisco Guillén-González, María Ángeles Rodríguez-Bellido, Diego Armando Rueda-Gómez

This work is devoted to study unconditionally energy stable and mass-conservative numerical schemes for the following repulsive-productive chemotaxis model: Find $u \geq 0$, the cell density, and $v \geq 0$, the chemical concentration, such that $$ \left\{ \begin{array} [c]{lll} \partial_t u - Δu - \nabla\cdot (u\nabla v)=0 \ \ \mbox{in}\ Ω,\ t>0,\\ \partial_t v - Δv + v = u \ \ \mbox{in}\ Ω,\ t>0, \end{array} \right. $$ in a bounded domain $Ω\subseteq \mathbb{R}^d$, $d=2,3$. By using a regularization technique, we propose three fully discrete Finite Element (FE) approximations. The first one is a nonlinear approximation in the variables $(u,v)$; the second one is another nonlinear approximation obtained by introducing ${\boldsymbolσ}=\nabla v$ as an auxiliary variable; and the third one is a linear approximation constructed by mixing the regularization procedure with the energy quadratization technique, in which other auxiliary variables are introduced. In addition, we study the well-posedness of the numerical schemes, proving unconditional existence of solution, but conditional uniqueness (for the nonlinear schemes). Finally, we compare the behavior of such schemes throughout several numerical simulations and provide some conclusions.

Francisco Guillén-González, Juan Vicente Gutiérrez-Santacreu

In this paper we deal with the numerical solution of a Hele--Shaw-like system via a cell model with active motion. Convergence of approximations is established for well-posed initial data. These data are chosen in such a way the time derivate is positive at the initial time. The numerical method is constructed by means of a finite element procedure together with the use of a closed-nodal integration. This gives rise to an algorithm which preserves positivity whenever a right-angled triangulation is considered. As a result, uniform-in-time a priori estimates are proven which allows us to pass to limit towards a solution to the Hele--Shaw problem.

Daniel Acosta-Soba, Francisco Guillén-González, J. Rafael Rodríguez-Galván

In this work, we develop a structure-preserving numerical scheme for a Cahn-Hilliard-Darcy model that describes tumor growth in a fluid-saturated porous medium. First, we derive a physically consistent model from the general framework proposed in [29] that guarantees mass conservation and pointwise bounds on the phase-field and nutrient variables, with a decreasing energy law. The resulting model couples the evolution of tumor cells via a Cahn-Hilliard equation with a diffusion equation for the nutrients thro chemotactic interactions and extends the model in [1] by introducing the effect of a surrounding fluid described by Darcy's law. Subsequently, we propose a fully discrete scheme that combines an upwind discontinuous Galerkin method in space and a convex splitting strategy in time, which inherits the fundamental properties of the continuous model: mass conservation, pointwise bounds and discrete energy law. Our theoretical analysis is accompanied by numerical experiments that demonstrate the robustness of the proposed scheme and show the influence of the surrounding fluid on the tumor evolution.

Francisco Guillén-González, Giordano Tierra

In this paper, we present energy-stable numerical schemes for a Smectic-A liquid crystal model. This model involve the hydrodynamic velocity-pressure macroscopic variables $({\bf u},p)$ and the microscopic order parameter of Smectic-A liquid crystals, where its molecules have a uniaxial orientational order and a positional order by layers of normal and unitary vector ${\bf n}$. We start from the formulation given in \cite{E} by using the so-called layer variable $ϕ$ such that ${\bf n}=\nabla ϕ$ and the level sets of $ϕ$ describe the layer structure of the Smectic-A liquid crystal. Then, a strongly non-linear parabolic system is derived coupling velocity and pressure unknowns of the Navier-Stokes equations $({\bf u},p)$ with a fourth order parabolic equation for $ϕ$. We will give a reformulation as a mixed second order problem which let us to define some new energy-stable numerical schemes, by using second order finite differences in time and $C^0$-finite elements in space. Finally, numerical simulations are presented for $2D$-domains, showing the evolution of the system until it reaches an equilibrium configuration. Up to our knowledge, there is not any previous numerical analysis for this type of models.