F. Guillén-González

R. C. Cabrales, F. Guillén-González, J. V. Gutiérrez-Santacreu

A numerical method is developed for solving a system of partial differential equations modeling the flow of a nematic liquid crystal fluid with stretching effect, which takes into account the geometrical shape of its molecules. This system couples the velocity vector, the scalar pressure and the director vector representing the direction along which the molecules are oriented. The scheme is designed by using finite elements in space and a time-splitting algorithm to uncouple the calculation of the variables: the velocity and pressure are computed by using a projection-based algorithm and the director is computed jointly to an auxiliary variable. Moreover, the computation of this auxiliary variable can be avoided at the discrete level by using piecewise constant finite elements in its approximation. Finally, we use a pressure stabilization technique allowing a stable equal-order interpolation for the velocity and the pressure. Numerical experiments concerning annihilation of singularities are presented to show the stability and efficiency of the scheme.

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

We consider the following repulsive-productive chemotaxis model: Let $p\in (1,2)$, find $u \geq 0$, the cell density, and $v \geq 0$, the chemical concentration, satisfying \begin{equation}\label{C5:Am} \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^p \ \ \mbox{in}\ Ω,\ t>0, \end{array} \right. \end{equation} in a bounded domain $Ω\subseteq \mathbb{R}^d$, $d=2,3$. By using a regularization technique, we prove the existence of solutions of this problem. Moreover, we propose three fully discrete Finite Element (FE) nonlinear approximations, where the first one is defined in the variables $(u,v)$, and the second and third ones by introducing ${\boldsymbolσ}=\nabla v$ as an auxiliary variable. We prove some unconditional properties such as mass-conservation, energy-stability and solvability of the schemes. Finally, we compare the behavior of the schemes throughout several numerical simulations and give some conclusions.

F. Guillén-González, M. V. Redondo-Neble

A first-order linear fully discrete scheme is studied for the incompressible time-dependent Navier-Stokes equations in three-dimensional domains. This scheme, based on an incremental pressure projection method, decouples each component of the velocity and the pressure, solving in each time step, a linear convection-diffusion problem for each component of the velocity and a Poisson-Neumann problem for the pressure. Using first-order \emph{inf-sup} stable $C^0$-finite elements, optimal error estimates of order $O(k+h)$ are deduced without imposing constraints on $h$ and $k$, the mesh size and the time step, respectively. Finally, some numerical results are presented according the theoretical analysis, and also comparing to other current first-order segregated schemes.

F. Guillén-González, M. V. Redondo-Neble

The purpose of this paper is the numerical analysis of a first order fractional-step time-scheme, using decomposition of theviscosity, and "inf-sup" stable finite element space-approximations for the Primitive Equations of the Ocean. The aim of the paper is twofold. Firstly, we prove that the scheme is unconditionally stable and convergent towards weak solutions of the Primitive Equations. Secondly, optimal error estimates for velocity and pressure are provided of order $O(k+h^l)$ for $l=1$ or $l=2$ when either first or second order finite-element approximations are considered ($k$ and $h$ being the time step and the mesh size, respectively). In both cases, these error estimates are obtained under the same constraint $k\le h^2$.