arXiv subjects

Diego Armando Rueda-Gómez

Publications and source records attributed to Diego Armando Rueda-Gómez.

4 recordsLinked to original sources

arXiv2021

Theoretical and numerical analysis for an interspecies competition model with chemoattraction-consumption in fluids

This work is devoted to the theoretical and numerical analysis of a two-species chemotaxis- Navier-Stokes system with Lotka-Volterra competitive kinetics in a bounded domain of Rd, d = 2, 3. First, we study the existence of global weak solutions and establish a regularity criterion which provides sufficient conditions to ensure the strong regularity of the weak solutions. After, we propose a finite element numerical scheme in which we use a splitting technique obtained by introducing an auxiliary variable given by the gradient of the chemical concentration and applying an inductive strategy, in order to deal with the chemoattraction terms in the two-species equations and prove optimal error estimates. For this scheme, we study the well-posedness and derive some uniform estimates for the discrete variables required in the convergence analysis. Finally, we present some numerical simulations oriented to verify the good behavior of our scheme, as well as to check numerically the optimal error estimates proved in our theoretical analysis.

math.NA↗

arXiv2020

Convergence and positivity of finite element methods for a haptotaxis model of tumoral invasion

In this paper, we consider a mathematical model for the invasion of host tissue by tumour cells in a $d$-dimensional bounded domain, $d\leq 3$. This model consists of a system of differential equations describing the evolution of cancer cell density, the extracellular matrix protein density and the matrix degrading enzyme concentration. We develop two fully discrete schemes for approximating the solutions based on the Finite Element (FE) method. For the first numerical scheme, we use a splitting technique to deal with the haptotaxis term, leading to introduce an equivalent system with a new variable given by the gradient of extracellular matrix. This scheme is well-posed and preserves the non-negativity of extracellular matrix and the degrading enzyme. We analyze error estimates and convergence towards regular solutions. The second numerical scheme is based on an equivalent formulation in which the cancer cell density equation is expressed in a divergence form through a suitable change of variables. This second numerical scheme preserves the non-negativity of all the discrete variables. Finally, we present some numerical simulations in agreement with the theoretical analysis.

math.NA↗

arXiv2020

Comparison of two finite element schemes for a chemo-repulsion system with quadratic production

In this paper we propose two fully discrete Finite Elements (FE) schemes for a repulsive chemotaxis model with quadratic production term. The first one (called scheme UV) corresponds to the backward Euler in time with FE in space approximation; while the second one (called scheme US$_\varepsilon$) is obtained as a modification of the scheme US proposed by [Guillén-González et al.], by applying a regularization procedure. We prove that the schemes UV and US$_\varepsilon$ have better properties than the FE scheme US. Specifically, we prove that, unlike the scheme US, the scheme UV is energy-stable in the primitive variables of the model, under a "compatibility" condition on the FE spaces. On the other hand, the scheme US$_\varepsilon$ is energy-stable with respect to the same modified energy of the scheme US, and an "approximated positivity" property holds (which is not possible to prove for the schemes US and UV). Additionally, we study the well-posedness of the schemes and the long time behaviour obtaining exponential convergence to constant states. Finally, we compare the numerical schemes throughout several numerical simulations.

math.DS↗

arXiv2018

Unconditionally energy stable fully discrete schemes for a chemo-repulsion model

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.

math.NA↗