arXiv subjects

Juan Vicente Gutiérrez-Santacreu

Publications and source records attributed to Juan Vicente Gutiérrez-Santacreu.

12 recordsLinked to original sources

arXiv2026

Toward generalized solutions of the Keller--Segel equations with singular sensitivity and signal absorption via an algebraic manipulation finite element algorithm

The paper that follows describes a numerical algorithm to solve the parabolic-parabolic Keller--Segel system characterized by singular sensitivity and signal absorption in such a manner that the numerical approximations converge towards a generalized solution on two-dimensional polygonal domains as the time and space discretization parameters tend to zero. The algorithm employs an algebraic manipulation finite element method for space, while time remains continuous, based on introducing a stabilized term. This term is constructed via a graph-Laplacian operator and a shock detector for detecting extrema in finite element functions. Furthermore, the cross-diffusion term also includes an algebraic manipulation, which is related to testing by a nodally interpolated, suitable nonlinear function involved in obtaining a discrete energy-like law leading to a priori estimates. This approach yields approximations that respect physical constraints at nodal points such as positivity and maximum principle, and maintaining mass properties as well. Compactness results are quite laborious due to the low regularity stemming from the a priori estimates and the discretization procedure itself. Finally, the passage to the limit rests on testing by the product of a positive test function and a renormalization of the numerical solution.

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arXiv2025

Finite element approximation and very weak solution existence in a two-dimensional, degenerate Keller-Segel model

This paper is devoted to the design and analysis of a numerical algorithm for approximating solutions of a degenerate cross-diffusion system, which models particular instances of taxis-type migration processes under local sensing mechanisms. The degeneracy leads to solutions that are very weak due to the low regularity themselves. Specifically, the solutions satisfy pointwise bounds (such as positivity and the maximum principle), integrability (such as mass conservation), and dual a priori estimates. The proposed numerical scheme combines a finite element spatial discretization with Euler time stepping. The discrete solutions preserve the above-mentioned properties at the discrete level, enabling the derivation of compactness arguments and the convergence (up to a subsequence) of the numerical solutions to a very weak solution of the continuous problem on two-dimensional polygonal domains.

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arXiv2024

Physics-based stabilized finite element approximations of the Poisson--Nernst--Planck equations

We present and analyze two stabilized finite element methods for solving numerically the Poisson--Nernst--Planck equations. The stabilization we consider is carried out by using a shock detector and a discrete graph Laplacian operator for the ion equations, whereas the discrete equation for the electric potential need not be stabilized. Discrete solutions stemmed from the first algorithm preserve both maximum and minimum discrete principles. For the second algorithm, its discrete solutions are conceived so that they hold discrete principles and obey an entropy law provided that an acuteness condition is imposed for meshes. Remarkably the latter is found to be unconditionally stable. We validate our methodology through numerical experiments.

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arXiv2024

Analysis of a fully discrete approximation for the classical Keller--Segel model: lower and a priori bounds

This paper is devoted to constructing approximate solutions for the classical Keller--Segel model governing \emph{chemotaxis}. It consists of a system of nonlinear parabolic equations, where the unknowns are the average density of cells (or organisms), which is a conserved variable, and the average density of chemoattractant. The numerical proposal is made up of a crude finite element method together with a mass lumping technique and a semi-implicit Euler time integration. The resulting scheme turns out to be linear and decouples the computation of variables. The approximate solutions keep lower bounds -- positivity for the cell density and nonnegativity for the chemoattractant density --, are bounded in the $L^1(Ω)$-norm, satisfy a discrete energy law, and have \emph{ a priori} energy estimates. The latter is achieved by means of a discrete Moser--Trudinger inequality. As far as we know, our numerical method is the first one that can be encountered in the literature dealing with all of the previously mentioned properties at the same time. Furthermore, some numerical examples are carried out to support and complement the theoretical results.

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arXiv2023

Exploring numerical blow-up phenomena for the Keller-Segel-Navier-Stokes equations

The Keller-Segel-Navier-Stokes system governs chemotaxis in liquid environments. This system is to be solved for the organism and chemoattractant densities and for the fluid velocity and pressure. It is known that if the total initial cell density mass is below $2π$ there exist globally defined generalised solutions, but what is less understood is whether there are blow-up solutions beyond such a threshold and its optimality. Motivated by this issue, a numerical blow-up scenario is investigated. Approximate solutions computed via a stabilised finite element method founded on a shock capturing technique are such that they satisfy \emph{a priori} bounds as well as lower and $L^1(Ω)$ bounds for the cell and chemoattractant densities. In particular, this latter properties are essential in detecting numerical blow-up configurations, since the non-satisfaction of these two requirements might trigger numerical oscillations leading to non-realistic finite-time collapses into persistent Dirac-type measures. Our findings show that the existence threshold value $2π$ encountered for the cell density mass may not be optimal and hence it is conjectured that the critical threshold value $4π$ may be inherited from the fluid-free Keller-Segel equations. Additionally it is observed that the formation of singular points can be neglected if the fluid flow is intensified.

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arXiv2022

Bound-preserving finite element approximations of the Keller-Segel equations

This paper aims to develop numerical approximations of the Keller--Segel equations that mimic at the discrete level the lower bounds and the energy law of the continuous problem. We solve these equations for two unknowns: the organism (or cell) density, which is a positive variable, and the chemoattractant density, which is a nonnegative variable. We propose two algorithms, which combine a stabilized finite element method and a semi-implicit time integration. The stabilization consists of a nonlinear artificial diffusion that employs a graph-Laplacian operator and a shock detector that localizes local extrema. As a result, both algorithms turn out to be nonlinear.Both algorithms can generate cell and chemoattractant numerical densities fulfilling lower bounds. However, the first algorithm requires a suitable constraint between the space and time discrete parameters, whereas the second one does not. We design the latter to attain a discrete energy law on acute meshes. We report some numerical experiments to validate the theoretical results on blowup and non-blowup phenomena. In the blowup setting, we identify a \textit{locking} phenomenon that relates the $L^\infty(Ω)$-norm to the $L^1(Ω)$-norm limiting the growth of the singularity when supported on a macroelement.

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arXiv2019

Numerical solution for an aggregation equation with degenerate diffusion

A numerical method for approximating weak solutions of an aggregation equation with degenerate diffusion is introduced. The numerical method consists of a stabilized finite element method together with a mass lumping technique and an extra stabilizing term plus a semi--implicit Euler time integration. Then we carry out a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that the sequence of finite element approximations converges toward the unique weak solution of the model at hands. In doing so, nonnegativity is attained due to the stabilizing term and the acuteness on partitions of the computational domain, and hence a priori energy estimates of finite element approximations are established. As we deal with a nonlinear problem, some form of strong convergence is required. The key compactness result is obtained via an adaptation of a Riesz--Fréchet--Kolmogorov criterion by perturbation. A numerical example is also presented.

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arXiv2018

From a cell model with active motion to a Hele-Shaw-like system. A numerical approach

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.

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arXiv2017

Two scenarios on a potential smoothness breakdown for the three-dimensional Navier-Stokes equations

In this paper we construct two families of initial data being arbitrarily large under any scaling-invariant norm for which their corresponding weak solution to the three-dimensional Navier-Stokes equations become smooth on either $[0,T_1]$ or $ [T_2,\infty)$, respectively, where $T_1$ and $T_2$ are two times prescribed previously. In particular, $T_1$ can be arbitrarily large and $T_2$ can be arbitrarily small. Therefore, possible formation of singularities would occur after a very long or short evolution time, respectively. We further prove that if a large part of the kinetic energy is consumed prior to the first (possible) blow-up time, then the global-in-time smoothness of the solutions follows for the two families of initial data.

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arXiv2017

Inf-sup stable finite-element methods for the Landau--Lifshitz--Gilbert and harmonic map heat flow equation

In this paper we propose and analyze a finite element method for both the harmonic map heat and Landau--Lifshitz--Gilbert equation, the time variable remaining continuous. Our starting point is to set out a unified saddle point approach for both problems in order to impose the unit sphere constraint at the nodes since the only polynomial function satisfying the unit sphere constraint everywhere are constants. A proper inf-sup condition is proved for the Lagrange multiplier leading to the well-posedness of the unified formulation. \emph{A priori} energy estimates are shown for the proposed method. When time integrations are combined with the saddle point finite element approximation some extra elaborations are required in order to ensure both \emph{a priori} energy estimates for the director or magnetization vector depending on the model and an inf-sup condition for the Lagrange multiplier. This is due to the fact that the unit length at the nodes is not satisfied in general when a time integration is performed. We will carry out a linear Euler time-stepping method and a non-linear Crank--Nicolson method. The latter is solved by using the former as a non-linear solver.

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arXiv2016

Convergence to suitable weak solutions for a finite element approximation of the Navier-Stokes equations with numerical subgrid scale modeling

In this work we prove that weak solutions constructed by a variational multiscale method are suitable in the sense of Scheffer. In order to prove this result, we consider a subgrid model that enforces orthogonality between subgrid and finite element components. Further, the subgrid component must be tracked in time. Since this type of schemes introduce pressure stabilization, we have proved the result for equal-order velocity and pressure finite element spaces that do not satisfy a discrete inf-sup condition.

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arXiv2014

On the approximation of turbulent fluid flows by the Navier-Stokes-$α$ equations on bounded domains

The Navier-Stokes-$α$ equations belong to the family of LES (Large Eddy Simulation) models whose fundamental idea is to capture the influence of the small scales on the large ones without computing all the whole range present in the flow. The constant $α$ is a regime flow parameter that has the dimension of the smallest scale being resolvable by the model. Hence, when $α=0$, one recovers the classical Navier-Stokes equations for a flow of viscous, incompressible, Newtonian fluids. Furthermore, the Navier-Stokes-$α$ equations can also be interpreted as a regularization of the Navier-Stokes equations, where $α$ stands for the regularization parameter. In this paper we first present the Navier-Stokes-$α$ equations on bounded domains with no-slip boundary conditions by means of the Leray regularization using the Helmholtz operator. Then we study the problem of relating the behavior of the Galerkin approximations for the Navier-Stokes-$α$ equations to that of the solutions of the Navier-Stokes equations on bounded domains with no-slip boundary conditions. The Galerkin method is undertaken by using the eigenfunctions associated with the Stokes operator. We will derive local- and global-in-time error estimates measured in terms of the regime parameter $α$ and the eigenvalues. In particular, in order to obtain global-in-time error estimates, we will work with the concept of stability for solutions of the Navier-Stokes equations in terms of the $L^2$ norm.

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