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1. A non-local perturbation of the logistic equation in IR N

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. Departamento de Matemática Aplicada II (ETSI), Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Universidad de Sevilla. TIC130: Investigación en Sistemas Dinámicos en Ingeniería, Delgado Delgado, Manuel, Molina Becerra, Mónica, Rodrigues dos Santos Júnior, Joao, Suárez Fernández, Antonio, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. Departamento de Matemática Aplicada II (ETSI), Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Universidad de Sevilla. TIC130: Investigación en Sistemas Dinámicos en Ingeniería, Delgado Delgado, Manuel, Molina Becerra, Mónica, Rodrigues dos Santos Júnior, Joao, and Suárez Fernández, Antonio

Abstract

A logistic equation in the whole space is considered. In this problem, a non-local perturbation is included. We establish a new sub-supersolution method forgeneral nonlocal elliptic equations and, consequently, we obtain the existence ofpositive solutions of a nonlocal logistic equation.

Published
2019

2. Heliostat field cleaning scheduling for Solar Power Tower plants: a heuristic approach

Author

Universidad de Sevilla. Departamento de Estadística e Investigación Operativa, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM329: Optimización, Universidad de Sevilla. FQM131: Ec.diferenciales, Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, Ashley, Thomas Ian, Carrizosa Priego, Emilio José, Fernández Cara, Enrique, Universidad de Sevilla. Departamento de Estadística e Investigación Operativa, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM329: Optimización, Universidad de Sevilla. FQM131: Ec.diferenciales, Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, Ashley, Thomas Ian, Carrizosa Priego, Emilio José, and Fernández Cara, Enrique

Abstract

Soiling of heliostat surfaces due to local climate has a direct impact on their optical efficiency and therefore a direct impact on the productivity of the Solar Power Tower plant. Cleaning techniques applied are dependent on plant construction and current schedules are normally developed considering heliostat layout patterns, providing sub-optimal results. In this paper, a method to optimise cleaning schedules is developed, with the objective of maximising energy generated by the plant. First, an algorithm finds a cleaning schedule by solving an integer program, which is then used as a starting solution in an exchange heuristic. Since the optimisation problems are of large size, a p-median type heuristic is performed to reduce the problem dimensionality by clustering heliostats into groups to be cleaned in the same period.

Published
2019

3. Convergence to equilibrium of global weak solutions for a Cahn-Hilliard-Navier-Stokes vesicle model

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales, Simulación Num.y Desarrollo Software, Climent Ezquerra, María Blanca, Guillén González, Francisco Manuel, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales, Simulación Num.y Desarrollo Software, Climent Ezquerra, María Blanca, and Guillén González, Francisco Manuel

Abstract

In this paper, we introduce a model describing the dynamic of vesicle membranes within an incompressible viscous fluid in 3D domains. The system consists of the NavierStokes equations, with an extra stress tensor depending on the membrane, coupled with a Cahn-Hilliard phase-field equation associated to a bending energy plus a penalization term related to the area conservation. This problem has a dissipative in time free-energy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the large-time behavior of the weak solutions. By using a modified Lojasiewicz-Simon’s result, we prove the convergence as time goes to infinity of each (whole) trajectory to a single equilibrium. Finally, the convergence of the trajectory of the phase is improved by imposing more regularity on the domain and initial phase.

Published
2019

4. Derivation of the Navier-Stokes-Poisson system with radiation for an accretion disk

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales, Simulación Num.y Desarrollo Software, Ducomet, Bernard, Nečasová, Šárka, Pokorný, Milan, Rodríguez Bellido, María Ángeles, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales, Simulación Num.y Desarrollo Software, Ducomet, Bernard, Nečasová, Šárka, Pokorný, Milan, and Rodríguez Bellido, María Ángeles

Published
2018

5. A new proof of the existence of suitable weak solutions and other remarks for the Navier-Stokes equations

Author

Universidad de Sevilla. FQM131: Ec.diferenciales, Simulación Num.y Desarrollo Software, Fernández Cara, Enrique, Marín Gayte, Irene, Universidad de Sevilla. FQM131: Ec.diferenciales, Simulación Num.y Desarrollo Software, Fernández Cara, Enrique, and Marín Gayte, Irene

Abstract

We prove that the limits of the semi-discrete and the discrete semi-implicit Euler schemes for the 3D Navier-Stokes equations upplemented with Dirichlet boundary conditions are suitable in the sense of Scheffer. This provides a new proof of the existence of suitable weak solutions, first established by Caffarelli, Kohn and Nirenberg. Our results are similar to the main result in Guermond, J.-L. (2007) Faedo-Galerkin Weak Solutions of the Navier-Stokes Equations with Dirichlet Boundary Conditions Are Suitable. Journal de Mathématiques Pures et Appliquées, 88, 87-106. We also present some additional remarks and open questions on suitable solutions.

Published
2018

6. Some remarks on the comparison principle in Kirchhoff equations

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Conselho Nacional de Desenvolvimento Científico e Tecnológico. Brasil, Ministerio de Economía y Competitividad (MINECO). España, Malcher Figueiredo, Giovany de Jesus, Suárez Fernández, Antonio, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Conselho Nacional de Desenvolvimento Científico e Tecnológico. Brasil, Ministerio de Economía y Competitividad (MINECO). España, Malcher Figueiredo, Giovany de Jesus, and Suárez Fernández, Antonio

Abstract

In this paper we study the validity of the comparison principle and the sub-supersolution method for Kirchhoff type equations. We show that these principles do not work when the Kirchhoff function is increasing, contradicting some previous results. We give an alternative sub-supersolution method and apply it to some models.

Published
2018

7. An inverse problem in elastography involving Lamé systems

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Universidad de Sevilla. FQM309: Control y Homogeneización de Ecuaciones en Derivadas Parciales, Ministerio de Ciencia, Innovación y Universidades (MICINN). España, Ministerio de Economía y Competitividad (MINECO). España, Fernández Cara, Enrique, Maestre Caballero, Faustino, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Universidad de Sevilla. FQM309: Control y Homogeneización de Ecuaciones en Derivadas Parciales, Ministerio de Ciencia, Innovación y Universidades (MICINN). España, Ministerio de Economía y Competitividad (MINECO). España, Fernández Cara, Enrique, and Maestre Caballero, Faustino

Abstract

This paper deals with some inverse problems for the linear elasticity system with origin in elastography: we try to identify the material coefficients from some extra information on (a part of) the boundary. In our main result, we assume that the total variation of the coefficient matrix is a priori bounded. We reformulate the problem as the minimization of a function in an appropriate constraint set. We prove that this extremal problem possesses at least one solution with the help of some regularity results. Two crucial ingredients are a Meyers-like theorem that holds in the context of linear elasticity and a nonlinear interpolation result by Luc Tartar. We also perform some numerical experiments that provide satisfactory results. To this end, we apply the Augmented Lagrangian algorithm, completed with a limited-memory BFGS subalgorithm. Finally, on the basis of these experiments, we illustrate the influence of the starting guess and the errors in the data on the behavior of the iterates.

Published
2018

8. Local null controllability of a 1D Stefan problem

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, Fernández Cara, Enrique, Hernández, Freddy, Limaco Ferrel, Juan, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, Fernández Cara, Enrique, Hernández, Freddy, and Limaco Ferrel, Juan

Abstract

The purpose of this article is to give a new proof of a null controllability result for a 1D free-boundary problem of the Stefan kind for a heat PDE. We introduce a method based on local inversion that, in contrast with other previous arguments, does not rely on any compactness property and can be generalized to higher dimensions.

Published
2018

9. Quantitative Fattorini-Hautus test and minimal null control time for parabolic problems

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, Ammar-Khodja, Farid, Benabdallah, Assia, González Burgos, Manuel, Morancey, Morgan, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, Ammar-Khodja, Farid, Benabdallah, Assia, González Burgos, Manuel, and Morancey, Morgan

Abstract

This paper investigates the link between the null controllability property for some abstract parabolic problems and an inequality that can be seen as a quantified Fattorini-Hautus test. Depending on the hypotheses made on the abstract setting considered we prove that this inequality either gives the exact minimal null control time or at least gives the qualitative property of existence of such a minimal time. We also prove that for many known examples of minimal time in the parabolic setting, this inequality recovers the value of this minimal time., Dans cet article nous étudions le lien entre la contrôlabilité à zéro d'un problème parabolique abstrait et la validité d'une inégalité qui est une version quantifiée du test de Fattorini–Hautus. Nous prouvons que cette inégalité permet de caractériser l'existence d'un temps minimal pour le problème de contrôlabilité à zéro et, selon les hypothèses considérées, d'obtenir la valeur de ce temps minimal. Nous prouvons aussi que dans la plupart des exemples connus de problèmes paraboliques ayant un temps minimal de contrôle à zéro, cette inégalité est une condition nécessaire et suffisante de contrôlabilité.

Published
2018

10. A projection-based time-splitting algorithm for approximating nematic liquid crystal flows with stretching

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER), Cabrales, Roberto Carlos, Guillén González, Francisco Manuel, Gutiérrez Santacreu, Juan Vicente, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER), Cabrales, Roberto Carlos, Guillén González, Francisco Manuel, and Gutiérrez Santacreu, Juan Vicente

Abstract

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.

Published
2017

11. An optimal control problem for a Kirchhoff-type equation

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Delgado Delgado, Manuel, Malcher Figueiredo, Giovany de Jesus, Gayte Delgado, María Inmaculada, Morales Rodrigo, Cristian, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Delgado Delgado, Manuel, Malcher Figueiredo, Giovany de Jesus, Gayte Delgado, María Inmaculada, and Morales Rodrigo, Cristian

Abstract

In this paper we study a control problem for a Kirchhoff-type equation. The method to obtain first order necessary optimality conditions is the Dubovitskii-Milyoutin formalism because the classical arguments do not work. We obtain a characterization of the optimal control by a partial differential system which is solved numerically.

Published
2017

12. An optimization tool to design the field of a solar power tower plant allowing heliostats of different sizes

Author

Universidad de Sevilla. Departamento de Estadística e Investigación Operativa, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM329: Optimización, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Carrizosa Priego, Emilio José, Domínguez Bravo, Carmen Ana, Fernández Cara, Enrique, Quero García, Manuel, Universidad de Sevilla. Departamento de Estadística e Investigación Operativa, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM329: Optimización, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Carrizosa Priego, Emilio José, Domínguez Bravo, Carmen Ana, Fernández Cara, Enrique, and Quero García, Manuel

Abstract

The design of a Solar Power Tower plant involves the optimization of the heliostat field layout. Fields are usually designed to have all heliostats of identical size. Although the use of a single heliostat size has been questioned in the literature, there are no tools to design fields with heliostats of several sizes at the same time. In this paper, the problem of optimizing the heliostat field layout of a system with heliostats of different sizes is addressed. We present an optimization tool to design solar plants allowing two heliostat sizes. The methodology is illustrated with a particular example considering different heliostat costs.

Published
2017

13. New results on the Stackelberg-Nash exact control of linear parabolic equations

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Dias Araruna, Fágner, Fernández Cara, Enrique, Guerrero Rodríguez, Sergio, Cardoso Santos, Mauricio, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Dias Araruna, Fágner, Fernández Cara, Enrique, Guerrero Rodríguez, Sergio, and Cardoso Santos, Mauricio

Abstract

This paper is concerned with Stackelberg-Nash strategies to control parabolic equations. We have one control, the leader, that is responsible for a null controllability property; additionally, we have a couple of controls, called the followers, that provides a Nash equilibrium for two cost functionals. This is a classical situation in many fields of science and, in mathematics, leads to a lot of interesting questions and open problems and possesses many applications. In the main result, we prove the existence of a leader such that the corresponding controlled system is driven to zero. This way, we improve some questions that were left open in previous works.

Published
2017

14. Combining linear and fast diffusion in a nonlinear elliptic equation

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Cintra da Silva, Willian, Morales Rodrigo, Cristian, Suárez Fernández, Antonio, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Cintra da Silva, Willian, Morales Rodrigo, Cristian, and Suárez Fernández, Antonio

Abstract

In this paper we analyse an elliptic equation that combines linearand nonlinear fast diffusion with a logistic type reaction function. We proveexistence and non-existence results of positive solutions using bifurcation the-ory and sub-supersolution method. Moreover, we apply variational methodsto obtain a pair of ordered positive solutions.

Published
2017

15. Linear spplitting schemes for a nematic-isotropic model with anchoring effects

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Guillén González, Francisco Manuel, Rodríguez Bellido, María Ángeles, Tierra Chica, Giordano, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Guillén González, Francisco Manuel, Rodríguez Bellido, María Ángeles, and Tierra Chica, Giordano

Published
2016

16. Study of a logistic equation with local and non-local reaction terms

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Delgado Delgado, Manuel, Malcher Figueiredo, Giovany de Jesus, Oliveira Pimenta, Marcos Tadeu de, Suárez Fernández, Antonio, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Delgado Delgado, Manuel, Malcher Figueiredo, Giovany de Jesus, Oliveira Pimenta, Marcos Tadeu de, and Suárez Fernández, Antonio

Abstract

In this work we examine a logistic equation with local and nonlocal reaction terms both for time dependent and steady-state problems. Mainly, we use bifurcation and monotonicity methods to prove the existence of positive solutions for the steady-state equation and sub-supersolution method for the long time behavior for the time dependent problem. The results depend strongly on the size and sign of the parameters on the local and non-local terms.

Published
2016

17. A posteriori error analysis of an augmented mixed method for the Navier-Stokes equations with nonlinear viscosity

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Gatica Pérez, Gabriel Nibaldo, Ruiz Baier, Ricardo, Tierra Chica, Giordano, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Gatica Pérez, Gabriel Nibaldo, Ruiz Baier, Ricardo, and Tierra Chica, Giordano

Abstract

In this work we develop the a posteriori error analysis of an augmented mixed finite element method for the 2D and 3D versions of the Navier-Stokes equations when the viscosity depends nonlinearly on the module of the velocity gradient. Two different reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary (convex or non-convex) polygonal and polyhedral regions are derived. Our analysis of reliability of the proposed estimators draws mainly upon the global inf-sup condition satisfied by a suitable linearization of the continuous formulation, an application of Helmholtz decomposition, and the local approximation properties of the Raviart-Thomas and Clément interpolation operators. In addition, differently from previous approaches for augmented mixed formulations, the boundedness of the Clément operator plays now an interesting role in the reliability estimate. On the other hand, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show their efficiency. Finally, several numerical results are provided to illustrate the good performance of the augmented mixed method, to confirm the aforementioned properties of the a posteriori error estimators, and to show the behaviour of the associated adaptive algorithm.

Published
2016

18. Controlling linear and semilinear systems formed by one elliptic and two parabolic PDEs with one scalar control

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, Fernández Cara, Enrique, Limaco Ferrel, Juan, Dias Bezerra de Menezes, Silvano, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, Fernández Cara, Enrique, Limaco Ferrel, Juan, and Dias Bezerra de Menezes, Silvano

Abstract

In this paper, we prove controllability results for some linear and semilinear systems where we find two parabolic PDEs and one elliptic PDE and we act through one locally supported in space scalar control. The arguments rely on a careful analysis of the linear case and an application of an inverse function theorem. The facts that we act through a single scalar control and one of the PDEs has no time derivative are the main novelties and introduce several nontrivial difficulties.

Published
2016

19. Field-design optimization with triangular heliostat pods

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. Departamento de Estadística e Investigación Operativa, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Universidad de Sevilla. FQM329: Optimización, Ministerio de Economía y Competitividad (MINECO). España, Junta de Andalucía, Domínguez Bravo, Carmen Ana, Bode, Sebastian James, Heiming, Gregor, Richter, Pascal, Carrizosa Priego, Emilio José, Fernández Cara, Enrique, Frank, Martin, Gauché, Paul, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. Departamento de Estadística e Investigación Operativa, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Universidad de Sevilla. FQM329: Optimización, Ministerio de Economía y Competitividad (MINECO). España, Junta de Andalucía, Domínguez Bravo, Carmen Ana, Bode, Sebastian James, Heiming, Gregor, Richter, Pascal, Carrizosa Priego, Emilio José, Fernández Cara, Enrique, Frank, Martin, and Gauché, Paul

Abstract

In this paper the optimization of a heliostat field with triangular heliostat pods is addressed. The use of structures which allow the combination of several heliostats into a common pod system aims to reduce the high costs associated with the heliostat field and therefore reduces the Levelized Cost of Electricity value. A pattern-based algorithm and two pattern-free algorithms are adapted to handle the field layout problem with triangular heliostat pods. Under the Helio100 project in South Africa, a new small-scale Solar Power Tower plant has been recently constructed. The Helio100 plant has 20 triangular pods (each with 6 heliostats) whose positions follow a linear pattern. The obtained field layouts after optimization are compared against the reference field Helio100.

Published
2016

20. Numerical methods for solving the Cahn-Hilliard equation and its applicability to mixtures of isotropic and nematic flows with anchoring effects

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales, Simulación Num.y Desarrollo Software, Rodríguez Bellido, María Ángeles, Guillén González, Francisco Manuel, Tierra Chica, Giordano, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales, Simulación Num.y Desarrollo Software, Rodríguez Bellido, María Ángeles, Guillén González, Francisco Manuel, and Tierra Chica, Giordano

Published
2016

21. On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, Centre national de la recherche scientifique (CNRS). France, Fernández Cara, Enrique, Münch, Arnaud, Araujo de Souza, Diego, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, Centre national de la recherche scientifique (CNRS). France, Fernández Cara, Enrique, Münch, Arnaud, and Araujo de Souza, Diego

Abstract

The aim of this work is to present some strategies to solve numerically controllability problems for the two-dimensional heat equation, the Stokes equations and the Navier-Stokes equations with Dirichlet boundary conditions. The main idea is to adapt the Fursikov-Imanuvilov formulation, see [A.V. Fursikov, O.Yu. Imanuvilov: Controllability of Evolutions Equations, Lectures Notes Series, Vol. 34, Seoul National University, 1996]; this approach has been followed recently for the onedimensional heat equation by the first two authors. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up near the final time. The associated optimality conditions can be viewed as a differential system in the three variables x1, x2 and t that is second–order in time and fourth–order in space, completed with appropriate boundary conditions. We present several mixed formulations of the problems and, then, associated mixed finite element Lagrangian approximations that are easy to handle. Finally, we exhibit some numerical experiments.

Published
2016

22. Controllability of non-scalar parabolic systems: Some recent results and phenomena

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, González Burgos, Manuel, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, and González Burgos, Manuel

Abstract

In this course we will deal with non-scalar systems which in fact are coupled parabolic scalar equations. We do not present results relating to the controllability problems of systems which come from fluid mechanics as Stokes, Navier-Stokes, ...

Published
2016

23. Null controllability of linear heat and wave equations with nonlocal spatial terms

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Fernández Cara, Enrique, Lü, Qi, Zuazua Iriondo, Enrique, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Fernández Cara, Enrique, Lü, Qi, and Zuazua Iriondo, Enrique

Abstract

In this paper, we study the null controllability of linear heat and wave equations with spatial nonlocal integral terms. Under some analyticity assumptions on the corresponding kernel, we show that the equations are controllable. We employ compactness-uniqueness arguments in a suitable functional setting, an argument that is harder to apply for heat equations because of its very strong time irreversibility. Some possible extensions and open problems concerning other nonlocal systems are presented.

Published
2016

24. A mixed finite element method for Darcy’s equations with pressure dependent porosity

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Gatica Pérez, Gabriel Nibaldo, Ruiz Baier, Ricardo, Tierra Chica, Giordano, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Gatica Pérez, Gabriel Nibaldo, Ruiz Baier, Ricardo, and Tierra Chica, Giordano

Abstract

In this work we develop the a priori and a posteriori error analyses of a mixed finite element method for Darcy’s equations with porosity depending exponentially on the pressure. A simple change of variable for this unknown allows to transform the original nonlinear problem into a linear one whose dual-mixed variational formulation falls into the frameworks of the generalized linear saddle point problems and the fixed point equations satisfied by an affine mapping. According to the latter, we are able to show the well-posedness of both the continuous and discrete schemes, as well as the associated Cea estimate, by simply applying a suitable combination of the classical Babuska-Brezzi theory and the Banach fixed point Theorem. In particular, given any integer k ≥ 0, the stability of the Galerkin scheme is guaranteed by employing Raviart-Thomas elements of order k for the velocity, piecewise polynomials of degree k for the pressure, and continuous piecewise polynomials of degree k+1 for an additional Lagrange multiplier given by the trace of the pressure on the Neumann boundary. Note that the two ways of writing the continuous formulation suggest accordingly two different methods for solving the discrete schemes. Next, we derive a reliable and efficient residualbased a posteriori error estimator for this problem. The global inf-sup condition satisfied by the continuous formulation, Helmholtz decompositions, and the local approximation properties of the Raviart-Thomas and Cl´ement interpolation operators are the main tools for proving the reliability. In turn, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of both methods, confirming the aforementioned properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are reported.

Published
2016

25. On the influence of wavy riblets on the slip behaviour of viscous fluids

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM309: Control y Homogeneización de Ecuaciones en Derivadas Parciales, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Bonnivard, Matthieu, Suárez Grau, Francisco Javier, Tierra Chica, Giordano, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM309: Control y Homogeneización de Ecuaciones en Derivadas Parciales, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Bonnivard, Matthieu, Suárez Grau, Francisco Javier, and Tierra Chica, Giordano

Abstract

In this work, we use the homogenization theory to investigate the capability of wavy riblet patterns to influence the behaviour of a viscous flow near a ribbed boundary. Starting from perfect slip conditions on the wall, we show that periodic oscillations of wavy riblets in the lateral direction may induce a friction effect in the direction of the flow, contrary to what happens with straight riblets. Finally, we illustrate this effect numerically by simulating riblet profiles that are widely used in experimental studies: the V -shape, U-shape, and blade riblets.

Published
2016

26. A mixed formulation for the direct approximation of L2-weighted controls for the linear heat equation

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Coordenação de Aperfeiçoamento de Pessoal de Nivel Superior. Brasil, Ministerio de Ciencia e Innovación (MICIN). España, Münch, Arnaud, Araujo de Souza, Diego, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Coordenação de Aperfeiçoamento de Pessoal de Nivel Superior. Brasil, Ministerio de Ciencia e Innovación (MICIN). España, Münch, Arnaud, and Araujo de Souza, Diego

Abstract

This paper deals with the numerical computation of null controls for the linear heat equation. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a given positive time. In [Fernandez-Cara & Münch, Strong convergence approximations of null controls for the 1D heat equation, 2013], a so-called primal method is described leading to a strongly convergent approximation of distributed control: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality conditions. In this work, we adapt the method to approximate the control of minimal square integrable-weighted norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner situation and is valid in any dimension.

Published
2016

27. Long-time behavior of a Cahn-Hilliard-Navier-Stokes vesicle-fluid interaction model

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Climent Ezquerra, María Blanca, Guillén González, Francisco Manuel, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Climent Ezquerra, María Blanca, and Guillén González, Francisco Manuel

Published
2015

28. Numerical solution of some geometric inverse problems

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Doubova Krasotchenko, Anna, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, and Doubova Krasotchenko, Anna

Published
2015

29. New phenomena in the null controllability of coupled parabolic systems

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, González Burgos, Manuel, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, and González Burgos, Manuel

Published
2015

30. Theoretical and numerical local null controllability of a Ladyzhenskaya-Smagorinsky model of turbulence

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Dirección General de Enseñanza Superior. España, Fernández Cara, Enrique, Limaco Ferrel, Juan, Dias Bezerra de Menezes, Silvano, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Dirección General de Enseñanza Superior. España, Fernández Cara, Enrique, Limaco Ferrel, Juan, and Dias Bezerra de Menezes, Silvano

Abstract

This paper deals with the control of a differential turbulence model of the Ladyzhenskaya–Smagorinsky kind. In the equations we find local and nonlocal nonlinearities: the usual transport terms and a turbulent viscosity that depends on the global in space energy dissipated by the mean flow. We prove that the system is locally null-controllable, with distributed controls locally supported in space. The proof relies on rather well known arguments. However, some specific difficulties are found here because of the occurrence of nonlocal nonlinear terms. We also present an iterative algorithm of the quasi-Newton kind that provides a sequence of states and controls that converge towards a solution to the control problem. Finally, we give the details of a numerical approximation and we illustrate the behavior of the algorithm with a numerical experiment.

Published
2015

31. The sub-supersolution method for Kirchhoff systems: applications

Author

Carvalho, Alexandre Nolasco, Ruf, Bernhard, Moreira dos Santos, Ederson, Gossez, Jean-Pierre, Monari Soares, Sergio Henrique, Cazenave, Thierry, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Malcher Figueiredo, Giovany de Jesus, Suárez Fernández, Antonio, Carvalho, Alexandre Nolasco, Ruf, Bernhard, Moreira dos Santos, Ederson, Gossez, Jean-Pierre, Monari Soares, Sergio Henrique, Cazenave, Thierry, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Malcher Figueiredo, Giovany de Jesus, and Suárez Fernández, Antonio

Abstract

In this paper we prove that the sub-supersolution method works for general Kirchhoff systems. We apply the cited method to prove the existence of positive solutions for some specific models.

Published
2015

32. Analysis of the hydrostatic Stokes problem and finite-element approximation in unstructured meshes

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Educación y Ciencia (MEC). España, Junta de Andalucía, Guillén González, Francisco Manuel, Rodríguez Galván, José Rafael, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Educación y Ciencia (MEC). España, Junta de Andalucía, Guillén González, Francisco Manuel, and Rodríguez Galván, José Rafael

Abstract

The stability of velocity and pressure mixed finite-element approximations in general meshes of the hydrostatic Stokes problem is studied, where two “inf-sup” conditions appear associated to the two constraints of the problem; namely incompressibility and hydrostatic pressure. Since these two constraints have different properties, it is not easy to choose finite element spaces satisfying both. From the analytical point of view, two main results are established; the stability of an anisotropic approximation of the velocity (using different spaces for horizontal and vertical velocities) with piecewise constant pressures, and the unstability of standard (isotropic) approximations which are stable for the Stokes problem, like the mini-element or the Taylor-Hood element. Moreover, we give some numerical simulations, which agree with the previous analytical results and allow us to conjecture the stability of some anisotropic approximations of the velocity with continuous piecewise linear pressure in unstructured meshes.

Published
2015

33. Controllability of linear parabolic systems: New phenomena

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, González Burgos, Manuel, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, and González Burgos, Manuel

Published
2015

34. On the regularity of the Q-Tensor depending on the data

Author

Ahusborde, Étienne, Amrouche, Chérif, López de Silanes Busto, María Cruz, Palacios Latasa, Manuel Pedro, Sanz Sáiz, Gerardo, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Guillén González, Francisco Manuel, Rodríguez Bellido, María Ángeles, Ahusborde, Étienne, Amrouche, Chérif, López de Silanes Busto, María Cruz, Palacios Latasa, Manuel Pedro, Sanz Sáiz, Gerardo, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Guillén González, Francisco Manuel, and Rodríguez Bellido, María Ángeles

Abstract

The coupled Navier-Stokes and Q-Tensor system is one of the models used to describe the behavior of the nematic liquid crystals, an intermediate phase between crystalline solids and isotropic fluids. These equations model the dynamics of the fluid via velocity and pressure (u, p) and the orientation of the molecules via a tensor Q. A review on the existence of weak solutions, maximum principle and a uniqueness criteria can be seen in [Guillen-González, F., and Rodríguez-Bellido, M. Á. Some properties on the Q-tensor system. Monogr. Mat. García Galdeano 39 (2014), 133–145] (the corresponding Cauchy problem in the whole R3 is analyzed by Zarnescu. However, the regularity of such solutions is only analyzed under some restrictive conditions: large viscosity or periodic boundary conditions. In this work, we study two different types of regularity for the Q-Tensor model: one inherited from the usual strong solution for the Navier-Stokes equations, and another one where (u, Q) and (∂tu, ∂tQ) have weak regularity (weak-t). This latter regularity is introduced due to the impossibility of obtaining local in time strong estimates for nonperiodic boundary conditions, where only the existence (and uniqueness) of local weak-t solution is obtained. Some regularity criteria for (u, Q) will also be given. In the particular case of Neumann boundary conditions for Q, the regularity criteria only must be imposed for the velocity u).

Published
2015

35. Liquid crystal and phase-field models are related by mathematics

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, Guillén González, Francisco Manuel, Climent Ezquerra, María Blanca, Gutiérrez Santacreu, Juan Vicente, Rodríguez Bellido, María Ángeles, Tierra Chica, Giordano, Rojas Medar, Marko Antonio, Cabrales, Roberto Carlos, Grün, Günther, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Economía y Competitividad (MINECO). España, Guillén González, Francisco Manuel, Climent Ezquerra, María Blanca, Gutiérrez Santacreu, Juan Vicente, Rodríguez Bellido, María Ángeles, Tierra Chica, Giordano, Rojas Medar, Marko Antonio, Cabrales, Roberto Carlos, and Grün, Günther

Published
2014

36. Some properties on the Q-Tensor system

Author

López de Silanes Busto, María Cruz, Palacios Latasa, Manuel Pedro, Sanz Sáiz, Gerardo, Amrouche, Chérif, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Guillén González, Francisco Manuel, Rodríguez Bellido, María Ángeles, López de Silanes Busto, María Cruz, Palacios Latasa, Manuel Pedro, Sanz Sáiz, Gerardo, Amrouche, Chérif, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Guillén González, Francisco Manuel, and Rodríguez Bellido, María Ángeles

Abstract

We study the coupled Navier-Stokes and Q-Tensor system (analyzed in cf. [Paicu, M., and Zarnescu, A. Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system. Arch. Ration. Mech. Anal. 203 (2012), 45–67] in the whole R3) in a bounded three-dimensional domain for several boundary conditions, rewriting the system in a way that properties as symmetry and null-trace for the tensor Q can be proved. We show some analytical results such as: the existence of global in time weak solution, a maximum principle for the Q-tensor, local in time strong solution (which is global assuming an additional regularity criterion for the velocity in the space-periodic boundary condition case), global in time strong solution imposing dominant viscosity (for the space-periodic or homogeneous Neumann boundary condition cases) and regularity criteria for uniqueness of weak solutions.

Published
2014

37. Type IV pili interactions promote intercellular association and moderate swarming of Pseudomonas aeruginosa

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, National Institutes of Health. United States, Anyan, Morgen E., Amiri, Aboutaleb, Harvey, Cameron W., Tierra Chica, Giordano, Morales Soto, Nydia, Driscoll, Callan M., Alber, Mark S., Shrout, Joshua D., Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, National Institutes of Health. United States, Anyan, Morgen E., Amiri, Aboutaleb, Harvey, Cameron W., Tierra Chica, Giordano, Morales Soto, Nydia, Driscoll, Callan M., Alber, Mark S., and Shrout, Joshua D.

Abstract

Pseudomonas aeruginosa is a ubiquitous bacterium that survives in many environments, including as an acute and chronic pathogen in humans. Substantial evidence shows that P. aeruginosa behavior is affected by its motility, and appendages known as flagella and type IV pili (TFP) are known to confer such motility. The role these appendages play when not facilitating motility or attachment, however, is unclear. Here we discern a passive intercellular role of TFP during flagellar-mediated swarming of P. aeruginosa that does not require TFP extension or retraction. We studied swarming at the cellular level using a combination of laboratory experiments and computational simulations to explain the resultant patterns of cells imaged from in vitro swarms. Namely, we used a computational model to simulate swarming and to probe for individual cell behavior that cannot currently be otherwise measured. Our simulations showed that TFP of swarming P. aeruginosa should be distributed all over the cell and that TFP−TFP interactions between cells should be a dominant mechanism that promotes cell−cell interaction, limits lone cell movement, and slows swarm expansion. This predicted physical mechanism involving TFP was confirmed in vitro using pairwise mixtures of strains with and without TFP where cells without TFP separate from cells with TFP. While TFP slow swarm expansion, we show in vitro that TFP help alter collective motion to avoid toxic compounds such as the antibiotic carbenicillin. Thus, TFP physically affect P. aeruginosa swarming by actively promoting cell-cell association and directional collective motion within motile groups to aid their survival.

Published
2014

38. Convergence to equilibrium for smectic-A liquid crystals in 3D domains

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Guillén González, Francisco Manuel, Climent Ezquerra, María Blanca, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Guillén González, Francisco Manuel, and Climent Ezquerra, María Blanca

Published
2014

39. Why (partial) differential equations? Why control problems? What for?

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Fernández Cara, Enrique, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, and Fernández Cara, Enrique

Published
2014

40. Nuevos fenómenos en la controlabilidad exacta a cero de sistemas de EDP parabólicas

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, González Burgos, Manuel, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, and González Burgos, Manuel

Published
2014

41. Some recent results concerning the theoretical and numerical controllability of PDEs

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Fernández Cara, Enrique, Münch, Arnaud, Araujo de Souza, Diego, Dias Araruna, Fágner, Cardoso Santos, Mauricio, Limaco Ferrel, Juan, Dias Bezerra de Menezes, Silvano, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Fernández Cara, Enrique, Münch, Arnaud, Araujo de Souza, Diego, Dias Araruna, Fágner, Cardoso Santos, Mauricio, Limaco Ferrel, Juan, and Dias Bezerra de Menezes, Silvano

Published
2014

42. On a initial-boundary Q-tensor problem related to liquid crystals

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Guillén González, Francisco Manuel, Rodríguez Bellido, María Ángeles, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Guillén González, Francisco Manuel, and Rodríguez Bellido, María Ángeles

Published
2013

43. Some inverse and control problems for fluids

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Dirección General de Investigación (DGI). España, Fernández Cara, Enrique, Horsin, Thierry, Kasumba, Henry, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Dirección General de Investigación (DGI). España, Fernández Cara, Enrique, Horsin, Thierry, and Kasumba, Henry

Abstract

This paper deals with some inverse and control problems for the Navier-Stokes and related systems. We will focus on some particular aspects that have recently led to interesting (theoretical and numerical) results: geometric inverse problems, Eulerian and Lagrangian controllability and vortex reduction oriented to shape optimization.

Published
2013

44. On energy-stable schemes for two Vesicle Membrane phase-field models

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Guillén González, Francisco Manuel, Tierra Chica, Giordano, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Guillén González, Francisco Manuel, and Tierra Chica, Giordano

Published
2013

45. The control of PDEs: some basic concepts, recent results and open problems

Author

Kaplický, Petr, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Fernández Cara, Enrique, Kaplický, Petr, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, and Fernández Cara, Enrique

Abstract

These Notes deal with the control of systems governed by some PDEs. I will mainly consider time-dependent problems. The aim is to present some fundamental results, some applications and some open problems related to the optimal control and the controllability properties of these systems. In Chapter 1, I will review part of the existing theory for the optimal control of partial differential systems. This is a very broad subject and there have been so many contributions in this field over the last years that we will have to limit considerably the scope. In fact, I will only analyze a few questions concerning some very particular PDEs. We shall focus on the Laplace, the stationary Navier-Stokes and the heat equations. Of course, the existing theory allows to handle much more complex situations. Chapter 2 is devoted to the controllability of some systems governed by linear time-dependent PDEs. I will consider the heat and the wave equations. I will try to explain which is the meaning of controllability and which kind of controllability properties can be expected to be satisfied by each of these PDEs. The main related results, together with the main ideas in their proofs, will be recalled. Finally, Chapter 3 is devoted to present some controllability results for other time-dependent, mainly nonlinear, parabolic systems of PDEs. First, we will revisit the heat equation and some extensions. Then, some controllability results will be presented for systems governed by stochastic PDEs. Finally, I will consider several nonlinear systems from fluid mechanics: Burgers, NavierStokes, Boussinesq, micropolar, etc. Along these Notes, a set of questions (some of them easy, some of them more intrincate or even difficult) will be stated. Also, several open problems will be mentioned. I hope that all this will help to understand the underlying basic concepts and results and to motivate research on the subject.

Published
2012

46. Some inverse problems in elastography

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Universidad de Sevilla. FQM309: Control y Homogeneización de Ecuaciones en Derivadas Parciales, Fernández Cara, Enrique, Maestre Caballero, Faustino, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Universidad de Sevilla. FQM309: Control y Homogeneización de Ecuaciones en Derivadas Parciales, Fernández Cara, Enrique, and Maestre Caballero, Faustino

Published
2011

47. Finite element approximation of nematic liquid crystal flows using a saddle-point structure

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Badia Rodríguez, Santiago, Guillén González, Francisco Manuel, Gutiérrez Santacreu, Juan Vicente, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Badia Rodríguez, Santiago, Guillén González, Francisco Manuel, and Gutiérrez Santacreu, Juan Vicente

Abstract

In this work, we propose finite element schemes for the numerical approximation of nematic liquid crystal flows, based on a saddle-point formulation of the director vector sub-problem. It introduces a Lagrange multiplier that allows to enforce the sphere condition. In this setting, we can consider the limit problem (without penalty) and the penalized problem (using a Ginzburg-Landau penalty function) in a unified way. Further, the resulting schemes have an stable behavior with respect to the value of the penalty parameter, a key difference with respect to the existing schemes. Two different methods have been considered for the time integration. First, we have considered an implicit algorithm that is unconditionally stable and energy preserving. The linearization of the problem at every time step value can be performed using a quasi-Newton method that allows to decouple fluid velocity and director vector computations for every tangent problem. Then, we have designed a linear semi-implicit algorithm (i.e. it does not involve nonlinear iterations) and proved that it is unconditionally stable, verifying a discrete energy inequality. Finally, some numerical simulations are provided.

Published
2011

48. On the exact controllability to trajectories of the nonlinear heat equation in unbounded domains

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, González Burgos, Manuel, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, and González Burgos, Manuel

Published
2011

49. An overview on numerical analyses of nematic liquid crystal flows

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Ciencia e Innovación (MICIN). España, Badia Rodríguez, Santiago, Guillén González, Francisco Manuel, Gutiérrez Santacreu, Juan Vicente, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, Ministerio de Ciencia e Innovación (MICIN). España, Badia Rodríguez, Santiago, Guillén González, Francisco Manuel, and Gutiérrez Santacreu, Juan Vicente

Abstract

The purpose of this work is to provide an overview of the most recent numerical developments in the field of nematic liquid crystals. The Ericksen-Leslie equations govern the motion of a nematic liquid crystal. This system, in its simplest form, consists of the Navier-Stokes equations coupled with an extra anisotropic stress tensor, which represents the effect of the nematic liquid crystal on the fluid, and a convective harmonic map equation. The sphere constraint must be enforced almost everywhere in order to obtain an energy estimate. Since an almost everywhere satisfaction of this restriction is not appropriate at a numerical level, two alternative approaches have been introduced: a penalty method and a saddle-point method. These approaches are suitable for their numerical approximation by finite elements, since a discrete version of the restriction is enough to prove the desired energy estimate. The Ginzburg-Landau penalty function is usually used to enforce the sphere constraint. Finite element methods of mixed type will play an important role when designing numerical approximations for the penalty method in order to preserve the intrinsic energy estimate. The inf-sup condition that makes the saddle-point method well-posed is not clear yet. The only inf-sup condition for the Lagrange multiplier is obtained in the dual space of H1(Ω). But such an inf-sup condition requires more regularity for the director vector than the one provided by the energy estimate. Herein, we will present an alternative inf-sup condition whose proof for its discrete counterpart with finite elements is still open.

Published
2011

50. Controllability of non-scalar parabolic systems

Author

Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, González Burgos, Manuel, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software, and González Burgos, Manuel

Published
2011

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