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In this work we study a fully discrete mixed scheme, based on continuous finite elements in space and a linear semi-implicit first-order integration in time, approximating an Ericksen–Leslie nematic liquid crystal model by means of a Ginzburg–Landau penalized problem. Conditional stability of this scheme is proved via a discrete version of the energy law satisfied by the continuous problem, and conditional convergence towards generalized Young measure-valued solutions to the Ericksen–Leslie problem is showed when the discrete parameters (in time and space) and the penalty parameter go to zero at the same time. Finally, we will show some numerical experiences for a phenomenon of annihilation of singularities.

In this article we propose different splitting procedures for the transient incompressible MHD system that are unconditionally stable. We consider two levels of splitting, on one side we perform the segregation of the fluid pressure and magnetic pseudo-pressure from the vectorial fields computation. At the second level, the fluid velocity and induction fields are also decoupled. This way, we transform a fully coupled indefinite multi-physics system into a set of smaller definite ones, clearly reducing the CPU cost.With regard to the finite element approximation, we stick to an unconditionally convergent stabilized finite element formulation, since it introduces convection stabilization, allows to circumvent inf-sup conditions (clearly simplifying implementation issues) and is able to capture nonsmooth solutions of the magnetic sub-problem. However, residual-based finite element formulations are not suitable for segregation, since they lose the skew-symmetry of the off-diagonal blocks. Therefore, in this work we have proposed a novel term-by-term stabilization of the MHD system based on projections that is still unconditionally convergent., Postprint (published version)

The goal of this paper is to present uniform-in-time error estimates by considering spectral Galerkin approximations of the Kazhikhov-Smagulov model for strong solutions. To be more precise, we derive an optimal uniform-in-time error bound in the boldsymbol{H}^1\times H^2$ norm for the velocity and density approximations being stated in Theorem 6.

Residual-based stabilized nite element techniques for the Navier-Stokes equations lead to numerical discretizations that provide convection stabilization as well as pressure stability without the need to satisfy an inf-sup condition. They can be motivated by using a variational multiscale framework, based on the decomposition of the uid velocity into a resolvable nite element component plus a modeled subgrid scale component. The subgrid closure acts as a large eddy simulation turbulence model, leading to accurate under-resolved simulations. However, even though variational multiscale formulations are increasingly used in the applied nite element community, their numerical analysis has been restricted to a priori estimates and convergence to smooth solutions only, via a priori error estimates. In this work we prove that some versions of these methods (based on dynamic and orthogonal closures) also converge to weak (turbulent) solutions of the Navier-Stokes equations. These results are obtained by using compactness results in Bochner-Lebesgue spaces. Navier-Stokes equations; stability; convergence; stabilized nite element methods; subgrid scales; variational multiscale methods., Preprint

Variational multiscale methods lead to stable finite element approximations of the Navier-Stokes equations, both dealing with the indefinite nature of the system (pressure stability) and the velocity stability loss for high Reynolds numbers. These methods enrich the Galerkin formulation with a sub-grid component that is modelled. In fact, the effect of the sub-grid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum. Thus, they also act as effective large-eddy simulation turbulence models and allow to compute flows without the need to capture all the scales in the system. In this article, we consider a dynamic sub-grid model that enforces the sub-grid component to be orthogonal to the finite element space in L2 sense.We analyze the long-term behavior of the algorithm, proving the existence of appropriate absorbing sets and a compact global attractor. The improvements with respect to a finite element Galerkin approximation are the long-term estimates for the sub-grid component, that are translated to effective pressure and velocity stability. Thus, the stabilization introduced by the sub-grid model into the finite element problem is not deteriorated for infinite time intervals of computation., Postprint (published version)

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.

We present error estimates of a linear fully discrete scheme for a threedimensional mass diffusion model for incompressible fluids (also called Kazhikhov– Smagulov model). All unknowns of the model (velocity, pressure and density) are approximated in space by C0-finite elements and in time an Euler type scheme is used decoupling the density from the velocity–pressure pair. If we assume that the velocity and pressure finite-element spaces satisfy the inf–sup condition and the density finite-element space contains the products of any two discrete veloci-ties, we first obtain point-wise stability estimates for the density, under the constraint lim(h,k)→0 h/k = 0 (h and k being the space and time discrete parameters, respectively), and error estimates for the velocity and density in energy type norms, at the same time. Afterwards, error estimates for the density in stronger norms are deduced. All these error estimates will be optimal (of order O(h + k)) for regular enough solu-tions without imposing nonlocal compatibility conditions at the initial time. Finally, we also study two convergent iterative methods for the two problems to solve at each time step, which hold constant matrices (independent of iterations).

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.

Variational multiscale methods lead to stable finite element approximations of the Navier–Stokes equations, dealing with both the indefinite nature of the system (pressure stability) and the velocity stability loss for high Reynolds numbers. These methods enrich the Galerkin formulation with a subgrid component that is modeled. In fact, the effect of the subgrid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum. Thus, they also act as effective large-eddy simulation turbulence models and allow one to compute flows without the need to capture all the scales in the system. In this article, we consider a dynamic subgrid model that enforces the subgrid component to be orthogonal to the finite element space in the L2 sense. We analyze the long-term behavior of the algorithm, proving the existence of appropriate absorbing sets and a compact global attractor. The improvements with respect to a finite element Galerkin approximation are the long-term estimates for the subgrid component, which are translated to effective pressure and velocity stability. Thus, the stabilization introduced by the subgrid model into the finite element problem does not deteriorate for infinite time intervals of computation.

En esta charla analizamos la aproximación numérica con elementos finitos en espacio y diferencias finitas en tiempo de un modelo de cristales líquidos nemáticos (de tipo Eriksen-Leslie) y de un modelo penalizado de tipo Ginzburg-Landau. Después de describir los principales antecedentes del tema, se propone un esquema lineal totalmente acoplado y condicionalmente estable. La convergente (respecto de los parámetros de discretización y del parámetro de penalización) hacia una solución débil del problema de Eriksen-Leslie queda como problema abierto.

We analyze two numerical schemes of Euler type in time and C0 finite-element type with P1-approximation in space for solving a phase-field model of a binary alloy with thermal properties. This model is written as a highly non-linear parabolic system with three unknowns: phase-field, solute concentration and temperature, where the diffusion for the temperature and solute concentration may degenerate. The first scheme is nonlinear, unconditionally stable and convergent. The other scheme is linear but conditionally stable and convergent. A maximum principle is avoided in both schemes, using a truncation operator on the L2 projection onto the P0 finite element for the discrete concentration. In addition, for the model when the heat conductivity and solute diffusion coefficients are constants, optimal error estimates for both schemes are shown based on stability estimates.

Variational multiscale methods lead to stable finite element approximations of the Navier-Stokes equations, both dealing with the indefinite nature of the system pressure stability) and the velocity stability loss for high Reynolds numbers. These methods enrich the Galerkin formulation with a sub-grid component that is modeled. In fact, the effect of the sub-grid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum. Thus, they also act as effective large-eddy simulation turbulence models and allow to compute flows without the need to capture all the scales in the system. In this article, we consider a dynamic sub-grid model that enforces the sub-grid component to be orthogonal to the finite element spacein L2 sense. We analyze the long-term behavior of the algorithm, proving the existence of appropriate absorbing setsand a compact global attractor. The improvements with respect to a finite element Galerkin approximation are thelong-term estimates for the sub-grid component, that are translated to effective pressure and velocity stability. Thus,the stabilization introduced by the sub-grid model into the finite element problem is not deteriorated for infinite time intervals of computation.

In this work we develop fully discrete (in time and space) numerical schemes for two-dimensional incompressible fluids with mass diffusion, also so-called Kazhikhov-Smagulov models. We propose at most H^1-conformed finite elements (only globally continuous functions) to approximate all unknowns (velocity, pressure and density), although the limit density (solution of continuous problem) will have H^2 regularity. A backward Euler in time scheme is considered decoupling the computation of the density from the velocity and pressure. Unconditional stability of the schemes and convergence towards the (unique) global in time weak solution of the models is proved. Since a discrete maximum principle cannot be ensured, we must use a different interpolation inequality to obtain the strong estimates for the discrete density, from the used one in the continuous case. This inequality is a discrete version of the Gagliardo-Nirenberg interpolation inequality in 2D domains. Moreover, the discrete density is truncated in some adequate terms of the velocity-pressure problem.

La finalidad de esta tesis es aportar un mayor grado de entendimiento, desde el aspecto numérico, de diversos sistemas en derivas parciales provenientes de la mecánica de fluidos. Más precisamente, estudiamos un modelo de Navier Stokes con difusión de masa y un modelo de cristales líquidos. Como puno final analizamos un modelo estático de campo de fase par un proceso de solidificación de una mezcla binaria con propiedades térmicas. Los dos primeros modelos son de tipo Navier Stokes incomprensible en formulación velocidad presión, acoplados con una nueva ecuación de estado (bien de la densidad o bien del vector orientación de los cristales líquidos). Además de las dificultades bien conocidas del modelo de Navier Stokes clásico (tratamiento del término no lineal convectivo, el acoplamiento velocidad presión y la restricción de incomprensibilidad), estos modelos presentan otras dificultades, como son: la nueva incógnita debe verificar un principio del máximo tiene un orden de regularidad mayor que la velocidad y aparecen nuevas no linealidades que acoplan la incógnita de la ecuación de estado con la velocidad del fluido.

En esta comunicación proponemos y analizamos un esquema numérico totalmente discreto con elementos finitos en espacio y diferencias finitas en tiempo para aproximar un modelo de cristales líquidos nemáticos (de tipo Eriksen-Leslie) por medio de un modelo penalizado de tipo Ginzburg-Landau. Se muestra la estabilidad y convergencia de dicho esquema hacia una solución débil del problema continuo, respecto de los parámetros de discretización y del parámetro de penalización.

In this work, we present some results for local and global in time solutions (defined in the time interval (0, T) with T < +∞ or T = +∞) for the model of mass diffusion by using the spectral semi-Galerkin approximations. We establish results related to the global existence of weak solutions, to the existence of strong solutions (local in time or global in time for small enough data in 3D domains or general data in 2D case), to the regularity of strong solutions and the effects of the exponential decay of the external force in the asymptotic behavior when t → +∞ for global solutions.