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The Navier-Stokes- 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 is a regime flow parameter that has the dimension of the smallest scale being resolvable by the model. Hence, when = 0, one recovers the classical Navier-Stokes equations for a flow of viscous, incompressible, Newtonian fluids. Furthermore, the Navier-Stokes- equations can also be interpreted as a regularization of the Navier- Stokes equations, where stands for the regularization parameter. In this paper we first present the Navier-Stokes- 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- 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 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 L2 norm.

The Navier-Stokes- 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 is a regime flow parameter that has the dimension of the smallest scale being resolvable by the model. Hence, when = 0, one recovers the classical Navier-Stokes equations for a flow of viscous, incompressible, Newtonian fluids. Furthermore, the Navier-Stokes- equations can also be interpreted as a regularization of the Navier- Stokes equations, where stands for the regularization parameter. In this paper we first present the Navier-Stokes- 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- 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 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 L2 norm. Ministerio de Educación y Ciencia JC2011-0418

Se presentará una introducción a la formulación variacional de problemas elípiticos lineales y a su resolución numérica mediante métodos de elementos finitos. Se discutirá la implementación del método y se presentarán sus propiedades de convergencia, estabilidad y orden. Se introducirá finalmente la extensión del método a problemas no lineales y de evolución. Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.

We consider a time-dependent diffusion-reaction model for two vector unknowns, satisfying a divergence-free constraint, and the associated scalar Lagrange multiplier. The motivation for studying such a model is provided by a plasma physics problem arising in the modeling of nuclear fusion devices (Braginskii equations), where the two vector unknowns represent ion and electron velocities, the scalar unknown is the electrostatic potential and the divergence-free constraint reflects the physical assumption of quasi-neutrality. We first recast the problem in a form reminiscent of the standard Stokes problem, which allows us to recognize the importance of using a compatible discretization for the vector and scalar unknowns, then propose and analyze a stable finite element formulation. Following this, we address some peculiar geometrical aspects of the model, showing how they can be naturally dealt with within our formulation, and finally discuss a solution procedure for the resulting linear system based on the classical Uzawa algorithm. Some numerical experiments complete the paper. European Commission No 633053

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. Ministerio de Economía y Competitividad Fondo Europeo de Desarrollo Regional

Ministerio de Economía y Competitividad

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. Ministerio de Educación y Ciencia

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.

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. Ministerio de Ciencia e Innovación

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.

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. Ministerio de Educación y Ciencia CGCI MECD-DGU Brazil/Spain