Juan Vicente Gutiérrez-Santacreu

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

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 a 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.

Long-Term Stability Estimates and Existence of a Global Attractor in a Finite Element Approximation of the Navier-Stokes Equations with Numerical Subgrid Scale Modeling

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

Unconditional stability and convergence of fully discrete schemes for

L^2

2D

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.

viscous fluids models with mass diffusion

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.

Conditional Stability and Convergence of a Fully Discrete Scheme for Three-Dimensional Navier-Stokes Equations with Mass Diffusion

We construct a fully discrete numerical scheme for three-dimensional incompressible fluids with mass diffusion (in density-velocity-pressure formulation), also called the Kazhikhov-Smagulov model. We will prove conditional stability and convergence, by using at most

A time-splitting finite-element stable approximation for the Ericksen-Leslie equations

C^0

Uniform-in-time error estimates for spectral Galerkin approximations of a mass diffusion model

-finite elements, although the density of the limit problem will have

Finite element discretization of a Stokes-like model arising in plasma physics

H^2

Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations

-regularity. The key idea of our argument is first to obtain pointwise estimates for the discrete density by imposing the constraint