Francisco Guillén-González

On linear schemes for a Cahn-Hilliard diffuse interface model

Numerical schemes to approximate the Cahn-Hilliard equation have been widely studied in recent times due to its connection with many physically motivated problems. In this work we propose two type of linear schemes based on different ways to approximate the double-well potential term. The first idea developed in the paper allows us to design a linear numerical scheme which is optimal from the numerical dissipation point of view meanwhile the second one allows us to design unconditionally energy-stable linear schemes (for a modified energy). We present first and second order in time linear schemes to approximate the CH problem, detailing their advantages over other linear schemes that have been previously introduced in the literature. Furthermore, we compare all the schemes through several computational experiments.

Second order schemes and time-step adaptivity for Allen-Cahn and Cahn-Hilliard models

Abstract In this paper, we focus on efficient second-order in time approximations of the Allen–Cahn and Cahn–Hilliard equations. First of all, we present the equations, generic second-order schemes (based on a mid-point approximation of the diffusion term) and some schemes already introduced in the literature. Then, we propose new ways of deriving second-order in time approximations of the potential term (starting from the main schemes introduced in Guillen-Gonzalez and Tierra (2013)), yielding to new second-order schemes. For these schemes and other second-order schemes previously introduced in the literature, we study the constraints on the physical and discrete parameters that can appear to assure the energy-stability, unique solvability and, in the case of nonlinear schemes, the convergence of Newton’s method to the nonlinear schemes. Moreover, in order to save computational cost we have developed a new adaptive time-stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step. Finally, we compare the behaviour of the schemes and the effectiveness of the adaptive time-stepping algorithm through several computational experiments.

On the approximate controllability of Stackelberg-Nash strategies for Stokes equations

Abstract. We study a Stackelberg strategy subject to the evolutionary Stokes equations, considering a Nash multi-objective equilibrium (not necessarily cooperative) for the “follower players” (as they are called in the economy field) and an optimal problem for the leader player with approximate controllability objective. We will obtain the following three main results: the existence and uniqueness of the Nash equilibrium and its characterization, the approximate controllability of the Stokes system with respect to the leader control and the associate Nash equilibrium, and the existence and uniqueness of the Stackelberg-Nash problem and its characterization.

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.

STABILITY FOR NEMATIC LIQUID CRYSTALS WITH STRETCHING TERMS

We study a nematic crystal model that appeared in [Liu et al., 2007], modeling stretching effects depending on the different shapes of the microscopic molecules of the material, under periodic boundary conditions. The aim of the present article is two-fold: to extend the results given in [Sun & Liu, 2009], to a model with more complete stretching terms and to obtain some stability and asymptotic stability properties for this model.

Weak Time Regularity and Uniqueness for a

The coupled Navier--Stokes and

Q

Q

-Tensor Model

-tensor system is one of the models used to describe the behavior of nematic liquid crystals. The existence of weak solutions and a uniqueness criterion have been already studied (see [M. Paicu and A. Zarnescu, Arch. Ration. Mech. Anal., 203 (2012), pp. 45--67] for a Cauchy problem in the whole

Mixed formulation, approximation and decoupling algorithm for a penalized nematic liquid crystals model

\mathbb{R}^3

Unconditional stability and convergence of fully discrete schemes for

, and [F. Guillen-Gonzalez and M. A. Rodriguez-Bellido, Nonlinear Anal., 112 (2015), pp. 84--104] for an initial-boundary problem in a bounded domain