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.
Abstract. In this paper we propose an unconditional energy-stable time-splitting finite-element scheme for approximating the Ericksen-Leslie equations governing the flow of nematic liquid crystals. These equations are to be solved for a velocity vector field and a scalar pressure as well as a director vector field representing the direction along which the molecules of the liquid crystal are oriented. The algorithm is designed at two levels. First, at the variational level, the velocity, pressure, and director are computed separately, but the director field has to be computed together with an auxiliary variable (associated to the equilibrium equation for the director) in order to deduce a priori energy estimates. Second, at the algebraic level, one can avoid computing such an auxiliary variable if this is approximated by a piecewise constant finite-element space. Therefore, these two steps give rise to a numerical algorithm that computes separately only the primary variables: velocity, pressure, and director vector. Moreover, we will use a pressure stabilization technique that allows a stable equal-order interpolation for the velocity and the pressure. Finally, some numerical simulations are performed in order to show the robustness and efficiency of the proposed numerical scheme and its accuracy.
Abstract. 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 GagliardoNirenberg interpolation inequality in 2D domains. Moreover, the discrete density is truncated in some adequate terms of the velocity-pressure problem.
SUMMARY In this article, we propose different splitting procedures for the transient incompressible magnetohydrodynamics (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 because it introduces convection stabilization, allows to circumvent inf‐sup conditions (clearly simplifying implementation issues), and is able to capture non‐smooth solutions of the magnetic subproblem. However, residual‐based finite element formulations are not suitable for segregation, because 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. Copyright © 2012 John Wiley & Sons, Ltd.
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 NavierStokes 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 H 1 (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.2000 Mathematics Subject Classification. 35Q35, 65M12, 65M60.
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