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.     

Unconditional stability and convergence of fully discrete schemes for 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 -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 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 domains. Moreover, the discrete density is truncated in some adequate terms of the velocity-pressure problem.

     

Error estimates of a linear decoupled Euler–FEM scheme for a mass diffusion model

We present error estimates of a linear fully discrete scheme for a three-dimensional 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 C ⁰-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 velocities, 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){\mathcal{O}(h+k)}) for regular enough solutions 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).     

Stability and convergence for a complete model of mass diffusion

We propose a fully discrete scheme for approximating a three-dimensional, strongly nonlinear model of mass diffusion, also called the complete Kazhikhov–Smagulov model. The scheme uses a C⁰ finite-element approximation for all unknowns (density, velocity and pressure), even though the density limit, solution of the continuous problem, belongs to H². A first-order time discretization is used such that, at each time step, one only needs to solve two decoupled linear problems for the discrete density and the velocity–pressure, separately.We extend to the complete model, some stability and convergence results already obtained by the last two authors for a simplified model where λ²-terms are not considered, λ being the mass diffusion coefficient. Now, different arguments must be introduced, based mainly on an induction process with respect to the time step, obtaining at the same time the three main properties of the scheme: an approximate discrete maximum principle for the density, weak estimates for the velocity and strong ones for the density. Furthermore, the convergence towards a weak solution of the density-dependent Navier–Stokes problem is also obtained as λ→0 (jointly with the space and time parameters).Finally, some numerical computations prove the practical usefulness of the scheme.