Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics

Abstract

<p style='text-indent:20px;'>We consider an unconditional fully discrete finite element scheme for a nematic liquid crystal flow with different kinematic transport properties. We prove that the scheme converges towards a unique critical point of the elastic energy subject to the finite element subspace, when the number of time steps go to infinity while the time step and mesh size are fixed. A Lojasiewicz type inequality, which is the key for getting the time asymptotic convergence of the whole sequence furnished by the numerical scheme, is also derived.</p>

Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation

“…Using this stability result and arguing as in [8], we obtain convergence to equilibrium for (61). This means that in Corollary 4.1, we may replace the word Theorem 4.2 by the word Theorem 4.3.…”

“…Several fully discrete Allen-Cahn equations were considered in [6]. Other fully discretized PDEs with a gradient-like flow structure were analyzed in [69,78,83,98,61]. Concerning the time semidiscretization, convergence to equilibrium for the backward Euler scheme applied to the Allen-Cahn equation was considered in [99] (see also [56]).…”

“…This is a nonlinear scheme with first order accuracy and which is unconditionally uniquely solvable and energy stable [51,133]. The mass is conserved, as can be seen by choosing v = 1 in (61). The stability result reads: 50,51]).…”

“…In Figure 3, we have represented the final state corresponding to the initial value (88) for the parameters θ = 0, π/4 and π/2. For the time resolution, we used the unconditionally stable convex splitting scheme (61) with time step τ = 0.04. By arguing as in Theorem 5.6, it is easy to prove that for every initial value, the sequence uniquely generated by the scheme converges to a steady state.…”

“…The idea is to combine the second-order secant scheme with a first order scheme in order to estimate the error due to the time discretization. The first-order scheme here is Eyre's convex splitting (61), which is unconditionnally energy stable. The values for the safety coefficient ρ and the tolerance tol are the same as in [58,Equation (29)], that is ρ = 0.9 and tol = 10 −3 .…”