Finite-element methods for fourth-order problems and smectic A liquid crystals

Hamdan, Abdalaziz (2022) Finite-element methods for fourth-order problems and smectic A liquid crystals. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

In recent years, energy-minimization finite-element methods have been proposed for the computational modelling of equilibrium states of several types of liquid crystals (LCs) [4, 34, 110]. This thesis is particularly interested in the models of smectic A liquid crystals, based on the free-energy functionals proposed by Pevnyi, Selinger, and Sluckin [112], and by Xia et al. [138]. The Euler-Lagrange equations for these models include fourth-order terms acting on the smectic order parameter (or density variation of the LC) and second-order terms acting on the Q-tensor or director field. Thus, we first focus extensively on finite-element methods for fourth-order problems. These methods include (i) C¹-continuous elements with a nonsymmetric Nitsche-type penalty method to weakly impose the essential boundary conditions, (ii) a nonsymmetric version of the C⁰ interior penalty method, where the nonsymmetric forms are used to guarantee optimal convergence rates in terms of h ≤ 1 and q ≈ 40, where h and q are the refinement level and the smectic wavenumber that prescribes a preferred wavelength for the solution of 2π=q respectively, and (iii) mixed finite-element methods based on introducing the gradient of the solution as an explicit variable and constraining its value using a Lagrange multiplier, that are symmetric and allow us to strongly impose the essential boundary conditions. Preliminary experiments show that the mixed formulations may be advantageous over the other methods, in the sense that we can construct efficient preconditioners for these discretizations. Therefore, we consider a four-field formulation for models of smectic A liquid crystals, approximating the smectic order parameter, its gradient, the Lagrange multiplier, and the Q-tensor. Then, we focus on the construction of solvers for the nonlinear systems that result from the discretization of these models. We consider a Newton-Krylov- Multigrid approach, using Newton's method to linearize the systems, and developing monolithic geometric multigrid preconditioners for the resulting saddle-point systems with vertex-based patch relaxation schemes.

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