Discontinuous Galerkin finite element methods for the Landau–de Gennes minimization problem of liquid crystals

Abstract

We consider a system of second-order nonlinear elliptic partial differential equations that models the equilibrium configurations of a two-dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin (dG) finite element methods are used to approximate the solutions of this nonlinear problem with nonhomogeneous Dirichlet boundary conditions. A discrete inf–sup condition demonstrates the stability of the dG discretization of a well-posed linear problem. We then establish the existence and local uniqueness of the discrete solution of the nonlinear problem. A priori error estimates in the energy and $\mathbf{L}^2$ norms are derived and a best approximation property is demonstrated. Further, we prove the quadratic convergence of the Newton iterates along with complementary numerical experiments.

Error Analysis of Nitsche’s and Discontinuous Galerkin Methods of a Reduced Landau–de Gennes Problem

“…L 2 ) norm for solutions, Ψ 𝜖 ∈ H 2 (Ω), accompanied by an analysis of the ℎ − 𝜖 dependency, where ℎ is the mesh size or the discretization parameter, along with some numerical experiments are discussed. However, there are no a posteriori error estimates in [24]. This paper builds on the results in [24] with several non-trivial generalisations and extensions.…”

“…However, there are no a posteriori error estimates in [24]. This paper builds on the results in [24] with several non-trivial generalisations and extensions. Define the admissible space X = w ∈ H 1 (Ω) : w = g on 𝜕Ω and we restrict attention to solutions Ψ 𝜖 ∈ X ∩ H 1+𝛼 (Ω) for 0 < 𝛼 ≤ 1, where 𝛼 is the index of elliptic regularity in this manuscript.…”

“…In [22], the authors study the numerical convergence of the diagonal and rotated solutions, in a conforming finite-element set-up, as a function of 𝜖. In [24], the authors carry out a rigorous a priori error analysis for the dGFEM approximation of H 2 -regular solutions of (1.2) in convex polygonal domains. Optimal linear (resp.…”

Variational and numerical analysis of a $\mathbf{Q}$-tensor model for smectic-A liquid crystals

“…We thus adopt the idea of C 0 -IP methods to solve the nonlinear fourth-order problem (P1) and derive a priori error estimates regarding u. We adapt the techniques of [20] to prove convergence rates with the use of familiar continuous Lagrange elements for the problem (P1). The weak form of (3.1) is defined as:…”

“…More specifically, Davis and Gartland [14] gave an abstract nonlinear finite element convergence analysis where an optimal H 1 error bound is proved on convex domains with piecewise linear polynomial approximations, but do not derive an error bound in the L 2 norm. Recently, Maity, Majumdar and Nataraj [20] analysed the discontinuous Galerkin finite element method for a two-dimensional reduced LdG free energy, where optimal a priori error estimates in the L 2 -norm with exact solutions in H 2 are achieved for a piecewise linear discretisation. Both works only focus on piecewise linear approximations.…”