Discontinuous Galerkin Finite Element Methods for the Landau-de Gennes Minimization Problem of Liquid Crystals

07/16/2019
by   Ruma Rani Maity, et al.
University of Bath
IIT Bombay
0

We consider a system of second order non-linear elliptic partial differential equations that models the equilibrium configurations of a two dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin finite element methods are used to approximate the solutions of this nonlinear problem with non-homogeneous Dirichlet boundary conditions. A discrete inf-sup condition demonstrates the stability of the discontinuous Galerkin discretization of a well-posed linear problem. We then establish the existence and local uniqueness of the discrete solution of the non-linear problem. An a priori error estimate in the energy norm is derived and a best approximation property is demonstrated. Further, we prove the quadratic convergence of Newton's iterates along with complementary numerical experiments.

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1 Introduction

Liquid crystals are a transitional phase of matter between the liquid and crystalline phases. They inherit versatile properties of both liquid phase (e.g. fluidity) and, crystalline phase (optical, electrical, magnetic anisotropy) which make them ubiquitous in practical applications ranging from wristwatch and computer displays, nanoparticle organizations to proteins and cell membranes. Liquid crystals present different phases as temperature varies. We consider the nematic phase formed by rod-like molecules that self-assemble into an ordered structure, such that the molecules tend to align along a preferred orientation. There are three main mathematical models for nematic liquid crystals studied in literature which are the Oseen-Frank model [30, 5, 2], Ericksen model [16] and Landau-De Gennes model [14, 34, 13].

We present some rigorous results for the discontinuous Galerkin (dG) formulation of a reduced two-dimensional Landau-de Gennes model, following the model problem studied in [32, 37]

. In the Landau-de Gennes theory, the state of the nematic liquid crystal is described by a tensor order parameter,

where and is the identity matrix. We refer to as the director or the locally preferred in-plane alignment direction of the nematic molecules; is the scalar order parameter that measures the degree of order about We can write as , where is the director angle in the plane. The general Landau-de Gennes Q-tensor order parameter is a symmetric traceless matrix and we can rigorously justify our -D approach in certain model situations (see [38]).

Since Q is a symmetric traceless tensor, the definition of Q leads to its representation as , where and . The stable nematic equilibria are minimizers of the Landau-de Gennes energy functional given by subject to appropriate boundary conditions; where the bulk energy , and being temperature and material dependent constants; the one-constant elastic energy with being an elastic constant, the surface anchoring energy with the anchoring strength on and prescribed Lipschitz continuous boundary function . The electrostatic energy with depending on vaccum permittivity and material dependent constants and E

being the electric field vector.

In the absence of surface effects () and external fields (), the Landau-de Gennes energy functional in the dimensionless form considered in this article is given by [32]

(1.1)

where and is a small parameter that depends on the elastic constant, bulk energy parameters and the size of the domain. We are concerned with the minimization of the functional for the Lipschitz continuous boundary function consistent with the experimentally imposed tangent boundary conditions. More precisely, the admissible space is The strong formulation of (1.1) seeks such that

(1.2)

The behavior of as has been studied by Brezis et al in [7] with the assumption The minimizers of the energy functional (1.1) for a given smooth boundary data with for a smooth bounded domain has been well-studied [7]. Further, the authors rigorously prove in [7] that as in , where is a harmonic map and is a solution of on on . We are interested in the dG finite element approximation of regular solutions of the boundary value problem (1.2). Let be the Ginzburg-Landau operator defined as . For small enough, the linearized operator is bijective when defined between standard spaces (e.g., ) [35], although the norm of its inverse blows up as .

Our motivation comes from the planar bistable nematic device reported in [32]. The device consists of a periodic array of shallow square or rectangular wells, filled with nematic liquid crystals, subject to tangent boundary conditions on the lateral surfaces. The vertical height of the well is much smaller than the cross-sectional dimensions and hence, it is reasonable to model the profile on the bottom cross-section, taking the domain to be a square as opposed to a three-dimensional square well. This model reduction can be rigorously justified using gamma-convergence techniques [22, 38]

. The tangent boundary conditions require that the nematic director, identified with the leading eigenvector

of the Landau-de Gennes Q-tensor order parameter, lies in the plane of the square and is tangent to the square edges. Indeed, this motivates the choice of the Dirichlet conditions for Q on the square edges as described below, which can also be phrased as surface anchoring energies. We model a single well as a square domain with Dirichlet boundary conditions on the square edges and analyse the convergence of numerical solutions in the discontinuous Galerkin framework to the experimentally observed regular solutions in micron-sized wells in [37].

This model problem has been studied using the conforming finite element method for (1.2) for a fixed in [32]. The variational formulation of a more generic Landau-de Gennes model with homogeneous Dirichlet boundary data has been studied [13] in which an abstract approach of the finite element approximation of nonsingular solution branches has been analyzed in the conforming finite element set up. However the analysis for the non-homogeneous boundary conditions has not been considered in this work. In [2] and [31], the authors have discussed a mixed finite element method for the Frank-Oseen and Ericksen-Leslie models for nematic liquid crystals, respectively.

It is well-known that Allen-Cahn equation [1] is a gradient flow with the Liapunov energy functional (1.1). Numerical approximations of the Allen-Cahn equation have been extensively investigated in the literature. A priori error estimate for the error bounds as a function of have been analyzed and shown to be of polynomial order in by Feng and Prohl in [19], for the conforming finite element approximation of the Allen-Cahn problem. A symmetric interior penalty discontinuous Galerkin method [18] and a posteriori error analysis [20, 6] (which has a low order polynomial dependence on ) have also been studied for Allen-Cahn equation. The problem considered in this article is different from the Allen Cahn equation; we have a system of two coupled nonlinear partial differential equations (PDEs) in a time independent scenario. The Allen Cahn work is relavant to time-dependent front propagation in nematic liquid crystals, in certain reduced symmetric situations [33].

In this article, we analyze discontinuous Galerkin finite element methods (dGFEMs) to approximate regular solutions of (1.2) for a fixed , in the context of the planar bistable nematic device. This involves a semilinear system of PDEs with cubic nonlinearity (see (1.2)) and non-homogeneous boundary conditions. As reported in [32, 37], there are six experimentally observed stable nematic equilibria, labeled as diagonal and rotated states, for this model problem. The nematic director roughly aligns along one of the square diagonals in the diagonal states whereas the director rotates by radians between a pair of parallel edges, in a rotated state. There are two diagonal states, since there are two square diagonals, and four rotated states related to each other by a rotation. The dGFEMs are attractive because they are element-wise conservative, are flexible with respect to local mesh adaptivity, are easier to implement than finite volume schemes, allow for non-uniform degrees of approximations for solutions with variable regularity over the computational domain and can handle non-homogeneous boundary condition in a natural way. These methods also relax the inter-element continuity requirement in conforming FEM. An a priori error analysis of dGFEMs for general elliptic problems has been derived in [36, 25, 24]. For a comprehensive study of several dGFEMs applied to elliptic problems, see [4]. The dGFEMs are also well studied for fourth order elliptic problems [11, 21]. Recently, dGFEMs have been studied for the von Kármán equations [11] that involves a quadratic non-linearity and homogeneous boundary conditions.

To the best of our knowledge, dGFEMs have not been analysed for the nonlinear system derived from the reduced two-dimensional Landau-de Gennes energy for nematic liquid crystals in (1.1) and this is a primary motivation for our study. Our contributions can be summarized as follows:

  1. an elegant representation of the nonlinear operator, convergence analysis with - dependency and a priori error estimate of dGFEMs for the nematic liquid crystal problems with non homogeneous boundary conditions;

  2. a best approximation result for regular solutions of the non-linear problem (1.2);

  3. quadratic convergence of the Newton’s iterates to the approximate solution obtained using dGFEM;

  4. numerical results that correspond to the six nematic equilibria using dGFEM which confirms the theoretical orders of convergence and rate of convergence as a function of and .

Throughout the paper, standard notations on Sobolev spaces and their norms are employed. The standard semi-norm and norm on for positive real numbers, are denoted by and . The standard inner product is denoted by . We use the notation (resp. ) to denote the product space . The standard norms () in the Sobolev spaces () defined by for all for all The norm on space is defined by for all Set and . The inequality abbreviates with the constant independent of mesh-size parameter and . The constants that appear in various Sobolev imbedding results in the sequel are denoted using a generic notation .

The paper is organised as follows. In Section 2, we present the model problem along with the weak formulation and some preliminary results. In Section 3, we state the dG finite element formulation of the problem in Subsection 3.1 and our main results are stated in Subsection 3.2. The existence and uniqueness of the discrete solution of the non-linear problem, error estimates, best approximation result and the convergence of Newton’s method are presented as main theorems. Section 4 contains some auxiliary results needed to prove the main results. A discrete inf-sup condition for a discrete bilinear form has been established. In Section 5, we prove the main theorems. A contraction map has been defined on the discrete space to use a fixed point argument for proving the existence and uniqueness of discrete solution. An alternative proof of the existence and uniqueness of the solution of the discrete problem using Newton-Kantorovich theorem has been given in this section. This is followed by a proof of the quadratic convergence of Newton’s method and numerical results that are consistent with the theoretical results in Section 6.

2 Preliminaries

In this section, we introduce the weak formulation for (1.2) and establish some boundedness results.
In the weak formulation of (1.2), we seek such that

(2.1)

where for all and

(2.2)
(2.3)
Remark 2.1.

When ,

The operator corresponds to the non-linear part of the system (1.2). Such representation of yields nice properties proven in Lemma 2.3 which makes the analysis elegant.

Lemma 2.2.

(Poincaré inequality)[28] Let be a bounded open subset of Then there exists a positive constant such that

The boundedness and coercivity of and the boundedness of given below can be easily verified. For all , ,

(2.4)
(2.5)

where depends on The next lemma establishes two boundedness results for .

Lemma 2.3.

(Boundedness of ) For all , , , ,

(2.6)

and for all , , ,

(2.7)
Proof.

It is enough to prove the results for ; then (2.6) and (2.7) follow from the definition of and a grouping of the terms. For , a use of Hölder’s inequality and the Sobolev imbedding result [17] leads to (2.6) as

For , , a use of the Sobolev imbedding result and the Cauchy-Schwarz inequality leads to

where in the last two inequalities above absorbs and the constant from . ∎

The existence of minimizers of (1.1) follows from the coercivity of and its convexity in in (1.1) and this implies the existence of a solution of the non-linear system in (2.1). The regularity result in the next lemma follows from arguments in [13, 23].

Lemma 2.4 (Regularity result).

Let be a open, bounded, convex and Lipschitz domain of Then any solution of (1.2) belongs to .

In this article, we approximate regular solutions [27] of (1.2) for a given . The regularity of solution implies that the linearized operator is invertible in the Banach space and is equivalent to the following inf-sup condition [15]

(2.8)

where and the inf-sup constant depends on . From now onwards, the subscript in is suppressed in the sequel for notational brevity.

3 Discrete formulation

In this section, we derive the dGFEM formulation for (1.2) and state our main results.

3.1 The dGFEM formulation

Let be a triangulation [12] of into triangles and let the discretization parameter associated with the partition be defined as where . Let ( resp. ) denote the interior (resp. boundary) edges of and . Also, the boundary of an element is denoted by and the unit normal vector outward from is denoted by For any interior edge shared by two triangles and , let the unit normal pointing from to be [see Figure 1]. We assume the triangulation be shape regular [12] in the sense that there exists such that if is the diameter of , then contains a ball of radius in its interior.

Figure 1: The neighboring triangles and and unit normal .

Define the broken Laplacian by For a positive integer , define the broken Sobolev space by equipped with the broken norm Denote to be the product space with the norm Follow the standard convention [36] for the jump and average. Consider the finite dimensional space that consists of piecewise linear polynomials defined by and equip it with the mesh dependent norm defined by where is the penalty parameter. Let be equipped with the product norm defined by

Remark 3.1.

The results can be extended to the case when higher order polynomials are used in the approximation.

The discontinuous Galerkin formulation corresponding to (1.2) seeks such that for all

(3.1)

where for ,

(3.2)

and for , ,

Remark 3.2.

The parameter values corresponds to symmetric interior penalty, incomplete interior penalty and non-symmetric interior penalty dG methods, respectively in the context of linear problems.

3.2 The main results

Our main results are given below. The proofs are presented in Sections 5 and 6.

Theorem 3.3 (Existence, uniqueness and dG norm error estimate).

Let be a regular solution of the non-linear system (2.1). For a given fixed sufficiently large and sufficiently small discretization parameter chosen as for , there exists a unique solution of the discrete non-linear problem (3.1) such that

where is a constant independent of and

Theorem 3.4 (Best approximation result).

Let be a regular solution of the non-linear system (2.1). For a given fixed sufficiently large and sufficiently small discretization parameter chosen as with , there exists a constant independent of and such that the unique discrete solution of (3.1) satisfies the best-approximation property

Remark 3.5.

Theorem 3.4 shows that discrete solution is "the best" approximation of in , up to the constant . Best approximation result is mainly motivated by "the best" approximation of discrete solution of linear PDEs in Céa’s lemma [12] for conforming FEM.

We use Newton’s method [27] for computation of discrete solutions. It is a standard and very effective root-finding method to approximate the roots of non-linear system of PDEs.

Theorem 3.6 (Convergence of Newton’s method).

Let be a regular solution of the non-linear system (2.1) and let solve (3.1). For a given fixed sufficiently large and sufficiently small discretization parameter chosen as with , there exists , independent of , such that for any initial guess with , it follows for all and the iterates of Newton’s method are well-defined and converges quadratically to that is, where is a constant independent of .

4 Auxiliary results

This section presents some auxiliary results needed to establish the main results in Subsection 3.2. The boundedness and ellipticity results for , and the boundedness result for , are proved and the inf-sup conditions for a discrete linearized bilinear form and a perturbed bilinear form are established.

Lemma 4.1.

(Poincaré type inequality)[29, 9] For , there exists a constant independent of and such that for ,

For boundedness and coercivity results of and the boundedness results of and , it is enough to prove the corresponding results for , and , respectively.

Lemma 4.2.

(Boundedness and coercivity of ) [36] For ,

where depends on the penalty parameter and the constant from trace inequality. For a sufficiently large parameter , there exists a positive constant such that

Lemma 4.3.

(Boundedness of and ) For , it holds that

(4.1)

and for all , , ,

(4.2)

where the hidden constant in depends on the constants from , and .

Proof.

We prove the boundedness results for and . Then a use of definitions of and , discrete Cauchy-Schwarz inequality and a grouping of the terms yields the required results. For and , a use of Holder’s inequality and Cauchy-Schwarz inequality along with Lemma 4.1 leads to

(4.3)

The proof of (4.2) follows analogous to that of the proof of (2.7) with a use of the imbedding result and Cauchy-Schwarz inequality. ∎

Remark 4.4.

A use of (4.3) leads to the following boundedness estimate

The next two lemmas describe the estimates for interpolation and enrichment operators that are crucial for the error estimates.

Lemma 4.5.

(Interpolation estimate)[10] For , there exists such that for any ,

for and some positive constant independent of .

Lemma 4.6.

(Enrichment operator). [9, 26] Let be an enrichment operator with being the Lagrange conforming finite element space associated with the triangulation . Then for any , the following results hold:

where and are constants independent of .

For all , define the discrete bilinear form by

Theorem 4.7.

Let be a regular solution of the non-linear system (2.1). For a given fixed a sufficiently large and a sufficiently small discretization parameter chosen such that , there exists a constant such that the following discrete inf-sup condition holds:

Proof.

For Lemma 4.3, (2.5) and (2.6) yield , and . Therefore, for a given with , there exist and that solve the linear systems equationparentequation

(4.4a)
(4.4b)

A use of Lemmas 4.1, 4.3 and elliptic regularity leads to

(4.5)

where the hidden constant in depends on , and . Subtract (4.4a) from (4.4b), choose and use (2.4) and (4.2) to obtain

(4.6)

A use of Lemma 4.6 in (4.6) yields

(4.7)

where the constant hidden in depends on and . Since is regular solution of (2.1), a use of (2.8) yields that there exists with such that

A use of (4.4b), (2.4), an introduction of intermediate terms and triangle inequality leads to

(4.8)

Since on and on , a use of second inequality in Lemma 4.6 and the triangle inequality yields

(4.9)

Since Lemma 4.2 implies that there exists with such that

(4.10)

A use of Lemmas 4.3 and 4.6 yields the estimates for the second and third terms in (4.10) as

(4.11)

A use of Lemmas 4.2, 4.5 and 4.6 leads to

(4.12)

A combination of (4.11) and (4.12) in (4.10) leads to

(4.13)

where includes a constant that depends on and A substitution of (4.13) in (4.9) and a use of Lemma