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1 SIAM J. NUMER. ANAL. Vol. 55, No. 3, pp c 7 Society for Industrial and Applied Matematics Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// A FINITE ELEMENT METHOD FOR NEMATIC LIQUID CRYSTALS WITH VARIABLE DEGREE OF ORIENTATION RICARDO H. NOCHETTO, SHAWN W. WALKER, AND WUJUN ZHANG Abstract. We consider te simplest one-constant model, put forward by J. Ericksen, for nematic liquid crystals wit variable degree of orientation. Te equilibrium state is described by a director field nn and its degree of orientation s, were te pair s, n) minimizes a sum of Frank-like energies and a double well potential. In particular, te Euler Lagrange equations for te minimizer contain a degenerate elliptic equation for n, wic allows for line and plane defects to ave finite energy. We present a structure preserving discretization of te liquid crystal energy wit piecewise linear finite elements tat can andle te degenerate elliptic part witout regularization, and we sow tat it is consistent and stable. We prove Γ-convergence of discrete global minimizers to continuous ones as te mes size goes to zero. We develop a quasi-gradient flow sceme for computing discrete equilibrium solutions and prove tat it as a strictly monotone energy decreasing property. We present simulations in two and tree dimensions to illustrate te metod s ability to andle nontrivial defects. Te following online video illustrates te role of numerical analysis for te simulation of liquid crystal penomena: [Walker, Sawn. Matematical Modeling and Simulation of Nematic Liquid Crystals A Montage). YouTube video, 4:3. Posted Marc 6, 6. ttp:// com/watc?v=pwww7 6cQ-U]. Key words. liquid crystals, finite element metod, Γ-convergence, gradient flow, line defect, plane defect AMS subject classifications. 65N3, 49M5, 35J7 DOI..37/5M3844X. Introduction. Complex fluids are ubiquitous in nature and industrial processes and are critical for modern engineering systems [3, 4, 5]. A critical difficulty in modeling and simulating complex fluids is teir inerent microstructure. Manipulating te microstructure via external forces can enable control of te mecanical, cemical, optical, or termal properties of te material. Liquid crystals [47, 5,, 4, 3, 3, 7, 33, 34, 5, 46] are a relatively simple example of a material wit microstructure tat may be immersed in a fluid wit a free interface [53, 5]. Several numerical metods for liquid crystals ave been proposed in [, 9, 3, 35, ] for armonic mappings and liquid crystals wit fixed degree of orientation; i.e., a unit vector field nx) called te director field) is used to represent te orientation of liquid crystal molecules. See [8, 36, 49] for metods tat couple liquid crystals to Stokes flow. We also refer te reader to te survey paper [6] for more numerical metods. In tis paper, we consider te one-constant model for liquid crystals wit variable Received by te editors September 8, 5; accepted for publication in revised form) February 7, 7; publised electronically June 7, 7. ttp:// Funding: Te first and tird autors were partially supported by NSF grant DMS-488. Te second autor was partially supported by NSF grants DMS and DMS-555. Te first autor was also supported by te Institut Henri Poincaré Paris). Te tird autor was also supported by te University of Maryland troug te Brin postdoctoral fellowsip. Department of Matematics and Institute for Pysical Science and Tecnology, University of Maryland, College Park, MD 74 rn@mat.umd.edu). Department of Matematics, Louisiana State University, Baton Rouge, LA walker@mat.lsu.edu). Department of Matematics, Rutgers University, Piscataway, NJ 8854 wujun@mat. rutgers.edu). 357 Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

2 358 R. H. NOCHETTO, S. W. WALKER, AND W. ZHANG Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// degree of orientation [6, 5, 47]. Te state of te liquid crystal is described by a director field nx) and a scalar function sx), / < s <, tat represents te degree of alignment tat molecules ave wit respect to n. Te equilibrium state is given by s, n), wic minimizes te so-called one-constant Ericksen s energy.). Despite te simple form of te one-constant Ericksen s model, its minimizer may ave nontrivial defects. If s is a nonvanising constant, ten te energy reduces to te Oseen Frank energy, wose minimizers are armonic maps tat may exibit point defects depending on boundary conditions) [4, 6,, 34, 33, 4]. If s is part of te minimization of.), ten s may vanis to allow for line and plane) defects in dimension d = 3 [5, 46], and te resulting Euler Lagrange equation for n is degenerate. However, in [34] it was sown tat bot s and u = sn ave strong limits, wic enabled te study of regularity properties of minimizers and te size of defects. Tis inspired te study of dynamics [] and corresponding numerics [8], wic are most relevant to our paper. However, in bot studies te model was regularized to avoid te degeneracy introduced by te s parameter. We design a finite element metod FEM) witout any regularization. We prove stability and convergence properties and explore equilibrium configurations of liquid crystals via quasi-gradient flows. Our metod builds on [, 9, ] and consists of a structure preserving discretization of.). Given a weakly acute mes T wit mes size see section.), we use te subscript to denote continuous piecewise linear functions defined over T, e.g., s, n ) is a discrete approximation of s, n). Our discretization of te energy is defined in.8) and requires tat T be weakly acute. Tis discretization preserves te underlying structure and converges to te continuous energy in te sense of Γ-convergence [7] as goes to zero. Next, we develop a quasi-gradient flow sceme for computing discrete equilibrium solutions. We prove tat tis sceme as a strictly monotone energy decreasing property. Finally, we carry out numerical experiments and sow tat our FEM and gradient flow allow for computing minimizers tat exibit line and plane defects. Te paper is organized as follows. In section, we describe Ericksen s model for liquid crystals wit variable degree of orientation and give details of our discretization. Section 3 sows te Γ-convergence of our numerical metod. A quasi-gradient flow sceme is given in section 4, were we also prove a strictly monotone energy decreasing property. Section 5 presents simulations in two and tree dimensions tat exibit nontrivial defects in order to illustrate te metod s capabilities.. Discretization of Ericksen s model. We review te model [6] and relevant analysis results from te literature. We ten develop our discretization strategy and sow tat it is stable. Te space dimension d can be arbitrary... Ericksen s one-constant model. Let te director field n : R d S d be a vector-valued function wit unit lengt, and let te degree of orientation s : R d [, ] be a real-valued function. Te case s = represents te state of perfect alignment in wic all molecules are parallel to n. Likewise, s = / represents te state of microscopic order in wic all molecules are ortogonal to te orientation n. Wen s =, te molecules do not lie along any preferred direction, wic represents te state of an isotropic distribution of molecules. Te equilibrium state of te liquid crystals is described by te pair s, n) minimizing a bulk-energy functional, wic in te simplest one-constant model reduces to Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

3 Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// E[s, n] := FEM FOR LIQUID CRYSTALS 359 κ s + s n ) dx } {{ } =:E [s,n] + ψs)dx, }{{} =:E [s] wit κ > and double well potential ψ, wic is a C function defined on / < s < tat satisfies te following properties:. lim s ψs) = lim s / ψs) =,. ψ) > ψs ) = min s [ /,] ψs) = for some s, ), 3. ψ ) = ; see [6]. Note tat wen te degree of orientation s equals a nonzero constant, te energy.) effectively reduces to te Oseen Frank energy n. Te degree of orientation s relaxes te energy of defects i.e., discontinuities in n), wic may still ave finite energy E[s, n] if te singular set.) S := {x, sx) = } is nonempty; in tis case, n / H ). By introducing an auxiliary variable u = sn [34, 3], we rewrite te energy as.3) E [s, n] = Ẽ[s, u] := κ ) s + u ) dx, wic follows from te ortogonal splitting u = n s + s n due to te constraint n =. Accordingly, we define te admissible class.4) A :={s, u) : /, ) R d : s, u) [H )] d+, u = sn, n S d }. We say tat te pair s, u) satisfies te structural condition for Ericksen s energy if.5) u = sn, / < s < a.e. in, and n S d a.e. in. Moreover, we may enforce boundary conditions on s, u) possibly on different parts of te boundary. Let Γ s, Γ u ) be open subsets of, were we set Diriclet boundary conditions for s, u). Ten we ave te restricted admissible class.6) Ag, r) := {s, u) A : s Γs = g, u Γu = r} for some given functions g, r) [W R d )] d+ tat satisfy te structural condition.5) on. We assume te existence of δ > sufficiently small suc tat.7) + δ gx), rx) ξ δ for all x R d, ξ R d, ξ =, and te potential ψ satisfies.8) ψs) ψ δ ) for s δ, ψs) ψ ) + δ Tis is consistent wit property of ψ. If we furter assume tat.9) g δ on, for s + δ. ten te function n is H in a neigborood of and satisfies n = g r on. Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

4 36 R. H. NOCHETTO, S. W. WALKER, AND W. ZHANG Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// Te existence of a minimizer s, u) Ag, r) is sown in [34, 3], but tis is also a consequence of our Γ-convergence teory. It is wort mentioning tat te constant κ in E[s, n] of.) plays a significant role in te occurrence of defects. Rougly speaking, if κ is large, ten κ s dx dominates te energy and s is close to a constant. In tis case, defects wit finite energy are less likely to occur. But if κ is small, ten s n dx dominates te energy, and s may become zero. In tis case, defects are more likely to occur. Tis euristic argument is later confirmed in te numerical experiments.) Since te investigation of defects is of primary interest in tis paper, we consider te most significant case to be < κ <. We now describe our finite element discretization E [s, n ] of te energy.) and its minimizer s, n )... Discretization of te energy. Let T = {T } be a conforming simplicial triangulation of te domain. We denote by N te set of nodes vertices) of T and by N te cardinality of N wit some abuse of notation). We demand tat T be weakly acute, namely,.) k ij := φ i φ j dx for all i j, were φ i is te standard at function associated wit node x i N. We indicate wit ω i = supp φ i te patc of a node x i i.e., te star of elements in T tat contain te vertex x i ). Condition.) imposes a severe geometric restriction on T [, 45]. We recall te following caracterization of.) for d =. Proposition. weak acuteness in two dimensions). For any pair of triangles T, T in T tat sare a common edge e, let α i be te angle in T i opposite to e for i =, ). If α + α π for every edge e, ten.) olds. Generalizations of Proposition. to tree dimensions, involving interior diedral angles of tetraedra, can be found in [3, 9]. We construct continuous piecewise affine spaces associated wit te mes, i.e.,.) S := {s H ) : s T is affine for all T T }, U := {u H ) d : u T is affine in eac component for all T T }, N := {n U : n x i ) = for all nodes x i N }. Let I denote te piecewise linear Lagrange interpolation operator on mes T wit values in eiter S or U. We say tat a pair s, u ) S U satisfies te discrete structural condition for Ericksen s energy if tere exists n N suc tat.) u = I [s n ], < s < in. We ten let g := I g and r := I r be te discrete Diriclet data, and we introduce te discrete spaces tat include Diriclet) boundary conditions S Γ s, g ) := {s S : s Γs = g }, U Γ u, r ) := {u U : u Γu = r }, along wit te discrete admissible class { }.3) A g, r ) := s, u ) S Γ s, g ) U Γ u, r ) :.) olds. In view of.9), we can also impose te Diriclet condition n = I [g r ] on. Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

5 FEM FOR LIQUID CRYSTALS 36 Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// In order to motivate our discrete version of E [s, n], note tat for all x i N, k ij = j= j= φ i φ j dx = because N j= φ j = in te domain ; te set of at functions {φ j } N j= is a partition of unity. Terefore, for piecewise linear s = N i= s x i )φ i, we ave s dx = k ii s x i ) i= i,j=,i j k ij s x i )s x j ), wence, exploiting k ii = j i k ij and te symmetry k ij = k ji, we get.4) were we define s dx = = k ij s x i ) s x i ) s x j ) ) i,j= i,j= k ij s x i ) s x j ) ) = ), k ij δij s i,j=.5) δ ij s := s x i ) s x j ), δ ij n := n x i ) n x j ). Wit tis in mind, we define te discrete energies to be.6) E [s, n ] := κ k ij δ ij s ) + i,j= and.7) E [s ] := i,j= ψs x))dx, s x i ) + s x j ) ) k ij δ ij n for s, u ) A g, u ). Te second summation in.6) does not come from applying te standard discretization of s n dx by piecewise linear elements. It turns out tat tis special form of te discrete energy preserves te key energy inequality Lemma.), wic allows us to establis our Γ-convergence analysis for te degenerate coefficient s witout regularization. Eventually, we seek an approximation s, u ) A g, r ) of te pair s, u) suc tat te discrete pair s, n ) minimizes te discrete version of te bulk energy.) given by.8) E [s, n ] := E [s, n ] + E [s ]. Te following result sows tat definition.6) preserves te key structure.3) of [3, 34] at te discrete level, wic turns out to be crucial for our analysis as well. We first introduce s := I s and two discrete versions of te vector field u,.9) u := I [s n ] U, ũ := I [ s n ] U. Note tat bot pairs s, u ), s, ũ ) S U satisfy.). Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

6 36 R. H. NOCHETTO, S. W. WALKER, AND W. ZHANG Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// Lemma. energy inequality). Let te mes T satisfy.). If s, u ) A g, r ), ten for any κ >, te discrete energy.6) satisfies.) E [s, n ] κ ) s dx + u dx =: Ẽ [s, u ] as well as.) Proof. Since E [s, n ] κ ) s dx + ũ dx =: Ẽ [ s, ũ ]. s x i )n x i ) s x j )n x j ) = s x i ) + s x j ) n x i ) n x j ) ) + s x i ) s x j ) ) n x i ) + n x j ), using te ortogonality relation n x i ) n x j ) ) n x i ) + n x j ) ) = n x i ) n x j ) = and.4) yields u dx = = i,j= k ij s x i )n x i ) s x j )n x j ) i,j= k ij s x i ) + s x j ) ) δ ijn + k ij δ ij s ) n x i ) + n x j ). i,j= Exploiting te relations n x i ) n x j ) + n x i ) + n x j ) = 4 and s x i ) + s x j ) ) = s x i ) + s x j ) ) s x i ) s x j ) ), we obtain.) u dx = wence we infer tat.3) E [s, n ] = + i,j= s x i ) + s x j ) k ij δ ij n k ij δ ij s ) i,j= k ij δ ij s ) n x i ) n x j ), i,j= κ ) s + u ) dx + k ij δ ij s ) δ ij n. i,j= Te inequality.) follows directly from k ij for i j. To prove.), we note tat.) still olds if we replace s, u ) wit s, ũ ):.4) ũ dx = + i,j= s x i ) + s x j ) k ij δ ij n k ij δ ij s ) i,j= k ij δ ij s ) n x i ) n x j ). i,j= Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

7 FEM FOR LIQUID CRYSTALS 363 Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// We finally find tat Ẽ [ s, ũ ] = + κ ũ + κ ) s ) dx = k ij δ ij s ) i,j= i,j= i,j= k ij δ ij s ) n x i ) n x j ) s x i ) + s x j ) k ij δ ij n E [s, n ], were we ave dropped te last term and used te triangle inequality δ ij s = s x i ) s x j ) s x i ) s x j ) = δij s along wit k ij to obtain.5) s L ) = Tis concludes te proof. k ij δ ij s ) i,j= k ij δ ij s ) = s L ). i,j= Remark.3 relation between.) and.)). Bot.) and.) account for te variational crime tat is committed wen enforcing u = s n and ũ = s n only at te vertices, and tat mimics.3). Since we ave a precise control of te consistency error for.), tis inequality will be used later for te consistency or lim-sup) step of Γ-convergence of our discrete energy.6) to te original continuous energy in.). On te oter and,.) as a suitable structure to prove te weak lower semicontinuity or lim-inf) step of Γ-convergence. Tis property is not obvious wen κ <, te most significant case for te formation of defects. 3. Γ-convergence of te discrete energy. In tis section, we sow tat our discrete energy.6) converges to te continuous energy.) in te sense of Γ- convergence. To tis end, we first let te continuous and discrete spaces be X := L ) [L )] d, X := S U. We next define E[s, n] as in.) for s, u) Ag, r) and E[s, u] = for s, n) X \ Ag, r). Likewise, we define E [s, n ] as in.8) for s, u ) A g, r ) and E [s, n] = for all s, u) X \ A g, r ). We split te proof of Γ-convergence into four subsections. In subsection 3., we use te energy Ẽ [s, u ] to sow te consistency property recall Remark.3), wereas we employ te energy Ẽ [ s, ũ ] in subsection 3. to derive te weak lower semicontinuity property. Furtermore, our functionals exibit te usual equi-coercivity property for bot pairs s, u) and s, ũ) but not for te director field n, wic is only well defined wenever te order parameter s. We discuss tese issues in subsection 3.3 and caracterize te limits s, u), s, ũ), and s, n). We eventually prove Γ-convergence in subsection 3.4 by combining tese results. 3.. Consistency or lim-sup property. We prove te following: if s, u) Ag, r), ten tere exist a sequence s, u ) A g, r ) converging to s, u) in H ) and a discrete director field n N converging to n in L \ S) suc tat 3.) E [s, n] lim sup E [s, n ] lim sup Ẽ [s, u ]. We observe tat if s, u) / Ag, r), ten E [s, n] = and 3.) is valid for any sequences s, u ) and s, n ) in ligt of.). We first sow tat we can always assume +δ o s δ for s, u) Ag, r). Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

8 364 R. H. NOCHETTO, S. W. WALKER, AND W. ZHANG Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// Lemma 3. truncation). Given s, u) Ag, r), let ŝ, û) be te truncations { ŝx) = min δ, max )} + δ, sx), ûx) = ŝx) nx) a.e. x. Ten ŝ, û) Ag, r) and E [ŝ, n] E [s, n], E [ŝ] E [s]. Te same assertion is true for any s, u ) A g, r ), except tat te truncations are defined nodewise, i.e., I ŝ, I û ) A g, r ). Proof. Te fact tat ŝ, û) satisfy te Diriclet boundary conditions is a consequence of.7). Moreover, ŝ, û) [H )] d+ and te structural property.5) olds by construction, wence ŝ, û) Ag, r). We next observe tat ŝ = χ s, := {x : } + δ sx) δ ; see [7, Cap. 5, Exercise 7]. Consequently, we obtain E [ŝ, n] = κ ŝ + ŝ n κ s + s n = E [s, n], as well as E [ŝ] = ψŝ) because of.8). Tis concludes te proof. ψs) = E [s], To construct a recovery sequence s, u ) A g, r ) we need point values of s, u) and tus a regularization procedure of functions in te admissible class Ag, r). We must enforce bot te structural property.5) and te Diriclet boundary conditions s = g and u = r; neiter is guaranteed by convolution. We are able to do tis provided Γ s = Γ u = and te Diriclet datum g satisfies.9). Proposition 3. regularization of functions in Ag, r)). Let Γ s = Γ u =, s, u) Ag, r), and let g satisfy.9). Given ɛ > tere exists a pair s ɛ, u ɛ ) Ag, r) [W )] d+ suc tat 3.) s, u) s ɛ, u ɛ ) H ) ɛ, 3.3) + δ s ɛ x), u ɛ x) ξ δ for all x, ξ R d, ξ =. Proof. We construct a two-scale approximation wit scales δ < σ, wic satisfies te boundary conditions exactly. We split te argument into several steps. Step : Regularization wit Diriclet condition. Extend s g H ) by zero to R d \. Let η δ be a smoot and nonnegative mollifier wit support contained in te ball B δ ) centered at wit radius δ. Define d δ : R d R by d δ x) := χ x) min{δ distx, ), }, wic is Lipscitz in R d, and observe tat d δ is supported in te boundary layer ω δ := {x : distx, ) δ}, Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

9 FEM FOR LIQUID CRYSTALS 365 Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// and tat d δ = δ χ ωδ. We consider te Lipscitz approximations of s, u) given by s δ := d δ s η δ ) + d δ ) g, uδ := d δ u η δ ) + d δ ) r. Since d δ vanises on, we readily see tat s δ, u δ ) = g, r) on. Moreover, te following properties are valid: 3.4) s δ s, u δ u, u δ u a.e. and in H ). Te last property is a consequence of te middle one via triangle inequality, and te first two are similar. It tus suffices to sow te first property for s. We simply write s δ s) = d δ s g) η δ + d δ g ηδ g ) + d δ s η δ s) + d δ ) s g). Since s g H ω δ ) and s g = on, we apply Poincaré s inequality to deduce wence s g L ω δ ) Cδ s g) L ω δ ), d δ s g) η δ L ) Cδ s g L ω δ ) C s g) L ω δ ) as δ. Likewise, a similar argument gives, for te fourt term, d δ ) s g) L ) C s g) L ω δ ) as δ. On te oter and, te estimate g η δ g L ) δ g L R d ) yields g η δ g L ω δ ) ω δ / δ g L R d ) Cδ 3 g L R d ), wic implies, for te second term above, dδ g ηδ g ) L ) Cδ g L R d ). Finally, for te tird term we recall tat s H R d ) equals g outside, and we exploit te convergence s η δ s in L ) to obtain d δ s η δ s) L ) as δ. Step : Structural condition. Unfortunately, te pair s δ, u δ ) does not satisfy te structural condition.5). We now construct a closely related pair tat satisfies.5). Recall tat g, r) [W R d )] d+ satisfy te bounds.7) in R d, and ence so do te extensions of s, u) because s = g, u = r outside. Tus, we can sow tat + δ s δ x), u δ x) ξ δ for all x, ξ R d, ξ = ; we only argue wit s δ because dealing wit u δ ξ is similar. We ave a := + δ s η δ =: b because η δ and te convolution preserves constants, wence s δ d δ b + d δ )b = b, s δ d δ a + d δ )a = a in. We next introduce te second parameter σ > δ and te Lipscitz approximation of te sign function ρ σ t) = min {, max{, t/σ} }, Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

10 366 R. H. NOCHETTO, S. W. WALKER, AND W. ZHANG Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// along wit te two-scale approximation of s, u), s σ,δ := ρ σ s δ ) u δ, u σ,δ := ρ σ s δ ) u δ. We note tat s σ,δ = u σ,δ by construction, wence.5) olds, and s σ,δ, u σ,δ ) = g, r) on because ρ σ s δ ) = on, for σ δ, according to.9); ence s σ,δ, u σ,δ ) Ag, r) [W )] d+. It remains to sow ow to coose δ and σ, wic we do next. Step 3: Convergence in H as δ. In view of 3.4) we readily deduce tat s σ,δ s σ := ρ σ s) s, u σ,δ u σ := ρ σ s) u a.e. and in L ). We now prove convergence also in H ). Since ρ σ s δ ) = σ χ { sδ σ} s δ, we get ρ σ s δ ) ρ σ s) = σ χ { sδ σ} χ { s σ} ) s + σ χ { sδ σ} sδ s ). Applying te Lebesgue dominated convergence teorem for te first term and 3.4) for te second term yields, as δ, 3.5) ρ σ s δ ) ρ σ s), ρ σ s δ ) ρ σ s) in L ). Te second convergence result is due to te fact tat f = sgn f) f for any f W ), were sgn f) is te sign function tat vanises at [7, Cap. 5, Exercise 7]. We next write s σ,δ s σ ) = ρ σ s δ ) ρ σ s) ) u δ u ) + ρ σ s) u δ u ) + ρ σ s δ ) u δ u ) + ρ σ s δ ) ρ σ s) ) u + ρ σ s δ ) ρ σ s) ) u and infer tat s σ,δ s σ ) as δ in L ) upon using again te Lebesgue dominated convergence teorem for te second and fift terms, togeter wit 3.4), 3.5), and u, u δ, ρ σ s δ ) for te oter terms. Step 4: Convergence in H as σ. It remains to prove s σ = ρ σ s) s s, u σ = ρ σ s) u u in H ). To tis end, we use again tat s = sgn s) s and write s σ s) = ρ σ s) s + ρ σ s) sgn s) ) s. Since ρ σ s) = σ χ { s <σ} s, we readily obtain as σ, ρ σ s) s L ) s L { s <σ}). On te oter and, ρ σ t) sgn t) for all t R, wence ρ σ s) sgn s) ) s L ) χ {s=} s L ) = as σ because χ {s=} s = a.e. in [7, Cap. 5, Exercise 7]. Recalling tat u = s a.e. in, and tus χ {u=} = χ {s=}, a similar argument sows tat u σ u. Step 5: Coice of σ and δ. Given ɛ >, we first use Step 4 to coose σ suc tat s, u) s σ, u σ ) H ) ɛ. We finally resort to Step 3 to select δ < σ, depending on σ, suc tat s σ, u σ ) s δ,σ, u δ,σ ) H ) ɛ. Terefore, we obtain te desired regularized pair, i.e., s ɛ := s δ,σ, u ɛ := u δ,σ satisfies s ɛ, u ɛ ) Ag, r) [W )] d+ along wit 3.) and 3.3). Te proof is complete. Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

11 FEM FOR LIQUID CRYSTALS 367 Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// We now fix ɛ > and let s ɛ,, u ɛ, ) X be te Lagrange interpolants of s ɛ, u ɛ ) Ag, r) given in Proposition 3., wic are well defined because s ɛ, u ɛ ) [W )] d+ and satisfy s ɛ,, u ɛ, ) = g, r ) on. For any node x i, we set { u ɛ, x i )/s ɛ, x i ) if s ɛ, x i ), n ɛ, x i ) = any unit vector oterwise and observe tat.) olds, wence s ɛ,, u ɛ, ) A g, r ). In view of te energy identity.3) and te property s ɛ,, u ɛ, ) s ɛ, u ɛ ) H ) as, to sow 3.) it suffices to prove tat te consistency term satisfies 3.6) C [s ɛ,, n ɛ, ] := ) k ij δij s δij ɛ, n ɛ, as. i,j= Heuristically, if n ɛ = u ɛ /s ɛ is in W ), ten te sum 3.6) would be of order s ɛ, dx, wic obviously converges to zero. However, suc an argument fails if te director field n ɛ lacks ig regularity, wic is te case wit defects. Since n ɛ is not regular in general wen s ɛ vanises, te proof of consistency requires a separate treatment of te region were n ɛ is regular and te region were n ɛ is singular. Te euristic argument carries over in te regular region, wile in te singular region we appeal to basic measure teory. Wit tis motivation in mind, we now prove te following lemma. Lemma 3.3 lim-sup inequality). Let s ɛ, u ɛ ) Ag, r) [W )] d+ be te functions constructed in Proposition 3., for any ɛ >, and let s ɛ,, u ɛ, ) A g, r ) be teir Lagrange interpolants. Ten E [s ɛ, n ɛ ] = lim E [s ɛ,, n ɛ, ] = lim Ẽ [s ɛ,, u ɛ, ] = Ẽ[s ɛ, u ɛ ]. Proof. Since ɛ is fixed, we simplify te notation and write s, n ) instead of s ɛ,, n ɛ, ). In order to prove tat C [s, n ] in 3.6), we coose an arbitrary δ > and divide te domain into two disjoint regions, S δ := {x : s ɛ x) < δ}, K δ := \ S δ, and split C [s, n ] into two parts, I K δ ) := ) δij s δ ij n, I S δ ) := x i,x j K δ k ij x i or x j S δ ) k ij δij s δ ij n. Step : Estimate on K δ. Since bot s ɛ and u ɛ are Lipscitz in, te set K δ is a compact set and te field n ɛ = s ɛ u ɛ is also Lipscitz in K δ wit a constant tat depends on ɛ and δ. Terefore, δ ij n = n x i ) n x j ) C ɛ,δ, because x i and x j are connected by a single edge of te mes, wence I K δ ) C ɛ,δ N i,j= k ij δ ij s ) as, because N i,j= k ijδ ij s ) = s L ) C s ɛ L ) <. Step : Estimate on S δ. If eiter x i or x j is in S δ, witout loss of generality, we assume tat x i S δ. Since s ɛ is Lipscitz, and s = I s ɛ is te Lagrange interpolant Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

12 368 R. H. NOCHETTO, S. W. WALKER, AND W. ZHANG Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// of s ɛ, tere is a mes size suc tat for any x in te star ω i of x i, s x) s x i ) C ɛ δ, wic implies tat ω i S δ. Since δ ij n, we get I S δ ) 4 k ij δ ij s ) 8 x i or x j S δ s dx 8 ω i s dx, S δ were te union ω i is taken over all nodes x i in S δ. If d < p <, we infer tat s dx C I s ɛ p dx S δ S δ ) p C ) s ɛ p p dx S δ as, in view of te stability of te Lagrange interpolation operator I in W p for p > d. Step 3: Te limit δ. Combining Steps and gives, for all δ >, lim i,j= ) ) ) k ij δij s δij n C s ɛ S p p dx = C s ɛ p p χ { sɛ δ}dx, δ were χ A is te caracteristic function of te set A. By virtue of te Lebesgue dominated convergence teorem, we obtain s ɛ p χ { sɛ δ}dx = s ɛ p χ {sɛ=}dx =, lim δ because s ɛ x) = for a.e. x {s ɛ = } [7, Cap. 5, Exercise 7]. Tis proves te lemma. 3.. Weak lower semicontinuity or lim-inf property. Tis property usually follows from convexity. Wile it is obvious tat te discrete energy Ẽ [s, u ] in.) is convex wit respect to u and s if κ, te convexity is not clear if < κ <. It is wort mentioning tat if κ <, te convexity of te continuous energy.3) is based on te fact tat u = s a.e. in, and ence te convex part u dx controls te concave part κ ) s dx [34]. However, for te discrete energy.), te equality u = s olds only at te vertices. Terefore, it is not obvious ow to establis te weak lower semicontinuity of Ẽ [s, u ]. Tis is wy we exploit te nodal relations s = s = u = ũ to derive an alternative formula for Ẽ [ s, ũ ]. Our next lemma inges on.) and makes te convexity of Ẽ [I s, ũ ] wit respect to ũ completely explicit. Lemma 3.4 weak lower semicontinuity). L w, w ) := κ ) I w + w, Te energy L w, w )dx, wit is well defined for any w U and is weakly lower semicontinuous in H ), i.e., for any weakly convergent sequence w w in te H ) norm, we ave lim inf L w, w )dx κ ) w + w dx. Proof. If κ, ten te assertion follows from standard arguments. Here, we only dwell upon < κ < and dimension d =, because te case d = 3 is similar. After extracting a subsequence not relabeled), we can assume tat w converges to w strongly in L ) and pointwise a.e. in. Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

13 FEM FOR LIQUID CRYSTALS 369 Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// Step : Equivalent form of L. We let T be any triangle in te mes T, label its tree vertices as x, x, x, and define e := x x and e := x x. After denoting w i = w x i ) for i =,,, a simple calculation yields w = w w ) e + w w ) e, I w = w w )e + w w )e, were {e i } i= is te dual basis of {e i} i=, tat is, e i e j = I ij, and I = I ij ) i,j= is te identity matrix. Assuming w i + w, we realize tat w i w = wi + w w i + w wi w ). We ten obtain I w = G w ) : w, were G w ) is te 3-tensor, G w ) := w + w w + w e e + w + w w + w e e, on T, and te contraction between a 3-tensor and a -tensor in dyadic form is given by Terefore, we ave g g g 3 ) : m m ) := g m )g m )g 3. L w, w ) = w + κ ) G w ) : w, wic expresses L w, w ) directly in terms of w and te nodal values of w. Step : Convergence of G w ). Given ɛ >, Egoroff s teorem [5] asserts tat w w uniformly on E ɛ, for some subset E ɛ and \E ɛ ɛ. We now consider te set A ɛ := { wx) ɛ} E ɛ and observe tat tere exists a sufficiently small ɛ suc tat for any x A ɛ, w x) ɛ for all ɛ. If Gw) := w w I, ten we claim tat 3.7) G w ) Gw) dx as. A ɛ For any x A ɛ, let {T } be a sequence of triangles suc tat x T. Since w x) ɛ and w is piecewise linear, tere exists a vertex of T, wic we label as x, suc tat w ɛ. To compare G w ) wit w x) w x) I, we use tat I = e e + e e : G w ) w x) w x) I = We define Hx, y) := G w ) w x) w x) I = i=, w i + w w i + w w x) w x) x+y x + y and observe tat for all x A ɛ, we ave i=, ) e i e i. Hw, w i ) Hw x), w x)) ) e i e i. Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

14 37 R. H. NOCHETTO, S. W. WALKER, AND W. ZHANG Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// Next, we estimate Hw, w) i Hw x), w x)) = w x) w + wi ) w + wi )w x) w + wi ) w x) w + wi w x) w + wi + w x) w )w x) w + wi ) w x) + w x) w i )w x) w + wi ) w x). Since w, w x) ɛ, and w x) w x i ) = w x x i ) for all x T, we ave Hw, w i ) Hw x), w x)) C ɛ w for all x A ɛ T. Integrating on A ɛ, we obtain G w ) w x) w x) I dx C ɛ w x) dx as. A ɛ A ɛ Since w w a.e. in and w w w w is bounded, applying te dominated convergence teorem, we infer tat w w w w as. A ɛ Combining tese two limits, we deduce 3.7). Step 3: Convexity. We now prove tat te energy density Lw, M) := M + κ ) Gw) : M is convex wit respect to any matrix M for any vector w; ereafter, Gw) = z I wit z = w w provided w, or z oterwise. Note tat Lw, M) is a quadratic function of M, so we need only sow tat Lw, M) for any M and w. Tus, it suffices to sow tat Gw) : M M. Assume tat M = i,j m ijv i v j, were {v i } i= is te canonical basis on R. Ten we ave M = i,j= m ij, and a simple calculation yields Gw) : M = i z i v i v v + v v ) : m kl v k v l k,l = z i m kl δ ik v l = i,k,l i,l were z = i= z iv i. Terefore, we obtain Gw) : M = Te Caucy Scwarz inequality yields ) z i m ij i= i= z i z i m il v l, ) z i m ij. j= ) i= i= m ij ) i= m ij ), Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

15 FEM FOR LIQUID CRYSTALS 37 Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// wic implies Gw) : M M and Lw, M) for any matrix M and vector w. A similar argument sows tat L w, M) for any matrix M and vector w. Step 4: Weak lower semicontinuity. Since G w ) Gw) in L A ɛ ) according to 3.7), Egoroff s teorem yields G w ) Gw) uniformly on B ɛ, were B ɛ A ɛ and A ɛ \ B ɛ ɛ. We claim tat 3.8) lim inf L w, w )dx Lw, w)dx. B ɛ Step 3 implies L w, w ) for all x. Hence, L w, w )dx B ɛ w + κ ) G w ) : w ) dx. A simple calculation yields L w, w )dx Lw, w )dx + κ )Q w, w ), B ɛ were Q w, w ) := [G w ) Gw)) : w ] t [G w ) : w ] B ɛ ) + Gw) : w ) t [G w ) Gw)) : w ] dx. Since Lw, w ) is convex wit respect to w Step 3), we ave [7, sect. 8.., p. 446] lim inf Lw, w )dx Lw, w)dx. B ɛ B ɛ To prove 3.8), it remains to sow tat Q w, w ) as. Since Gw) and G w ) are bounded and w x) dx is uniformly bounded, we ave Q w, w ) C G w ) Gw) w dx B ɛ C max G w ) Gw) w dx as, B ɛ B ɛ due to te uniform convergence of G w ) to Gw) in B ɛ. Terefore, we infer tat lim inf L w, w )dx B ɛ Lw, w)dx. Since te inequality above olds for arbitrarily small ɛ, taking ɛ yields lim inf L w, w )dx Lw, w)dx = Lw, w)dx, \{wx)=} were te last equality follows from w = a.e. in te set {wx) = } [7, Cap. 5, Exercise 7, p. 9]. Finally, noting tat Gw) : w = w, we get te assertion. Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

16 37 R. H. NOCHETTO, S. W. WALKER, AND W. ZHANG Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// Equi-coercivity. We now prove uniform H -bounds for te pairs s, u ) and s, ũ ), wic enables us to extract convergence subsequences in L ) and pointwise a.e. in. We ten caracterize and relate te limits of suc sequences. Lemma 3.5 coercivity). For any s, u ) A g, r ), we ave { } E [s, n ] min{κ, } max u dx, s dx as well as { } E [s, n ] min{κ, } max ũ dx, I s dx. Proof. Inequality.) of Lemma. sows tat 3.9) E [s, n ] κ ) s dx + u dx. If κ, ten E [s, n ] obviously controls te H -norm of u wit constant. If < κ <, ten combining.6) wit.) yields E [s, n ] κ + κ k ij δ ij s ) i,j= i,j= s x i ) + s x j ) ) k ij δ ij n κ u dx, wence E [s, n ] min{κ, } u dx as asserted. Te same argument, but invoking.) and.4), leads to a similar estimate for ũ dx. Finally, we note tat.6) implies E [s, n ] κ i,j= k ij δ ij s ) = κ s dx. Upon recalling s = I s and noting δ ij s δ ij s and k ij, we deduce s L ) s L ) and complete te proof. Lemma 3.6 caracterizing limits). s, u ) A g, r ) satisfy 3.) E [s, n ] Λ for all >, Let {T } satisfy.), and let a sequence wit a constant Λ > independent of. Ten tere exist subsequences not relabeled) s, u ) X and s, ũ ) X weakly converging in [H )] d+ suc tat s, u ) converges to s, u) [H )] d+ in L ) and a.e. in ; s, ũ ) converges to s, ũ) [H )] d+ in L ) and a.e. in ; te limits satisfy s = s = u = ũ a.e. in ; tere exists a director field n defined in suc tat n converges to n in L \ S) and a.e. in \ S and u = sn, ũ = sn a.e. in. Proof. Te sequences s, u ) and s, ũ ) are uniformly bounded in H ) according to Lemma 3.5 coercivity). Terefore, since H ) is compactly embedded Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

17 FEM FOR LIQUID CRYSTALS 373 Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// in L ) [], tere exist subsequences not relabeled) tat converge in L ) and a.e. in to pairs s, u) [H )] d+ and s, ũ) [H )] d+, respectively. Since s s and s s as, invoking te triangle inequality yields s s s s + s s + s s as, wic is a consequence of interpolation teory and.5), namely, s s L ) = I s s L ) C s L ) C s L ) CΛ. A similar argument sows ũ I ũ L ) C ũ L ) C ũ L ) CΛ. Since I ũ = s s and ũ ũ as, we deduce ũ = s a.e. in. Likewise, arguing instead wit te pair s, u ), we infer tat u = s a.e. in. We now define te limiting director field n in \ S to be n = s u and see tat n = a.e. in \S; we define n in S to be an arbitrary unit vector. In order to relate n wit n, we observe tat bot s and n are piecewise linear. Applying te classical interpolation teory on eac element T of T, we obtain s n I [s n ] L T ) C s n L T ). Summing over all T T, we get s n I [s n ] L ) C s n L ) C s L ) n L ). An inverse estimate gives n L ) C, because n. Hence, s n I [s n ] L ) CΛ as. Since u = I [s n ] u as, we discover tat also s n u a.e. in as. Consequently, for a.e. x \ S we ave s x) sx), wence s x) is well defined for small and n x) = s x)n x) s x) ux) sx) = nx) as. Since n, te Lebesgue dominated convergence teorem yields n χ \S nχ \S L ) as. It remains only to prove ũ = sn a.e. in. Te same argument employed above gives s n I [ s n ] L ) CΛ as, wence s n ũ. Tis implies tat s x) is well defined for a.e. x \ S and Tis completes te proof. n x) = s x)n x) s x) ũx) sx) = nx) as Γ-convergence. We are now in te position to prove te main result, namely te convergence of global discrete minimizers. Te proof is a minor variation of te standard one [8, 4]. Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

18 374 R. H. NOCHETTO, S. W. WALKER, AND W. ZHANG Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// Teorem 3.7 convergence of global discrete minimizers). Let {T } satisfy.). If s, u ) A g, r ) is a sequence of global minimizers of E [s, n ] in.8), ten every cluster point is a global minimizer of te continuous energy E[s, n] in.). Proof. In view of.8), assume tere is a constant Λ > suc tat lim inf E [s, n ] = lim inf E [s, n ] + E [s ] ) Λ; oterwise, tere is noting to prove. Applying Lemma 3.6 yields subsequences not relabeled) s, ũ ) s, ũ) and s, u ) s, u) converging weakly in [H )] d+, strongly in [L )] d+, and a.e. in. Using Lemma 3.4, we deduce Ẽ [ s, ũ] = κ ) s + ũ dx lim inf Ẽ [ s, ũ ] lim inf E [s, n ], were te last inequality is a consequence of.). Since s converges a.e. in to s, so does ψs ) to ψs). Now apply Fatou s lemma to arrive at E [s] = ψs) = lim ψs ) lim inf ψs ) = lim inf E [s ]. Consequently, we obtain Ẽ [ s, ũ] + E [s] lim inf E [s, n ] lim sup E [s, n ]. Moreover, te triple s, u, n) given by Lemma 3.6 satisfies te structure property.5). In view of Proposition 3., given ɛ > arbitrary, we can always find a pair t ɛ, v ɛ ) Ag, r) [W )] d+ suc tat Ẽ [t ɛ, v ɛ ] + E [t ɛ ] = E [t ɛ, m ɛ ] + E [t ɛ ] inf E[t, m] + ɛ E[s, n] + ɛ, t,m) Ag,r) were m ɛ := t ɛ v ɛ if t ɛ ; oterwise, m ɛ is an arbitrary unit vector. Apply Lemma 3.3 to t ɛ, v ɛ ) and m ɛ to find t ɛ,, v ɛ, ) A g, r ), m ɛ, N suc tat E [t ɛ, m ɛ ] = lim E [t ɛ,, m ɛ, ]. On te oter and,.8) and 3.3) imply tat ψt ɛ, ) max{ψ + δ ), ψ δ )}, and we can invoke te Lebesgue dominated convergence teorem to infer tat E [t ɛ ] = lim ψt ɛ,) = lim ψt ɛ, ) = lim E [t ɛ, ]. Terefore, collecting te preceding estimates, we arrive at Ẽ [ s, ũ] + E [s] lim sup E [s, n ] lim E [t ɛ,, m ɛ, ] E[s, n] + ɛ. We now prove tat Ẽ[ s, ũ] = E [s, n]. We exploit te relation ũ = sn a.e. in wit n =, togeter wit te fact tat n admits a weak gradient in \ S, to find te ortogonal decomposition ũ = s n + s n a.e. in \ S. Hence, Ẽ [ s, ũ] = κ ) s + ũ dx = κ s + s n dx \S = \S \S κ s + s n dx E [s, n], Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

19 FEM FOR LIQUID CRYSTALS 375 Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// because s = s and s L \S) = s L \S). Note tat te singular set S does not contribute, because s L S) = s n L S) =. Finally, letting ɛ, we see tat te pair s, n) is a global minimizer of E as asserted. If te global minimizer of te continuous energy E[s, n] is unique, ten Teorem 3.7 readily implies tat te discrete energy minimizer s, n ) converges to te unique minimizer of E[s, n]. Tis teorem is about global minimizers only, bot discrete and continuous. In te next section, we design a quasi-gradient flow to compute discrete local minimizers and sow its convergence see Teorem 4.). In general, convergence to a global minimizer is not available, and neiter are rates of convergence, due to te lack of continuous dependence results. However, if local minimizers of E[s, n] are isolated, ten tere exist local minimizers of E [s, n ] tat Γ-converge to s, n) [8, 4]. 4. Quasi-gradient flow. We consider a gradient flow metodology consisting of a gradient flow in s and a minimization in n as a way to compute minimizers of.) and.8). We begin wit its description for te continuous system and verify tat it as a monotone energy decreasing property. We ten do te same for te discrete system. 4.. Continuous case. We introduce te following subspace to enforce Diriclet boundary conditions on open subsets Γ of : 4.) H Γ) = {v H ) : v = on Γ}. Let te sets Γ s, Γ n satisfy Γ n = Γ u Γ s, and let.9) be valid on Γ s. Terefore, te traces n = q := g r and n = q := I [g r ] are well defined on Γ n First order variation. Consider te bulk energy E[s, n], were te pair s, u), wit u = sn, is in te admissible class Ag, r) defined in.4). We take a variation z H ) of s and obtain δ s E[s, n; z] = δ s E [s, n; z] + δ s E [s; z], te first variation of E in te direction z, were δ s E [s, n; z] = s z + n sz) dx and δ s E [s; z] = ψ s)z dx. Next, we introduce te space of tangential variations of n: 4.) V n) = { v H ) d : v n = a.e. in }. In order to satisfy te constraint n =, we take a variation v V n) of n and get δ n E[s, n; v] = δ n E [s, n; v] = s n v) dx. Note tat variations in V n) preserve te unit lengt constraint up to second order accuracy [47]: n + tv = + t v and n + tv for all t R Quasi-gradient flow. We consider an L -gradient flow for E wit respect to te scalar variable s: t sz dx := δ s E [s, n; z] δ s E [s; z] for all z HΓ s ); Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

20 376 R. H. NOCHETTO, S. W. WALKER, AND W. ZHANG Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// ere, we enforce stationary Diriclet boundary conditions for s on te set Γ s, wence z = on Γ s. A simple but formal integration by parts yields t sz dx = s + n s + ψ s) ) z dx for all z HΓ s ), were we use te implicit Neumann condition ν s = on \ Γ s, and ν is te outer unit normal on. Terefore, s satisfies te nonlinear) parabolic PDE: 4.3) t s = s n s ψ s). Given s satisfying.9) on Γ s, let n satisfy n = a.e. in, te stationary Diriclet boundary condition n = q on te open set Γ n, and te following degenerate minimization problem: E[s, n] E[s, m] for all m = a.e., wit te same boundary condition as n. Tis implies 4.4) δ n E[s, n; v] = for all v V n) H Γ n ) d Formal energy decreasing property. Differentiating te energy wit respect to time, we obtain t E[s, n] = δ s E[s, n; t s] + δ n E[s, n; t n]. By virtue of 4.3) and 4.4), we deduce tat 4.5) t E[s, n] = δ s E[s, n; t s] = t s dx. Hence, te bulk energy E is monotonically decreasing for our quasi-gradient flow. 4.. Discrete case. Let s k S Γ s, g ) and n k N Γ n, q ) denote finite element functions wit Diriclet conditions s k = g on Γ s and n k = q on Γ n, were k indicates a time-step index see section 4.. for te discrete gradient flow algoritm). To simplify notation, we use te following: 4.6) s k i := s k x i ), n k i := n k x i ), z i := z x i ), v i := v x i ) First order variation. First, we introduce te discrete version of 4.), V n ) = {v U : v x i ) n x i ) = for all nodes x i N }. Next, te first order variation of E in te direction v V nk ) H Γ n ) at te director variable n k reads ) s k δ n E [s k, n k i ) + s k j 4.7) ; v ] = k ) ij δ ij n k ) δ ij v ), i,j= wereas te first order variation of E in te direction z S HΓ s ) at te degree of orientation variable s k consists of two terms, δ s E [s k, n k s k ; z ] = κ k ij δij s k ) δij z ) + k ij δ ij n k i z i + s k j z ) j 4.8). i,j= i,j= Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

21 FEM FOR LIQUID CRYSTALS 377 Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// To design an unconditionally stable sceme for te discrete gradient flow of E [s ], we employ te convex splitting tecnique in [5, 43, 44]. We split te double well potential into a convex and concave part: let ψ c and ψ e be bot convex for all s /, ) so tat ψs) = ψ c s) ψ e s), and set 4.9) δ s E [s k+ ; z ] := [ ψ c s k+ ) ψ es k ) ] z dx. Lemma 4. convex-concave splitting). For any s k and sk+ in S, we ave ψs k+ )dx ψs k )dx δ s E [s k+ ; s k+ s k ]. Proof. A simple calculation, based on te mean-value teorem and te convex splitting ψ = ψ c ψ e, yields ψs k+ ) ψs k ) ) dx = δ s E [s k+ ; s k+ s k ] + T, were T = + [ ψ c s k + θs k+ s k )) ψ cs k+ ) ] s k+ s k ) dθ dx [ ψ e s k ) ψ es k + θs k+ s k )) ] s k+ s k ) dθ dx. Te convexity of bot ψ c and ψ e implies T, as desired Discrete quasi-gradient flow algoritm. Our sceme for minimizing te discrete energy E [s, n ] is an alternating direction metod, wic minimizes wit respect to n and evolves s separately in te steepest descent direction during eac iteration. Terefore, tis algoritm is not a standard gradient flow. Algoritm discrete quasi-gradient flow): Given s, n ) in S Γ s, g ) N Γ n, q ), iterate Steps a) c) for k. Step a): Minimization. Find t k V nk ) H Γ n ) suc tat n k + tk minimizes te energy E [s k, nk + v ] for all v in V nk ) H Γ n ), i.e., t k satisfies δ n E [s k, n k + t k ; v ] = Step b): Projection. Normalize n k+ i Step c): Gradient flow. Using s k, nk+ ), find s k+ s k+ s k δt z dx= δ s E [s k+, n k+ for all v V n k ) H Γ n ). := nk i +tk i n k i +tk i at all nodes x i N. in S Γ s, g ) suc tat ; z ] δ s E [s k+ ; z ] for all z S HΓ s ). We impose Diriclet boundary conditions on bot s k and nk. Note tat te sceme as no restriction on te time step tanks to te implicit Euler metod in Step c) Energy decreasing property. Te quasi-gradient flow sceme in section 4.. as a monotone energy decreasing property, wic is a discrete version of 4.5), provided te mes T is weakly acute, namely it satisfies.) [, 45]. Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

22 378 R. H. NOCHETTO, S. W. WALKER, AND W. ZHANG Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// Teorem 4. energy decrease). Let T satisfy.). Te iterate s k+, n k+ of te algoritm discrete quasi-gradient flow) of section 4.. exists and satisfies E [s k+, n k+ ] E [s k, n k ] s k+ s k δt ) dx. ) Equality olds if and only if s k+, n k+ ) = s k, nk ) equilibrium state). Proof. Steps a) and b) are monotone, wereas Step c) decreases te energy. Step a): Minimization. Since E is convex in n k for fixed sk, tere exists a tangential variation t k wic minimizes E [s k, nk +vk ] among all tangential variations v k. Te fact tat E is independent of te director field n k implies E [s k, n k + t k ] E [s k, n k ]. Step b): Projection. Since te mes T is weakly acute, we claim tat n k+ = nk + tk n k + tk E [ s k, n k+ ] [ E s k, n k + t k ]. We follow [, 9]. Let v = n k + tk and w = v v and observe tat v and w is well defined. By.6) definition of discrete energy), we need only sow tat s k i k ) + s k j ) ij w x i ) w x j ) s k i k ) + s k j ) ij v x i ) v x j ) for all x i, x j N. Because k ij for i j, tis is equivalent to sowing tat w x i ) w x j ) v x i ) v x j ). Tis follows from te fact tat te mapping a a/ a defined on {a R d : a } is Lipscitz continuous wit constant. Note tat equality above olds if and only if n k+ = n k or, equivalently, tk =. Step c): Gradient flow. Since E is quadratic in terms of s k, and since s k+ s k+ s k ) = s k+ s k ) + s k+ s k, reordering terms gives were E [s k+, n k+ ] E [s k, n k+ ] = R E [s k+ s k, n k+ ] R, On te oter and, Lemma 4. implies E [s k+ ] E [s k ] = ψs k+ )dx R := δ s E [s k+, n k+ ; s k+ s k ]. ψs k )dx R := δ s E [s k+ ; s k+ s k ]. Combining bot estimates and invoking Step c) of te algoritm yields E [s k+, n k+ ] E [s k, n k+ ] R + R = s k+ s k δt ) dx, wic is te assertion. Note finally tat equality occurs if and only if s k+ = n k, wic corresponds to an equilibrium state. Tis completes te proof. n k+ = s k and Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.

23 FEM FOR LIQUID CRYSTALS 379 Downloaded 6/8/7 to Redistribution subject to SIAM license or copyrigt; see ttp:// 5. Numerical experiments. We present computational experiments to illustrate our metod, wic was implemented wit te MATLAB/C++ finite element toolbox FELICITY [48]. For all tree-dimensional simulations, we used te algebraic multigrid solver AGMG [39, 37, 38, 4] to solve te linear systems in Steps a) and c) of te quasi-gradient flow algoritm. In two dimensions, we simply used te backslas command in MATLAB. 5.. Tangential variations. Solving Step a) of te algoritm requires a tangential basis for te test function and te solution. However, forming te matrix system is easily done by first ignoring te tangential variation constraint i.e., arbitrary variations), followed by a simple modification of te matrix system. Let At k = B represent te linear system in Step a), and suppose d = 3. Multiplying by a discrete test function v, we ave v T At k = v T B for all v R dn. Next, using n k, find r, r suc tat {n k, r, r } forms an ortonormal basis of R 3 at eac node x i, i.e., find an ortonormal basis of V nk ). Next, expand tk = Φ r + Φ r and make a similar expansion for v. After a simple rearrangement and partitioning of te linear system, one finds it decouples into two smaller systems: one for Φ and one for Φ. After solving for Φ, Φ, define te nodal values of t k by te formula t k = Φ r + Φ r. 5.. Point defect in two dimensions. For te classic Frank energy n, a point defect in two dimensions as infinite energy [47]. Tis is not te case for te energy.), because s can go to zero at te location of te point defect, so te term s n remains finite. We simulate te gradient flow evolution of a point defect moving to te center of te domain is te unit square). We set κ = and take te double well potential to ave te following splitting: ψs) = ψ c s) ψ e s) = 63.s 6.s s s ), wit local minimum at s = and global minimum at s = s :=.755 see section., and note tat a vertical sift makes ψs ) = witout affecting te gradient flow). We impose te following Diriclet boundary conditions for s and n on Γ s = Γ n = : 5.) s = s, n = x, y), ) x, y), ). Initial conditions on for te gradient flow are s = s and a regularized point defect away from te center. Figure sows te evolution of te director field n and te scalar degree of orientation parameter s. One can see te regularizing effect of s. We note tat an L -gradient flow sceme, instead of te quasi- weigted) gradient flow we use, yields a muc slower evolution to equilibrium Plane defect in tree dimensions. Next, we simulate te gradient flow evolution of te liquid crystal director field toward a plane defect in te unit cube =, ) 3. Tis is motivated by an exact solution found in [47, sect. 6.4]. We set κ =. and remove te double well potential. We impose te following mixed boundary Copyrigt by SIAM. Unautorized reproduction of tis article is proibited.