Abstract
This paper focuses on the optimal error estimates of a linearized semi-implicit scheme for the nematic liquid crystal flows, which is used to describe the time evolution of the materials under the influence of both the flow velocity and the microscopic orientation configurations of rod-like liquid crystal flows. Optimal error estimates of the scheme are proved without any restriction of time step by using an error splitting technique proposed by Li and Sun. Numerical results are provided to confirm the theoretical analysis and the stability of the semi-implicit scheme.
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1 Introduction
In this paper, we consider the following hydrodynamics system modeling the flow of nematic liquid crystal material (see [28]):
at \(x\in \Omega \) and \(t\in [0,T]\) for some positive constant \(T>0\). Here \(\Omega \subset {\mathbb {R}}^2\) is a bounded and convex domain with a smooth boundary \(\partial \Omega \). The unknown \(\mathbf{u }(x,t):\Omega \times [0,T]\longrightarrow {\mathbb {R}}^2\) and \( p(x,t):\Omega \times [0,T]\longrightarrow {\mathbb {R}}\) represent the velocity and the pressure of the flows, respectively. The unknown \(\mathbf{b }(x,t):\Omega \times [0,T]\longrightarrow {\mathbb {S}}\), where \({\mathbb {S}}\) is the unit circle in \({\mathbb {R}}^2\), represents the macroscopic molecular orientation of the liquid crystal material. The vector \(\mathbf{f }:\Omega \times [0,T]\longrightarrow \mathbb {R}^2\) represents a body force on the flow. The constants \(\mu , \lambda \) and \(\gamma \) denote the viscosity, the competition between kinetic and potential energy, and the microscopic elastic relaxation time for the molecular orientation field, respectively. The term \(\nabla \mathbf{b }\odot \nabla \mathbf{b }\) is a \(2\times 2\) matrix whose (i, j)-the entry is given by \((\nabla _i\mathbf{b })\cdot (\nabla _j\mathbf{b })\) for \(1\le i,j\le 2\). It is noteworthy that if \(\mathbf{b }\) is a constant map, the system (1.1)–(1.3) reduces to the incompressible Navier–Stokes equations [35]. If \(\mathbf{u }=0\), the system (1.1)–(1.3) reduces to the heat flow of harmonic maps [8]. In addition, the above system (1.1)–(1.3) should be completed by an appropriate initial and boundary condition. For the sake of simplicity, we consider the following initial and boundary conditions:
where \(\mathbf{n }\) denotes the unit outward normal vector on \(\partial \Omega \). Here, we require that the initial vector functions \(\mathbf{u }_0\) and \(\mathbf{b }_0\) satisfy the compatibility condition \(\text{ div } \mathbf{u }_0=0\) and \(|\mathbf{b }_0|=1\).
The system (1.1)–(1.3) was firstly derived by Lin [28] as a simplified version of Ericksen–Leslie model for the hydrodynamics of nematic liquid crystal flows developed by Ericksen [10, 11] and Leslie [22]. It is the macroscopic continuum description of the time evolution of the materials under the influence of both the flow velocity \(\mathbf{u }\) and the microscopic orientation configurations \(\mathbf{b }\) of rod-like liquid crystal flows. There have two major difficulties in studying the above system. One is the presence of \(\text{ div } (\nabla \mathbf{b }\odot \nabla \mathbf{b })\) such that the system (1.1)–(1.3) becomes a strongly nonlinear coupled system. The other comes from the nonlinear constraint \(|\mathbf{b }|=1\).
The mathematical analysis for (1.1)–(1.5) was initiated by Lin and Liu [29, 30]. The nonlinear constraint \(|\mathbf{b }|=1\) was relaxed by introducing a Ginzburg–Landau penalty function \({(1-|\mathbf{b }|^2)\mathbf{b }}/{\varepsilon ^2}\) to replace \(|\nabla \mathbf{b }|^2\mathbf{b }\) in (1.2), where \(\varepsilon >0\) is a small penalty parameter. For Ginzburg–Landau approximation problem, Lin and Liu [29] proved the local existence of the strong solution and the global existence of the weak solution. It also has been shown that the global strong solution exists in the case of the large viscosity \(\mu \). The partial regularity for the suitable weak solution was proved in [30]. Some regularity criterions for the global weak solution were studied for 3D bounded and smooth domain [15]. However, as pointed in [29, 30], since the estimates and arguments heavily depend on \(\varepsilon \), it is still an open and challenging problem to study the limiting case as \(\varepsilon \) tends to zero. Recently, some new theoretical analysis for the original problem (1.1)–(1.5) have been studied. For example, Xu and Zhang [37] proved the global existence and regularity of weak solution for 2D Cauchy problem if \(||\mathbf{u }_0||_{L^2}\) and \(||\nabla \mathbf{b }_0||_{L^2}\) are small enough. For 3D model in a bounded and smooth domain, Li and Wang [27] established the existence and uniqueness of the local strong solution with large initial data and the global strong solution with small initial data. For the compressible nematic liquid crystal flows, Huang et al. [21] proved the local existence and uniqueness of strong solution provided that the initial data are sufficiently smooth and the pressure is a local Lipschitz continuous function with respect to the density function.
The numerical methods for Ginzburg–Landau approximation problem have been investigated in some previous works. For example, the first numerical method was studied by Liu and Walkington [31], where \({\mathbb {Q}}_3\) Hermite finite element was used for the approximation of the director. To avoid using Hermite finite element, a mixed finite element method was subsequently studied in [32]. In their works, the fully nonlinear implicit schemes were proposed. Although these schemes are unconditional stable, however, one has to solve a nonlinear problem by Newton’s iteration scheme at each time step. Becker et al. [6] studied a new mixed method by introducing \(\mathbf{w }=-\Delta \mathbf{b }+\displaystyle \frac{|\mathbf{b }|^2-1}{\varepsilon ^2}\mathbf{b }.\) The fully discrete scheme proposed in [6] was nonlinear at the step of solving \(\mathbf{w }\) numerically, and allowed them to establish a discrete energy law. With the help of this energy law, the authors showed the unconditional convergence of the numerical solution to the solution of Ginzburg–Landau approximation problem as h and \(\tau \) tend to zero. But no error estimates were derived in [6]. Motivated by Becker–Feng–Prohl, Girault and González-Santacreu [14] introduced the auxiliary variable \(\mathbf{w }=-\Delta \mathbf{b }\) to design a semiexplicit Euler scheme where the Ginzburg–Landau penalty function was explicitly discretizated. But the error estimate derived in [14] heavily depends on \(O(e^{-1/\varepsilon ^2})\). Other mixed FEM fully discrete schemes were developed by González-Santacreu [7, 16] in views of a fully explicit time integration of the potential term and the projection time-stepping method for Navier–Stokes equations [9, 35]. Based on a saddle-point strategy, Badia–González–Santacreu suggested a fully implicit scheme and a semi-implicit scheme for Ginzburg–Landau approximation problem in [3]. We observe that no error estimates are derived in [3, 7, 16]. The reader is referred to [2] for a survey of numerical methods on nematic liquid crystal flows and the Ginzburg–Landau approximation.
Instead of using the Ginzburg–Landau penalty function to relax the nonlinear constraint \(|\mathbf{b }|=1\), there have some existing works in studying the fully discrete schemes which directly approximate the original system (1.1)–(1.5). Becker et al. [6] investigated a fully discrete scheme which was conditionally stable under the time step restriction \(\tau \le O(h^3)\). The constraint \(|\mathbf{b }|=1\) are derived in the sense of \(\mathbf{L }^2\)-norm by the convergence of the numerical solution as h and \(\tau \) tend to zero. The same approximation also has been used for Landau–Lifshitz equation in [12]. Inspired by the projection time-stepping method for Navier–Stokes equations [9, 35] and Landau-Lifshitz equation [36], a time-stepping/projection scheme for the approximation of (1.1)–(1.5) is proposed by Prohl [33]. However, the error estimates derived in [33] are not optimal.
In this paper, we will propose a linearized semi-implicit finite element scheme for the approximation of the original system (1.1)–(1.5) and prove the optimal error estimates of this scheme. The derivations of the optimal error estimates are based upon the recent works by Li and Sun [23, 24] (also see [20, 25]), where the error estimates are split into the temporal error, the spatial error and the projection error by introducing a corresponding time-discrete parabolic system (or elliptic system). A key issue is that the regularities of the solutions to the discrete parabolic system need to be proved such that the uniform boundedness in different norms hold. With this boundedness, we can show that the spatial error analysis are bounded by \(Ch^\sigma \) for some \(\sigma >0\) and \(C>0\) independent of h and \(\tau \), from which the time step restriction can be removed. Meanwhile, optimal error estimates in the discrete \(\mathbf{L }^\infty (0,T;\mathbf{L }^2(\Omega ))\)-norm and \(\mathbf{L }^\infty (0,T;\mathbf{H }^1(\Omega ))\)-norm for \(\mathbf{u }\) and in the discrete \(\mathbf{L }^\infty (0,T;\mathbf{H }^1(\Omega ))\)-norm for \(\mathbf{b }\) are established without any time step restriction.
The rest of the paper is organized as follows. In Sect. 2, we introduce some notations and recall some known results for the nematic liquid crystal model (1.1)–(1.5). The uncoupled and linearized semi-implicit Euler finite element scheme and the main results in this paper are presented in Sect. 3. Meanwhile, the discrete parabolic system corresponding to the original system is introduced. Moreover, from the regularity of \((\mathbf{u }, p)\), the lowest-order \(P_1-P_0\) stabilized finite elements are used to approximate the velocity and the pressure in (1.1). The temporal error and the spatial error are shown in Sects. 4 and 5, respectively. The regularities of the solution to the discrete parabolic system are established in Sect. 4. The numerical results are presented in Sect. 7 to confirm our theoretical analysis. The conclusions are summarized in final section.
2 Preliminaries
Standard Sobolev space notations are used in this paper [1]. We use the boldface Sobolev spaces \(\mathbf{H }^m(\Omega ), \mathbf{W }^{m,p}(\Omega )\) and \(\mathbf{L }^p(\Omega )\) to denote the vector Sobolev spaces \(H^m(\Omega )^2\), \(W^{m,p}(\Omega )^2\) and \(L^p(\Omega )^2\) for \(m\in {\mathbb {N}}\) and \(1\le p\le +\infty \), respectively. In particular, \((\cdot ,\cdot )\) denotes the \(\mathbf{L }^2(\Omega )\) inner product. The symbols \(C, C_0, C_1, C_2, \ldots \) are used to denote a generic positive constant which may depends on \(\mathbf{u }, p, \mathbf{b }, \mathbf{f }\) and \(\mu , \lambda , \gamma \) and is independent of the mesh size h and the time step \(\tau \).
For the mathematical setting of the nematic liquid system model (1.1)–(1.5), we introduce the following spaces:
It is well known that the norm \(||\nabla \mathbf{v }||_{L^2}\) is equivalent to the standard \(\mathbf{H }^1\) norm for any \(\mathbf{v }\in \mathbf{V }\) due to Poincaré inequality. Define the following continuous bilinear forms \(a(\cdot ,\cdot )\) and \(d(\cdot ,\cdot )\) on \(\mathbf{V }\times \mathbf{V }\) and \(\mathbf{V }\times M\), respectively, by
and a trilinear form on \(\mathbf{X }\times \mathbf{X }\times \mathbf{X }\) by
Integrating by part, it is easy to check that
Corresponding to (1.1), we recall Stokes operator A. Introduce the orthogonal projection operator \({\mathbb {P}}_{\mathbf{H }}\) from \(\mathbf{L }^2(\Omega )\) onto \(\mathbf{H }\) which satisfies (cf. [35])
Then Stokes operator A is defined by (cf. [35])
Now, we recall some known inequalities frequently used in this paper [1, 35]:
Next, we give a regularity result for the solution to the problem (1.1)–(1.5) established in [21].
Theorem 2.1
Let \(\mathbf{u }_0\in \mathbf{D }(A)\) and \(\mathbf{b }_0\in \mathbf{H }^3(\Omega )\) with \(|\mathbf{b }_0|=1\) in \(\Omega \). For given \(\mathbf{f }\in \mathbf{L }^\infty (0,T;\mathbf{H })\cap \mathbf{L }^2(0,T;\mathbf{L }^4(\Omega ))\), then there exists some \(T^\star <T\) such that the problem (1.1)–(1.5) admits a unique local strong solution \((\mathbf{u },p,\mathbf{b })\) satisfying
Remark 2.1
Although the authors investigated the compressible nematic liquid crystal model in [21], the regularity results derived in [21] also hold for the incompressible nematic liquid crystal model (1.1)–(1.3) with the initial and boundary conditions (1.4)–(1.5). The regularity (2.14) for the pressure is not derived in [21] because the pressure p depends on the density in the compressible nematic liquid crystal model. But it can be easily proved by using (2.10)–(2.12) and inf-sup condition.
Remark 2.2
We require that the initial value \(\mathbf{u }_0\) and \(\pi \) is the solution to the following Stokes problem
where \(\mathbf{f }_0=\mathbf{f }(x,0)\in \mathbf{H }\). Then by the regularity result for Stokes problem [35], the solution \((\mathbf{u }_0, \pi )\) belongs to \(\mathbf{D }(A)\times H^1(\Omega )\cap M\).
Suppose that \(\mathbf{u }_0\in \mathbf{D }(A)\) satisfies (2.15). Under the following non-local compatibility conditions:
where \(p_0\in H^1(\Omega )\cap M\) is the weak solution to
the following regularities can be derived:
under \(\mathbf{f }_t\in \mathbf{L }^\infty (0,T; \mathbf{V }^\prime )\cap \mathbf{L }^2(0,T;\mathbf{H })\), where \(\mathbf{V }_0^\prime \) is the dual space of \(\mathbf{V }_0\). The proof of (2.16)–(2.17) is given in Appendix.
3 Main Results
In this paper, we assume that \(\Omega \) is a bounded and convex domain with a smooth boundary \(\partial \Omega \). Let \(T_h\) be a family of quasi-uniform triangular partition of \(\Omega \). The corresponding ordered triangles are denoted by \(K_1,K_2,\ldots ,K_M\). Let \(h_{i}=\text{ diam }(K_i), i=1,\ldots ,M\). Then we denote by \(h=\max \{h_1,h_2,\ldots ,h_M\}\) the mesh size. For a triangle \(K_j\) with two nodes on the boundary, we use \(\widetilde{K}_j\) to denote the triangle with one curved edge with the same nodes as \(K_j\). For interior element, we simply set \(\widetilde{K}_j\) as \(K_j\) itself. Let \(\Omega _h=\bigcup _1^M K_j\) and \(x=G(\widetilde{x})\) be a map from \(\Omega _h\) to \(\Omega \) such that G and \(G^{-1}\) both are Lipschitz continuous, and G is the identity mapping for interior element \(K_j\), and G maps \(K_j\) onto \(\widetilde{K}_j\) smoothly for \(K_j\) at the boundary [13, 38]. For a given partition of \(\Omega \), we define
where \(J_G\) denotes the Jacobian of G, and \(P_{r}(K)\) denotes the space of the polynomials on K of degree at most r for every \(K\in T_h\) and a nonnegative integer r. For \(x\in \Omega \), we define an operator \({\mathcal {G}}_{X}\) on \(\widehat{\mathbf{X }}_h\) by \({\mathcal {G}}_{X}\mathbf{v }_h(x)=\mathbf{v }_h(G^{-1}(x))\), and an operator \({\mathcal {G}}_{V}\) on \(\widehat{\mathbf{V }}_h\) by \({\mathcal {G}}_{V}\mathbf{w }_h(x)=\mathbf{w }_h(G^{-1}(x))\), and an operator \({\mathcal {G}}_{M}\) on \(\widehat{M}_h\) by \({\mathcal {G}}_{M} q_h(x)=q_h(G^{-1}(x))\), and an operator \({\mathcal {G}}_{\widetilde{M}}\) on \(\widehat{\widetilde{M}}_h\) by \({\mathcal {G}}_{\widetilde{M}} \phi _h(x)=\phi _h(G^{-1}(x))\). Then the finite element spaces are defined by
It is clear that \(\mathbf{X }_h\) is a finite element subspace of \(\mathbf{X }\) and \(\mathbf{V }_h\) is a finite element subspace of \(\mathbf{V }\). Moreover, there holds \(\displaystyle \int _\Omega {\mathcal {G}}_M q_h(x)dx=\displaystyle \int _\Omega {\mathcal {G}}_{\widetilde{M}}\phi _h(x)dx=0\). Thus, \(M_h\) and \(\widetilde{M}_h\) both are finite element spaces of M. For any \(\mathbf{v }\in \mathbf{X }\), we define \(\Pi _h^0\mathbf{v }={\mathcal {G}}_X\widehat{\Pi }_h{\mathcal {G}}_X^{-1}\mathbf{v }\), where \(\widehat{\Pi }_h:{\mathcal {C}}(\Omega _h)\rightarrow \widehat{X}_h\) is the Lagrange interpolation operator. Then for any \(\mathbf{v }\in \mathbf{H }^3(\Omega )\), (cf. [13, 26])
Similarly, For any \(\mathbf{w }\in \mathbf{V }\), we define \(\mathbf{R }_h^0\mathbf{v }={\mathcal {G}}_V\widehat{\mathbf{R }}_h{\mathcal {G}}_V^{-1}\mathbf{w }\), where \(\widehat{\mathbf{R }}_h:{\mathcal {C}}(\Omega _h)\rightarrow \widehat{V}_h\) is the Lagrange interpolation operator. Then for any \(\mathbf{w }\in \mathbf{H }^2(\Omega )\cap \mathbf{V }\), (cf. [34])
Observe that \(P_1-P_0\) finite element space for velocity and pressure does not satisfy the so-called discrete inf-sup condition. Here, we use a stabilized technique proposed by Bochev et al. [4] for Stokes problem. Introduce the generalized bilinear forms defined by
for all \((\mathbf{w }_h,r_h), (\mathbf{v }_h,q_h)\in \mathbf{V }_h\times M_h \). Here \(\alpha >0\) is the stable parameter. The stable term \(G(r_h,q_h)\) is defined by
where \(\Pi _1\) is a continuous projection operator from \(M_h\) to \(\widetilde{M}_h\).
Suppose that the solution \((\mathbf{u },p,\mathbf{b })\) satisfies the regularities (2.10)–(2.14) and (2.16)–(2.17) in [0, T] for some \(T>0\). Let \(0=t_0<t_1<\cdots <t_N=T\) be a uniform partition of the time interval [0, T] with time step \(\tau =T/N\) and \(t_n=n\tau \) for \(0\le n\le N\). Let
For any sequence \(\{g^n\}_{n=0}^N\), denote \(D_\tau g^{n+1}=\displaystyle \frac{g^{n+1}-g^n}{\tau }\) for \(0\le n\le N-1\).
Under the above notations, we propose a linearized semi-implicit Euler finite element scheme for the nematic liquid crystal model (1.1)–(1.5), which is to find \(\mathbf{B }^{n+1}_h\in \mathbf{X }_h\) and \((\mathbf{U }^{n+1}_h, P^{n+1}_h)\in \mathbf{V }_h\times M_h\) for \(n=0,1,\ldots ,N-1\), such that
with \(\mathbf{B }^{0}_h=\Pi _h^0\mathbf{b }_0\in \mathbf{X }_h\), \(\mathbf{U }_h^0=\mathbf{R }_h^0\mathbf{u }_0\in \mathbf{V }_h\) and
The emphasis of this paper is to show optimal error estimates for the semi-implicit Euler scheme (3.3)–(3.4). The main result derived in this paper is presented in the following theorem.
Theorem 3.1
Suppose \(\mathbf{u }_0\in \mathbf{V }_0\cap \mathbf{H }^2(\Omega )\), \(\mathbf{b }_0\in \mathbf{H }^3(\Omega )\) with \(|\mathbf{b }_0|=1\), \(\mathbf{f }\in \mathbf{L }^\infty (0,T;\mathbf{H })\cap \mathbf{L }^2(0,T; \mathbf{L }^4(\Omega ))\) and \(\mathbf{f }_t\in \mathbf{L }^\infty (0,T;\mathbf{V }_0^\prime )\cap \mathbf{L }^2(0,T;\mathbf{H })\). Then the finite element semi-implicit discrete system (3.3)–(3.4) exists a unique solution \((\mathbf{U }^n_h, P_h^n, \mathbf{B }_h^n)\in \mathbf{V }_h\times M_h\times \mathbf{X }_h\) for \(n=1,\ldots , N\). Moreover, there have two constants \(h_0>0\) and \(\tau _0>0\) such that when \(h<h_0\) and \(\tau <\tau _0\), the following optimal error estimates hold:
To prove Theorem 3.1 by using the temporal-spatial error splitting method proposed by Li and Sun in [23, 24], for \(\mathbf{B }^0=\mathbf{b }_0\) and \(\mathbf{U }^0=\mathbf{u }_0\) and \(n=0,1,\ldots ,N-1\), we define \(\mathbf{B }^{n+1}\) and \((\mathbf{U }^{n+1}, P^{n+1})\) to be the solutions of the following discrete parabolic (or elliptic) system corresponding to the time-dependent system (1.1)–(1.5):
with homogeneous Neumann boundary condition \(\partial _n\mathbf{B }^{n+1}|=0\) on \(\partial \Omega \), and
with homogeneous boundary condition \(\mathbf{U }^{n+1}=0\) on \(\partial \Omega \).
Remark 3.1
For given \(\mathbf{U }^n\) with \(\text{ div } \mathbf{U }^n=0\) and \(\mathbf{B }^n\), the existence and uniqueness of the weak solution \(\mathbf{B }^{n+1}\) to the linear elliptic problem (3.8) with Neumann boundary condition follows from Lax–Milgram theorem by using \(b(\mathbf{U }^n,\mathbf{B }^{n+1},\mathbf{B }^{n+1})=0\). For given \(\mathbf{U }^n, \mathbf{B }^n\) and \(\mathbf{B }^{n+1}\), the existence and uniqueness of the weak solution \((\mathbf{U }^{n+1}, P^{n+1})\) to the linearized Navier–Stokes equations (3.9) with homogeneous boundary condition follows from the classical existence and uniqueness theorem for steady Navier–Stokes problem [35].
Let us denote
and for \(1\le n\le N\),
where \(\mathbf{R }_h, Q_h\) and \(\Pi ^n_h\) are projection operators defined in Sect. 5. The proof of (3.6)–(3.7) is based upon the following error splitting in some norm \(||\cdot ||\):
Here \(||\mathbf{e }_{\mathbf{u }}^n||, ||\mathbf{e }_{\mathbf{b }}^n||, ||e_{p}^n||\) are temporal errors, and \(||\mathbf{e }_{\mathbf{u }h}^n||, ||\mathbf{e }_{\mathbf{b }h}^n||, ||e_{p h}^n||\) are spatial errors, and \(||\mathbf{E }_{\mathbf{u }}^n||, ||\mathbf{E }_{\mathbf{b }}^n||, || E_{p}^n||\) are projection errors.
Before the proof of Theorem 3.1, we recall the following inverse inequality which holds for \(\mathbf{v }_h\in \mathbf{V }_h\) or \(\mathbf{v }_h\in \mathbf{X }_h\) [5]:
Finally, a discrete version of Gronwall’s inequality established in [19] is frequently used in this paper.
Lemma 3.1
Let \(a_k, b_k, c_k\) and \(\gamma _k\), for integers \(k\ge 0\), be the nonnegative numbers such that
Suppose that \(\tau \gamma _k<1\), for all k, and set \(\sigma _k=(1-\tau \gamma _k)^{-1}\). Then
Remark 3.2
If the first sum on the right in (3.11) extends only up to \(n-1\), then the estimate (3.12) holds for all \(\tau >0\) with \(\sigma _k=1\).
4 Temporal Error Analysis
In this section, we begin to estimate temporal errors \(\mathbf{e }_\mathbf{u }^n, \mathbf{e }_\mathbf{b }^n\) and \(\mathbf{e }_p^n\) for \(1\le n\le N\). Meanwhile, some regularities of solutions \(\mathbf{B }^n\) and \((\mathbf{U }^n, P^n)\) to the discrete parabolic system (3.8)–(3.9) are derived. For \(0\le n\le N-1\), we take \(t=(n+1)\tau \) at (1.1)–(1.5) to yield
with boundary conditions \(\mathbf{u }^{n+1}=0\) and \(\partial _\mathbf{n }\mathbf{b }^{n+1}=0\) on \(\partial \Omega \), where
It follows from Hölder’s inequality and (2.17) that
Subtracting (3.8) from (4.2), and (3.9) from (4.1), (4.3) leads to
and
with boundary conditions \(\mathbf{e }_\mathbf{u }^{n+1}=0\) and \(\partial _\mathbf{n }\mathbf{e }_\mathbf{b }^{n+1}=0\) on \(\partial \Omega \).
First, we prove the following temporal errors.
Lemma 4.1
Suppose that the solution \((\mathbf{u },p,\mathbf{b })\) to (1.1)–(1.5) satisfies the regularities (2.10)–(2.14) and (2.16)–(2.17) in [0, T]. For \(0\le n\le N-1\), there exists some \(\tau _1>0\) such that when \(\tau <\tau _1\), there hold
Proof
Due to \(\mathbf{u }^0=\mathbf{U }^0=\mathbf{u }_0\) and \(\mathbf{b }^0=\mathbf{B }^0=\mathbf{b }_0\), the inequalities (4.7)–(4.9) obviously hold for \(m=0\). Suppose that (4.7)–(4.9) hold for \(m\le n\), then we need to show that these inequalities also hold for \(m\le n+1\). Multiplying (4.5) by \(\mathbf{e }_\mathbf{b }^{n+1}\) and integrating over \(\Omega \), we obtain
Rewrite \(|\nabla \mathbf{b }^{n+1}|^2\mathbf{b }^{n+1}-|\nabla \mathbf{B }^{n}|^2\mathbf{B }^{n}\) as
By using (2.12), (2.13), (2.6), (4.7) for \(m\le n\), Hölder’s inequality and Young’s inequality, \(I_1\) is bounded by
where \(C_2>0\) is independent of \(C_0\). Moreover, we use
From Hölder’s inequality and Young’s inequality again, we estimate \(I_2\) as
An alternative to \(I_3\) is
Then, from (2.11) and (2.12), we get
where we use
Then combining these estimates for \(I_1, I_2\) and \(I_3\) into (4.10), we obtain
Testing (4.6) by \(\mathbf{e }_\mathbf{u }^{n+1}\) yields
From Hölder’s inequality and Young’s inequality, \(I_4\) satisfies
Similarly, it is easy to show that
We rewrite \(I_6\) as
Then \(I_6\) is bounded by
where we use (4.7)–(4.8) for \(m\le n\) and
Combining these estimates into (4.14), we obtain
Testing (4.5) by \(-\Delta \mathbf{e }_\mathbf{b }^{n+1}\) and using a similar argument for (4.13), we can get
Taking sufficiently small \(\tau \) to satisfy
and summing up the inequalities (4.13), (4.16), (4.17) and using discrete Gronwall’s inequality (see Lemma 5.1 in [19]), there exists some \(C_3>0\) and \(\tau _{11}>0\) such that when \(\tau <\tau _{11}\), there holds
Then temporal errors of the director in (4.7) hold if we choose \(C_0>4\exp (TC_3)\).
For \(1\le k\le N\), by the definition \(\mathbf{e }_\mathbf{b }^k\), we have
Following (2.13) and (4.18), we get
For \(0\le n\le N-1\), rewrite (3.8) as
where \(\mathbf{F }^n=\gamma |\nabla \mathbf{B }^n|^2\mathbf{B }^n-(\mathbf{U }^n\cdot \nabla )\mathbf{B }^{n+1}-D_\tau \mathbf{B }^{n+1}\). Then we have
where we use Sobolev imbedding \(\mathbf{H }^2(\Omega )\subset \mathbf{W }^{1,p}(\Omega )\) for any \(1\le p<+\infty \). Thus, by classical regularity theory of elliptic problem, we have \(||\mathbf{B }^{n+1}||_{W^{2,4}}\le C\), which implies \(||\mathbf{B }^{n+1}||_{W^{1,\infty }}\le C\) if we use Sobolev imbedding \(\mathbf{W }^{2,4}(\Omega )\subset \mathbf{W }^{1,\infty }(\Omega )\). In this case, \(\nabla \mathbf{F }^n\) in \(\mathbf{L }^2\)-norm can be bounded by
Thus \(\mathbf{F }^n\in \mathbf{H }^1(\Omega )\). From classical regularity theory of elliptic problem, again, we obtain
The above estimate with (2.12) implies that
Testing (4.6) by \(A\mathbf{e }_\mathbf{u }^{n+1}\), we get
An alternative to \(I_7\) is
which can be bounded by
From Young’s inequality, we bound \(I_8\) as
An alternative to \(I_9\) is
By Hölder’s inequality and (4.20), all terms in the right-hand side of the above inequality can be estimated, respectively, by
Combining these estimates into (4.21), and using Young’s inequality and the discrete Gronwall’s inequality, we conclude that there exists some \(C_5>0\) and \(\tau _{12}>0\) such that when \(\tau <\tau _{12}\), there holds
Thus, (4.7) holds if we choose \(C_0>4\max \{\exp (TC_3),\exp (TC_5)\}\). As a direct result of (4.7), we have
From the arguments in estimating \(I_1\) to \(I_3\), we have
In terms of (4.23) and \(\mathbf{H }^2\) regularity for linear elliptic problem, it can be shown \( ||\mathbf{e }_\mathbf{b }^{n+1}||_{H^2}\le C_1\). Thus, we complete the proof of
Other estimates in (4.8) and (4.9) are from (4.7) by a simple calculation. \(\square \)
For \(0\le n\le N-1\), in order to estimate \(||e_p^{n+1}||_{L^2}\) by the inf-sup condition, we need to estimate \(||D_\tau \mathbf{e }_{\mathbf{u }}^{n+1}||_{L^2}\). Testing (4.6) by \(\tau D_\tau \mathbf{e }_\mathbf{u }^{n+1}\) and using a similar proof for (4.21), we can easily obtain
As a direct consequence of (4.7) and inf-sup condition, we immediately obtain
Other regularities of \(\mathbf{B }^n\) and \((\mathbf{U }^n, P^n)\) are derived in next lemma.
Lemma 4.2
Under the assumptions in Lemma 4.1, we have
Proof
Following (2.13) and (4.8), we get
This completes the proof of (4.27). The proof of (4.28) can be easily completed by using (2.16) and (4.9). From (4.19), we have
and
By using \(\mathbf{f }\in \mathbf{L }^\infty (0,T;\mathbf{H })\cap \mathbf{L }^2(0,T;\mathbf{L }^4(\Omega ))\), we derive
from the regularity result for steady Navier–Stokes equations. For \(0\le n\le N-1\), the estimate for \(P^{n+1}\) is derived from (3.9) by
Observing \(||p||_{L^2}\le C||\nabla p||_{L^2}\) for any \(p\in M\), we complete the proof of this lemma. \(\square \)
5 Spatial Error Analysis
In this section, for \(1\le n\le N\), we begin to estimate spatial errors \(\mathbf{e }_{\mathbf{u }h}^n, \mathbf{e }_{\mathbf{b }h}^n\) and \(\mathbf{e }_{ph}^n\) under the regularities of \(\mathbf{U }^n, \mathbf{B }^n\) and \(P^n\) derived in Lemma 4.2. In order to derive the spatial error estimates, we need to introduce the following projection \((\mathbf{R }_h, Q_h):\mathbf{V }\times M \longrightarrow \mathbf{V }_h\times M_h\) defined by
for each \((\mathbf{w },r)\in \mathbf{V }\times M\) and all \((\mathbf{w }_h, r_h)\in \mathbf{V }_h\times M_h\). According to Theorem 4.1 in [4], it is easy to check that \((\mathbf{R }_h\mathbf{w }, Q_hr)\) is well defined. By the definition of \({\mathcal {B}}_h\) and \({\mathcal {B}}\), there holds
For \(1\le n\le N\), by a classical argument as in [17] the following approximation properties hold:
For given \(1\le n\le N\) and \(\mathbf{U }^{n-1}\in \mathbf{V }\), define \(\Pi _h^{n}\mathbf{B }^{n} \in \mathbf{X }_h\) by
From the classical finite element theory for elliptic problem [5], we have
Multiplying (3.8) by \(\phi _h\in \mathbf{X }_h\) and (3.9) by \((\mathbf{v }_h,q_h)\in \mathbf{V }_h\times M_h\), and subtracting the resulting equations from (3.3) and (3.4), respectively, we get
and
where we use the definitions of \(\Pi ^n_h\) and \((\mathbf{R }_h,Q_h)\).
The main results in this section are summarized in the following lemma.
Lemma 5.1
Under the assumptions in Theorem 3.1 and Lemma 4.1, for \(0\le n\le N-1\), there exist some \(h_2>0\) and \(\tau _2>0\) such that when \(h<h_2\) and \(\tau <\tau _2\), there hold
Proof
Since \(\mathbf{e }_{\mathbf{u }h}^0=\mathbf{R }_h^0\mathbf{U }^0-\mathbf{U }_h^0=0\) and \(\mathbf{e }_{\mathbf{b }h}^0=\Pi ^0_h\mathbf{B }^0-\mathbf{B }_h^0=0\), the inequalities (5.13) and (5.15) obviously hold for \(m=0\). Now, we suppose that (5.13) and (5.15) hold for \(m\le n\). Then we need to show that these inequalities also hold for \(m\le n+1\). By inverse inequality (3.10), we have
Taking \(\phi _h=\mathbf{e }_{\mathbf{b }h}^{n+1}\) in (5.11) leads to
By using (5.7), (5.8) and (4.19), it is easy to bound \(J_1\) and \(J_2\), respectively, as
and
From (3.5), (5.3), (5.9) and (5.14) for \(m\le n\), \(J_3\) satisfies
We rewrite \(J_4\) according to the following equation:
Then from the following estimates:
\(J_4\) can be bounded by
For sufficiently small h such that \((\gamma C_9^2C_0^2h^{4/3}+C_9^4C_0^4h^{10/3})<\gamma /4\). Then combining these estimates into (5.17) leads to
Taking \(\mathbf{v }_h=\mathbf{e }_{\mathbf{u }h}^{n+1}\) and \(q_h=e_{ph}^{n+1}\) in (5.12) leads to
From (5.4) and (4.28), \(J_5\) is bounded by
An alternative to \(J_{6}\) is
Then we have
Observing the following identity
and using the following estimates:
we estimate \(J_{7}\) as
Combining these estimates into (5.19) leads to
To prove (5.15), we set \(\phi _h=D_\tau \mathbf{e }_{\mathbf{b }h}^{n+1}\) in (5.11) to get
By the similar arguments for \(J_1\) and \(J_2\), we have
and
An alternative to \(J_{10}\) is
which are estimated, respectively, by
Then \(J_{10}\) is bounded by
To estimate \(J_{11}\), we rewrite it as
Then all terms in \(J_{11}\) are bounded by
For sufficiently small h such that \(\displaystyle \frac{C_0^2C_6^2}{16} h^2<1\), then \(J_{11}\) satisfies
Combining these estimates into (5.21) leads to
Summing up three inequalities (5.18), (5.20), (5.22), for sufficiently small \(\tau \) such that
then from discrete Gronwall’s inequality we derive
which proves Lemma 5.1 if we take \(C_0^2>16C_{16}\exp (TC_{17})\). \(\square \)
6 Proof of Theorem 3.1
First, we show the existence and uniqueness of the solution to the finite element discrete system (3.3)–(3.4). For \(0\le n\le N-1 \), an alternative to (3.3) is
Taking \(\phi _h=\mathbf{B }_h^{n+1}\) in the above equation leads to
Then, the existence and uniqueness of \(\mathbf{B }_h^{n+1}\) for \(0\le n\le N-1\) follows from Lax–Milgram theorem. The existence and uniqueness of the solution to (3.4) can be shown by using Lax–Milgram theorem for \(\mathbf{U }^{n+1}_h\) and Theorem 4.1 in [4] for \(P_h^{n+1}\). Error estimates \(||\mathbf{b }^n-\mathbf{B }_h^n||_{H^1}\) and \(||\mathbf{u }^n-\mathbf{U }^n_h||_{L^2}\) follow from the error splitting and (4.7), (5.3), (5.7), (5.14) and (5.15). By inverse inequality (3.10), (4.7) and (5.3), we derive
for sufficiently large \(C_0\). On the other hand, by using
we have
for sufficiently large \(C_0\). We completes the proof of Theorem 3.1 if we take \(\tau _0=\min \{\tau _1, \tau _2\}\) and \(h_0=h_2\).
7 Numerical Results
In this section, we present the numerical results by using the linearized semi-implicit scheme (3.5)–(3.6) to verify the optimal error estimates derived in Theorem 3.1. All programs are implemented by the free finite element software FreeFem++ [18]. We consider the nematic liquid crystal model in the unit circle \(\Omega =\{(x,y): \ x^2+y^2<1\}\). The initial data are taken as
Parameters are set as \(\alpha =\lambda =\gamma =1\) and \(\mu =1\). We take a uniform triangular partition with M nodes on \(\partial \Omega \). Then a class of uniform meshes of the unit circle is made by a mesh generator in FreeFem++; see Fig. 1 for illustration.
The FEM meshes of the unit circle with \(M=50\)
\(H^1\) and \(L^2\) errors of the director
Evolution of director fields: \(T=0\) (top left), \(T=0.6\) (top right), \(T=1.5\) (bottom left), \(T=2.74375\) (bottom right)
Since no exact solution exists, to verify the optimal convergence rates, the reference solution is taken as the numerical solution corresponding to \(M=300\). The time step \(\tau \) is required to satisfy \(\tau =O(1/M^2)\). Therefore, from the error estimates derived in Theorem 3.1, we have the second-order convergence rate for the errors \(||\mathbf{b }^n-\mathbf{B }^n_h||_{H^1}\) and \(||1-|\mathbf{B }_h^n|^2||_{L^2}\). To verify the optimal convergence rates, we use several mesh pairs \(M=50, 100\) and 150 with different time step \(\tau =9/M^2\) and \(1/M^2\). The numerical results are displayed in Tables 1 and 2, from which we can see that the numerical convergence rates for orientation of molecular coincide with the ones predicted by theoretical analysis in Theorem 3.2, although the numerical errors with \(\tau =9/M^2\) seem not very accurate.
Evolution of velocity fields: \(T=0.1\) (top left), \(T=0.6\) (top right), \(T=1.5\) (bottom left), \(T=2.74375\) (bottom right)
To confirm the temporal error is of the first-order convergence rate, we take the reference solution corresponding to \(\tau =3.125\times 10^{-6}\) and \(M=120\). For different time step \(\tau _{i+1}=0.5\tau _i\) for \(i=1,2,4\) with \(\tau _1=10^{-3}\), the numerical errors of velocity, director and pressure are displayed in Table 3. It can be observed that the semi-implicit scheme gives the convergence rates of the order \(O(\tau )\) on the temporal errors which coincide with the ones predicted in Lemma 4.1.
To confirm the stability of the semi-implicit scheme without any restriction of the time step \(\tau \), the reference solution is taken as the numerical solution corresponding to \(\tau =3.125\times 10^{-6}\) and \(M=200\). We solve the semi-implicit scheme (3.3) and (3.4) with three different time step \(\tau =1.25\times 10^{-4}, 1.00\times 10^{-4}\) and \(6.25\times 10^{-5}\) on gradually refined meshes with \(M=20i, i=2,\ldots ,9\). The \(H^1\) errors \(||\mathbf{b }(\cdot ,0.5)-\mathbf{B }_h^J||_{H^1}\) and \(L^2\) errors \(||1-|\mathbf{B }_h^J|^2||_{L^2}\) are displayed in Fig. 2, from which we can see that for a fixed \(\tau \), when refining the mesh gradually, the \(H^1\) errors converge to a small constant and the proposed semi-implicit scheme (3.3) and (3.4) is stable and convergent without any restriction of the time step.
Fix \(M=100\) and \(\tau =1/M^2\). The evolutions of the director fields and the velocity fields at different times are displayed in Figs. 3 and 4, where \(T=2.74375\) is the stoping time of iteration. The stopping criterion used is to require \(||\mathbf{U }^n_h-\mathbf{U }^{n-1}_h||_{L^2}+||\mathbf{B }_h^n-\mathbf{B }_h^{n-1}||_{L^2}\) to be less than \(10^{-6}\).
8 Conclusion
In this paper, we show optimal error estimates for an linearized semi-implicit Euler finite element scheme for the approximation of the nematic liquid crystals flows. To the best of our knowledge, no optimal error estimates have been given in previous works. More important, the semi-implicit scheme proposed in this paper is linear. Optimal error estimates are proved without any restriction of time step \(\tau \) by using the error splitting technique. The numerical results show the efficiency of the scheme and confirm our theoretical analysis. In addition, the technique presented in this paper can be applied to analyze linearized semi-implicit Galerkin FEM for 3D nematic liquid crystal flows.
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Acknowledgements
Rong An was supported by Zhejiang Provincial Natural Science Foundation with Grant No. LY16A010017. Jian Su was supported by National Natural Science Foundation of China with Grant No. 91330117.
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Appendix
Appendix
To get the regularities (2.16)–(2.17) of \(\mathbf{u }_{tt}, p_t\) and \(\mathbf{b }_{tt}\), where \((\mathbf{u }, p, \mathbf{b })\) is the solution to the problem (1.1)–(1.5) and satisfies (2.10)–(2.14), we need to show that \(||\nabla \mathbf{u }_t(x,t)||_{L^2}\) and \(||\mathbf{b }(x,t)||_{H^1}\) remain bounded as \(t\longrightarrow 0\). In this case, a non-local compatibility condition is needed and can be derived as follows. First, we begin to show that the problem
exists a unique \(p_0\in H^1(\Omega )\cap M\). In fact, from \(\mathbf{f }_0\in \mathbf{H }, \mathbf{u }_0\in \mathbf{D }(A)\) and \(\mathbf{b }_0\in \mathbf{H }^3(\Omega )\), it can be shown that \(\text{ div } (\mathbf{f }_0-(\mathbf{u }_0\cdot \nabla )\mathbf{u }_0-\lambda \text{ div } (\nabla \mathbf{b }_0\odot \nabla \mathbf{b }_0))\in L^2(\Omega )\) and \(\mu \Delta \mathbf{u }_0+\mathbf{f }_0-(\mathbf{u }_0\cdot \nabla )\mathbf{u }_0-\lambda \text{ div } (\nabla \mathbf{b }_0\odot \nabla \mathbf{b }_0)\in \mathbf{H }(\text{ div } ,\Omega )\). Thus, one has \((\mu \Delta \mathbf{u }_0+\mathbf{f }_0-(\mathbf{u }_0\cdot \nabla )\mathbf{u }_0-\lambda \text{ div } (\nabla \mathbf{b }_0\odot \nabla \mathbf{b }_0))\cdot \mathbf{n }|_{\partial \Omega }\in H^{-1/2}(\Omega )\). Finally, we note that the following compatibility condition is satisfied:
due to the fact \(\text{ div } (\Delta \mathbf{u }_0)=0\). From these observations, it follows that the problem (9.1) exists a unique weak solution \(p_0\in H^1(\Omega )\cap M\). Moreover, the solution \(p_0\) is the limit of the pressure p(x, t) in \(H^1(\Omega )\cap M\) as \(t\longrightarrow 0\). To make our point precise, we give the following lemma:
Lemma 9.1
Let the initial values \(\mathbf{u }_0\in \mathbf{D }(A)\) and \(\mathbf{b }_0\in \mathbf{H }^3(\Omega )\) with \(|\mathbf{b }_0|=1\) in \(\Omega \). Suppose that the solution \((\mathbf{u }, p, \mathbf{b })\) to the problem (1.1)–(1.5) satisfies \(||A\mathbf{u }(x,t)-A\mathbf{u }_0(x)||_{L^2}\longrightarrow 0\) and \(||\mathbf{b }(x,t)-\mathbf{b }_0(x)||_{H^3}\longrightarrow 0\) as \(t\longrightarrow 0\). Then the pressure p(x, t) tends to the solution \(p_0\) to the problem (9.1) in the sense that
Proof
For \(t>0\), it follows from (1.1) and \(\text{ div } \mathbf{u }=0\) that the pressure \(p\in H^1(\Omega )\cap M\) is the weak solution to the problem
From \(||A\mathbf{u }(x,t)-A\mathbf{u }_0(x)||_{L^2}\longrightarrow 0\) and \(||\mathbf{b }(x,t)-\mathbf{b }_0(x)||_{H^3}\longrightarrow 0\) as \(t\longrightarrow 0\), we can see that
in \(L^2(\Omega )\), and
in \(H^{-1/2}(\partial \Omega )\) as \(t\longrightarrow 0\). These facts imply the desired result. \(\square \)
Let \(p_0\in H^1(\Omega )\cap M\) be defined by the problem (9.1). Then the non-local compatibility conditions are concluded in the following lemma.
Lemma 9.2
Under the assumptions of Lemma 9.1, if \(||\nabla \mathbf{u }_t(x,t)||_{L^2}\) and \(||\mathbf{b }(x,t)||_{H^1}\) remain bounded as \(t\longrightarrow 0\), then there must hold
Proof
As \(t\longrightarrow 0\), it follows from \(||A\mathbf{u }(x,t)-A\mathbf{u }_0(x)||_{L^2}\longrightarrow 0\) and \(||\mathbf{b }(x,t)-\mathbf{b }_0(x)||_{H^3}\longrightarrow 0\) that
If \(||\nabla \mathbf{u }_t(x,t)||_{L^2}\) and \(||\mathbf{b }(x,t)||_{H^1}\) remain bounded as \(t\longrightarrow 0\), then the convergences (9.5) and (9.6) hold weakly in \(\mathbf{H }^1(\Omega )\) and \(\mathbf{H }^2(\Omega )\), respectively. Thus,
hold weakly in \(\mathbf{H }^{1/2}(\partial \Omega )\) as \(t\longrightarrow 0\). The facts \(\mathbf{u }_t|_{\partial \Omega }=0\) and \(\nabla \mathbf{b }_t\cdot \mathbf{n }|_{\partial \Omega }=0\) for any \(t>0\) imply the desired non-local compatibility conditions (9.3) and (9.4).\(\square \)
Under these non-local compatibility conditions, we can estimate \(\mathbf{u }_t(0)\) and \(\mathbf{b }_t(0)\) in \(\mathbf{H }^1(\Omega )\) as follows.
Lemma 9.3
Let \(\mathbf{f }_0\in \mathbf{H }\) and \((\mathbf{u }_0, \pi )\in \mathbf{D }(A)\times H^1(\Omega )\cap M\) be determined by the Stokes problem (2.15). Under the assumptions of Lemma 9.1, there holds that \(\mathbf{u }_t(0)\) and \(\mathbf{b }_t(0)\) belong to \(\mathbf{H }^1(\Omega )\).
Proof
Taking \(t=0\) at (1.1) deduces to
Since \((\mathbf{u }_0, \pi )\in \mathbf{D }(A)\times H^1(\Omega )\cap M\) is determined by the Stokes problem (2.15), then (9.7) can be rewritten as
By using the following formula:
and applying \({\mathbb {P}}_{\mathbf{H }}\) to the resulting equation, we obtain
It follows from (2.2) that
Note \(\mathbf{u }_0\in \mathbf{D }(A)\) and \(\mathbf{b }_0\in \mathbf{H }^3(\Omega )\). Then we have
and
The above two estimates imply \(\mathbf{u }_t(0)\in \mathbf{H }^1(\Omega )\). Taking \(t=0\) in (1.2) leads to
By using a similar method, we can prove
and
which imply \(\mathbf{b }_t(0)\in \mathbf{H }^1(\Omega )\). \(\square \)
Based on the results derived in Lemma 9.3, we can show some regularities of \(\mathbf{u }_{tt}, p_t\) and \(\mathbf{b }_{tt}\) in next theorem.
Theorem 9.1
Under the assumptions of Theorem 2.1 and Lemma 9.3, suppose \(\mathbf{f }_t\in \mathbf{L }^\infty (0,T; \mathbf{V }_0^\prime )\cap \mathbf{L }^2(0,T;\mathbf{H })\), then we have
where \(T^\star \) is defined in Theorem 2.1.
Proof
Differentiating (1.1) with respect to t, we get
Multiplying (9.8) by \(\mathbf{u }_{tt}\) and integrating over \(\Omega \), we have
where we use \(\text{ div } \mathbf{u }_{tt}=0\). It follows from (2.10)–(2.13) that \(\mathbf{u }_{tt}\in \mathbf{L }^2(0,T^\star ;\mathbf{H })\) and \(\mathbf{u }_t\in \mathbf{L }^\infty (0,T^\star ;\mathbf{V })\) if we integrate the above inequality with respect to t from 0 to \(t\le T^\star \) and note that \(\mathbf{u }_t(0)\in \mathbf{H }^1(\Omega )\). By using a similar method, multiplying (9.8) by \(A\mathbf{u }_{t}\) and integrating over \(\Omega \), we have
which implies \(\mathbf{u }_t\in L^2(0,T^\star ;\mathbf{D }(A))\). Multiplying (9.8) by \(\mathbf{v }\in \mathbf{V }_{0}\) leads to
Then it is easy to show
which implies \(\mathbf{u }_{tt}\in \mathbf{L }^\infty (0,T^\star ;\mathbf{V }_0^\prime )\). To estimate \(\nabla p_t\), we use (9.8) to deduce that
From \(\mathbf{u }_{tt}\in \mathbf{L }^2(0,T^\star ;\mathbf{H })\) and \(\mathbf{u }_t\in L^2(0,T^\star ;\mathbf{D }(A))\) shown in the previous paragraph, it is easily seen that \(\nabla p_t\in \mathbf{L }^2(0,T^\star ; \mathbf{L }^2(\Omega ))\) after integrating the above inequality from 0 to \(t\le T^\star \) and using (2.10)–(2.13).
Differentiating (1.2) with respect to t yields
It follows from (2.4)–(2.9) that
which completes the proof of \(\mathbf{b }_{tt}\in \mathbf{L }^2(0,T^\star ;\mathbf{L }^2(\Omega ))\) by using (2.10)–(2.13). \(\square \)
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An, R., Su, J. Optimal Error Estimates of Semi-implicit Galerkin Method for Time-Dependent Nematic Liquid Crystal Flows. J Sci Comput 74, 979–1008 (2018). https://doi.org/10.1007/s10915-017-0479-7
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DOI: https://doi.org/10.1007/s10915-017-0479-7
Keywords
- Nematic liquid crystal model
- Linearized semi-implicit scheme
- Finite element method
- Optimal error estimates