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Research article

The error analysis for the Cahn-Hilliard phase field model of two-phase incompressible flows with variable density

  • Received: 20 September 2023 Revised: 10 November 2023 Accepted: 15 November 2023 Published: 23 November 2023
  • MSC : 35K51, 35K55, 65M12, 65M70

  • In this paper, we consider the numerical approximations of the Cahn-Hilliard phase field model for two-phase incompressible flows with variable density. First, a temporal semi-discrete numerical scheme is proposed by combining the fractional step method (for the momentum equation) and the convex-splitting method (for the free energy). Second, we prove that the scheme is unconditionally stable in energy. Then, the L2 convergence rates for all variables are demonstrated through a series of rigorous error estimations. Finally, by applying the finite element method for spatial discretization, some numerical simulations were performed to demonstrate the convergence rates and energy dissipations.

    Citation: Mingliang Liao, Danxia Wang, Chenhui Zhang, Hongen Jia. The error analysis for the Cahn-Hilliard phase field model of two-phase incompressible flows with variable density[J]. AIMS Mathematics, 2023, 8(12): 31158-31185. doi: 10.3934/math.20231595

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  • In this paper, we consider the numerical approximations of the Cahn-Hilliard phase field model for two-phase incompressible flows with variable density. First, a temporal semi-discrete numerical scheme is proposed by combining the fractional step method (for the momentum equation) and the convex-splitting method (for the free energy). Second, we prove that the scheme is unconditionally stable in energy. Then, the L2 convergence rates for all variables are demonstrated through a series of rigorous error estimations. Finally, by applying the finite element method for spatial discretization, some numerical simulations were performed to demonstrate the convergence rates and energy dissipations.



    In this paper, we focus on the following hydrodynamically consistent Cahn-Hilliard phase field model for incompressible two-phase flows with variable density

    ρt+(ρu)=0, (1.1a)
    ϕt+uϕγΔw=0, (1.1b)
    w=Δϕ+1ε2(ϕ3ϕ), (1.1c)
    ρ(ut+uu)ηΔu+pλwϕ=0, (1.1d)
    u=0, (1.1e)

    where ϕ, w, u and p are the phase field function, the chemical potential, the velocity of flow and pressure respectively. Moreover, ρ=12(ρ1+ρ2)+ϕ2(ρ1ρ2) is the density, where ρ1 and ρ2 are the densities of the two fluids. The parameters γ, ε, η and λ are the mobility parameter related to the relaxation time scale, the interface thickness [1], the viscosity of the field, and the mixing energy density respectively. The system (1.1) is supplemented with appropriate boundary and initial conditions which are given as

    ϕ(x,0)=ϕ0(x), ρ(x,0)=ρ0(x), u(x,0)=u0(x),ϕ|Ω=0,         u|Ω=0.

    This model is also called the Cahn-Hilliard-Navier-Stokes model when the density is constant, which has many practical applications in physical and engineering, such as wetting, coating, and painting. By considering the influence of variable density, this model has broad applicability, which include highly stratified flows, interfaces between fluids of different densities and some problems of inertial confinement.

    The phase field method which has a wide range of applications is one of the main methods to deal with the fluid interface in two-phase flow modeling; see [2,3,4,5] and the references therein. It was initially developed to simulate solid-liquid phase transitions, where the interface is treated as a thin, smooth transition layer[6,7,8] to remove the singularities between two phases. The basic framework is to use phase field variables to represent the volume fraction of fluid components and then adopt a variational form to derive the model. Recently, it has increasingly attracted researchers' interest, mainly because the phase field method is superior to other available methods in some aspects of two-phase flow [9]. Various material properties or complex interface behaviors can be simulated directly by introducing suitable energy functions. Numerical solutions play a crucial role in their study and applications because the analytic solutions are usually not available.

    A few works have been devoted to the design, analysis and implementation of numerical schemes for the phase field model, although this model is very classical and canonical. We briefly review the available methods used in the phase field model. It is worth noting that there are projection/gauge/penalty methods [10,11,12], scalar auxiliary variables [13,14], linear stability [15,16], convex splitting [17,18,19], invariable energy quadratization(IEQ) [20,21], nonlinear quadratic [22], exponential time differencing [23], etc. In practical applications, we often couple the flow-field equation with the phase field equation. Typically two-phase incompressible flow models are coupled to phase-field models. The Cahn-Hilliard model is taken into account in this paper, because it is effective in the following two aspects: (ⅰ) Cahn-Hilliard models can accurately conserve the volume and dynamics; (ⅱ) the equation is one of the most important models in mathematical physics. Because of these reasons, here we use the Cahn-Hilliard equation developed in [24] to couple the incompressible flows with variable density.

    There also have been a lot of works on numerical approximates for the Cahn-Hilliard phase field model for incompressible two-phase flows with variable density. Hohenberg and Halperin proposed the model in [25] to simulate two incompressible viscous fluids with constant density. In [26], Gurtin et al. obtained the equal model by using the framework of rational continuum mechanics. A fully adaptive energy stabilization scheme is proposed in [27]. An efficient Picard iteration procedure was designed in [28] to further decouple the model. In the last years, many authors have been concerned with designing incompressible two-phase flow models with variable densities. Several efficient and energy-stable time discretization schemes for the coupled nonlinear Cahn-Hilliard phase field system with variable density are constructed; see [29]. Yang and Dong presented an energy-stable scheme in [30] for the numerical approximation of the two-phase governing equations with variable density and viscosity for the two fluids by introducing a scalar-valued variable related to the total of the kinetic energy and the potential free energy. A second-order accurate, coupled, energy-stable schemeis proposed in [31], where the Crank-Nicolson method and the IEQ method were used. In [32], the conservation scheme of the first-order energy law was established, in which the Cahn-Hilliard solver was used to decouple from the two-phase incompressible flows solver through the use of the fractional step method. Ye et al.[33] have designed a fully-decoupled type scheme to solve the Cahn-Hilliard phase field model for a two-phase incompressible fluid flow system with constant density. They only give a detailed practical implementation method and also prove the solvability. None of the various existing schemes have been subjected to error analysis, where the main difficulty lies in the delicate treatment of a several of nonlinear terms. Rigorous error estimates of models with variable densities, using an optimal order error bound, may seem to be a difficult prospect, but is a very interesting direction for future research.

    In this paper, we finally arrive at an unconditionally stable in energy, first-order time-accurate scheme for the incompressible Cahn-Hilliard two phase flows with variable density by coming up with a fractional step method. This method has an advantage over the projection method in that the original boundary conditions of the problem can be implemented in all substeps of the scheme. The popular approach to discretizing the Cahn-Hilliard phase field model (1.1b) and (1.1c) in time is based on the convex-splitting of the free energy functional, i.e., an idea that can be traced back to [32]. In the convex-splitting framework, one treats the contribution from the convex part implicitly and the contribution from the concave part explicitly. This treatment promotes the energy stability of the scheme and this property is unconditional in terms of time steps. We also give a rigorous proof of the convergence results and error estimates in the theoretical analysis. The main contribution of this paper is a rigorous error analysis, particularly under the condition that energy stability is available. To the best of the authors' knowledge, the proof developed in this article is the first to have the description of error estimation. The accuracy and stability are also demonstrated through the simulation of various numerical examples, where the challenge is in creating an efficient and easy to implement numerical scheme that preserves the energy dissipation law.

    The rest of this article is organized as follows. In Section 2, we construct an efficient time discrete scheme for variable density and derive unconditional energy stability. In Section 3, the error analysis of the semi-discrete scheme in time is provided. Some numerical experimentations are given in Section 4. Finally, conclusions are drawn in Section 5.

    For the sake of simplicity, some notations are needed for the following content. We assume that the domain ΩR2 is open, sufficiently smooth and bounded. For any two functions ϕ(x) and ψ(x), their L2 inner product on Ω is denoted by (ϕ,ψ)=Ωϕ(x)ψ(x)dx, and the L2 norm of ϕ(x) is denoted by ϕ=(ϕ,ϕ)12. Let τ>0 be the time step size and set tn=nτ for 0nN with T=Nτ. Moreover, we introduce the following spaces,

    H={uL2(Ω)|divu in Ω, un=0 on Ω},V=H10, V0={uV|divu=0 in Ω},M=L20(Ω)={qL2(Ω)|Ωqdx=0},

    where n is the outward normal vector of Ω. Next, we reformulate the system (1.1) as follows:

    ρn+1ρnτ+ρn+1un=0, (2.1a)
    ϕn+1ϕnτ+˜un+1ϕnγΔwn+1=0, (2.1b)
    wn+1=1ε2((ϕn+1)3ϕn)Δϕn+1, (2.1c)
    ρn˜un+1unτηΔ˜un+1+ρn+1(un)˜un+1λwn+1ϕn+14ρn+1(un)˜un+1=0, (2.1d)
    ρn+1un+1˜un+1τη(Δun+1Δ˜un+1)+pn+1=0, (2.1e)
    un+1=0 (2.1f)

    Remark 1. In (2.1d), the term 14ρn+1(un)˜un+1 is added to obtain the unconditional stability, as it is 0 if un=0.

    Theorem 1. (Stabilityofρ) For any τ>0 and any sequence {un}n=0,,N satisfying the boundary condition unn=0 on Ω, the solution {ρn}n=1,,N to (2.1a) satisfies

    ρN2+N1n=1ρn+1ρN2=ρ02. (2.2)

    Proof. Testing (2.1a) with 2τρn+1 gives

    ρn+12ρn2+ρn+1ρn2+2τΩ(unρn+1+12ρn+1un)ρn+1dx=0. (2.3)

    Owing to the boundary conditions on un, we note that

    Ω(unρn+1+12ρn+1un)ρn+1dx=12Ω(|ρn+1|2un)dx=12Ω|ρn+1|2unndx=0. (2.4)

    Thus, we get

    ρn+12+ρn+1ρn2=ρn2.

    Summing all indices n ranging 0 to N1, the proof is completed.

    Next, the stability of (2.1b)–(2.1e) will be proved in the theorem below. Moreover, since the kinetic energy of the fluid is 12ρnun2, it is more suitable to establish a bound based on ρnun2 than on the velocity itself. For simplicity, let us say that σn=ρn for all 1nN and σ0=ρ0.

    Theorem 2. (Stabilityofenergy) For any τ>0, (2.1b)–(2.1e) satisfy the the conditions of following energy estimates:

    σNuN2+λϕN2λε2ϕN2+λ2ε2ϕN4L4+2τγλN1n=0wn+12+λN1n=0((ϕn+1ϕn)2+12ε2ϕn+1ϕn2)+λ2ε2N1n=0(ϕn+12ϕn)2+N1n=0(σn(˜un+1un)2+σn+1(un+1˜un+1)2)+ητN1n=0(un+12+˜un+12+(un+1˜un+1)2)+λε2N1n=0(ϕn+1(ϕn+1ϕn)2)=σ0u02+λϕ02λε2ϕ02+λ2ε2ϕ04L4.

    Proof. Multiplying (2.1b) by 2τλwn+1 and integrating over Ω, we get

    2λ(ϕn+1ϕn,wn+1)+2τλγwn+12+2τλΩ˜un+1ϕnwn+1dx=0. (2.5)

    Multiplying (2.1c) by 2λ(ϕn+1ϕn) yields

    2λ(ϕn+1ϕn,wn+1)=λ2ε2(ϕn+14L4ϕn4L4)+λ2ε2(ϕn+12ϕn2)2+λε2ϕn+1(ϕn+1ϕn)2+λε2(ϕn2ϕn+12+ϕn+1ϕn2)+λ(ϕn+12ϕn2+(ϕn+1ϕn)2), (2.6)

    where we use the following identity:

    2a(ab)=a2b2+(ab)2,a3(ab)=14(a4b4+(a2b2)2+2a2(ab)2). (2.7)

    Testing (2.1a) with τ|˜un+1|2 leads to

    σn+1˜un+12σn˜un+12+τΩρn+1un|˜un+1|2dx+τ2Ωρn+1(un)|˜un+1|2dx=0. (2.8)

    By taking the inner product of (2.1d) with 2τ˜un+1, we have

    σn˜un+12σnun2+σn(˜un+1un)22τλΩwn+1ϕn˜un+1dx+2τη˜un+12+τΩρn+1(un)|˜un+1|2dx+τ2Ωρn+1(un)|˜un+1|2dx=0. (2.9)

    Testing (2.1e) with 2τun+1 yields

    σn+1un+12σn+1˜un+12+σn+1(un+1˜un+1)2+ητun+12+ητ˜un+12+ητ(un+1˜un+1)2=0. (2.10)

    Taking into account the boundary condition on un and using integration by parts, we have

    Ωρn+1un|˜un+1|2dx+Ωρn+1(un)|˜un+1|2dx+Ωρn+1un|˜un+1|2dx=Ω(ρn+1un|˜un+1|2)dx=Ωρn+1un|˜un+1|2ndx=0. (2.11)

    Summing the above inequality, we arrive at

    2τλγwn+12+λ2ε2(ϕn+14L4ϕn4L4)+λ2ε2(ϕn+12ϕn2)2+λε2ϕn+1(ϕn+1ϕn)2+λε2(ϕn2ϕn+12+ϕn+1ϕn2)+λ(ϕn+12ϕn2+(ϕn+1ϕn)2)+σn+1un+12σnun2+σn(˜un+1un)2+σn+1(un+1˜un+1)2+ητun+12+ητ˜un+12+ητ(un+1˜un+1)2=0. (2.12)

    Adding up the above inequality from n=0 to N1, we obtain Theorem 2.3.

    The bound on the pressure p is proved in the following theorem.

    Theorem 3. (Stabilityofp) For any τ>0, the solution pn+1 to (2.1e) satisfies the following inequality:

    τ2N1n=0pn+12C(σ0u02+λϕ02λε2ϕ02+λ2ε2ϕ04L4)(ρ0+1). (2.13)

    Proof. Under the inf-sup condition, there exists a positive constant β such that

    βpn+1supvV,v0(v,pn+1)v. (2.14)

    Testing (2.14) with all vV leads to

    (v,pn+1)=η((un+1˜un+1),v)+τ1(ρn+1(un+1˜un+1),v)η(un+1+˜un+1)v+τ1σn+1(un+1˜un+1)L2σn+1L3vL6. (2.15)

    Given that ρnρ0 for all 1nN, and by using Hölder's inequality, we have

    σn+1L3=(pn+132)13Cρn+112Cρ012. (2.16)

    Then, by the Sobolev embedding inequality vL6CvL2 for any vV, we get

    (v,pn+1)η(un+1+˜un+1)v+Cτ1σn+1(un+1˜un+1)ρ012v. (2.17)

    Substituting the above inequalities into Eq (2.14), we obtain

    βpn+1η(un+1+˜un+1)+Cτ1σn+1(un+1˜un+1)ρ012. (2.18)

    And by using Theorem 2.2, we get the desired result.

    In this section, we will give the time error estimates and show that the scheme has a first-order convergence rate. Although we verified that the scheme (2.1) is unconditionally stable in the previous chapter, we need to make the following assumptions[34,35] when conducting temporal error analysis:

    {ρn}n=0,,N is uniformly bounded in L, (3.1)
    for all n=0,,N, it holds that ρnχ a.e. in Ω, (3.2)

    where χ is a number in (0,ρmin0].

    We assume that the exact solution (ρ,u,ϕ,w,p) is sufficiently smooth. To be more precise,

    ρH2(0,T;L2(Ω))L2(0,T;W1,(Ω)),ϕL(0,T;H3(Ω))W1,(0,T;H2(Ω))W2,(0,T;H1(Ω))W3,(0,T;L2(Ω)),wL(0,T;H2(Ω))W1,(0,T;L2(Ω)),uH2(0,T;L2(Ω))L(0,T;VH2(Ω)),pW2,(0,T;H1(Ω)). (3.3)

    We denote

    enϕ=ϕ(tn)ϕn,enw=w(tn)wn,enρ=ρ(tn)ρn,enu=u(tn)un,˜enu=u(tn)˜un,enp=p(tn)pn.

    In (1.1), taking t=tn+1 and subtracting from (2.1), we get the following error equations:

    en+1ρenρτ+en+1ρu(tn+1)+ρn+1(u(tn+1)u(tn))+ρn+1enu=Rn+1ρ, (3.4a)
    en+1ϕenϕτγen+1w+u(tn+1)ϕ(tn+1)˜un+1ϕn=Rn+1ϕ, (3.4b)
    en+1w+en+1ϕ=1ε2(ϕ(tn+1)3(ϕn+1)3ϕ(tn+1)+ϕn), (3.4c)
    ρn˜en+1uenuτη˜en+1u+p(tn+1)+ρ(tn+1)(u(tn+1))u(tn+1)ρn+1(un)˜un+1λw(tn+1)ϕ(tn+1)+λwn+1ϕn=Rn+1u, (3.4d)
    ρn+1en+1u˜en+1uτη(en+1u˜en+1u)pn+1=0, (3.4e)
    en+1u=0, (3.4f)

    where

    Rn+1ϕ=ϕ(tn+1)ϕ(tn)τϕt(tn+1),Rn+1ρ=ρ(tn+1)ρ(tn)τρt(tn+1),Rn+1u=ρnu(tn+1)u(tn)τρ(tn+1)ut(tn+1).

    If the exact solution is sufficiently smooth, it is easy to establish the following estimate of the truncation error.

    Lemma 1. Under the regularity assumptions given by (3.3), the truncation errors satisfy:

    Rn+1ρ2Cτtn+1tnρtt(t)2dtCτ2,Rn+1ϕ2Cτtn+1tnϕtt(t)2dtCτ2,Rn+1u2Cτtn+1tn(utt(t)2+ρt(t)2)dt+Cenρ2Cτ2+Cenρ2,

    for all 0nN1.

    Proof. By using the integral residual of the Taylor formula, we have

    Rn+1ρ=1τtn+1tn(ttn)ρtt(t)dt. (3.5)

    By Hölder's inequality, we can derive

    Rn+1ρ2=Ω(1τtn+1tn(ttn)ρtt(t)dt)2dx1τ2Ωtn+1tn(ttn)2dt122tn+1tnρtt(t)2dt122dx1τ2(τ33tn+1tnΩρtt(t)2dxdt)Cτtn+1tnρtt(t)2dtCτ2. (3.6)

    Similarly, we can prove the inequality of Rn+1ϕ. For Rn+1u, we can rewrite

    Rn+1u=ρn(u(tn+1)u(tn)τut(tn+1))(enρ+tn+1tnρt(t)dt)ut(tn+1)=ρnτtn+1tn(ttn)utt(t)dt(enρ+tn+1tnρt(t)dt)ut(tn+1). (3.7)

    Using Rn+1ρ estimation and Hölder's inequality can yield the result for Rn+1u.

    We introduce the following Gronwall's inequality, which will frequently be used in error estimates.

    Lemma 2. Let ak, bk, ck and γk, for integers k0, be the nonnegative numbers such that

    an+τnk=0bkτnk=0γkak+τnk=0ck+B  for  0.

    Suppose that τγk<1, for all k, and set σk=(1τγk)1. Then,

    an+τnk=0bkexp(τnk=0γkσk)(τnk=0ck+B)  for  0.

    We verify Lemma 5 by the following lemma.

    Lemma 3. Define

    Gn+1c=(ϕ(tn+1))3(ϕn+1)3=3(ϕ(tn+1))2en+1ϕ3ϕ(tn+1)(en+1ϕ)2+(en+1ϕ)3. (3.8)

    Then, for n<Tτ1, we have

    Gn+1cCen+1ϕH1,en+1ϕH2C(τ+en+1w+en+1ϕH1+enϕ). (3.9)

    Proof. For Gn+1c, we use (3.8) to conclude that

    Gn+1cC(en+1ϕϕ(tn+1)2L+en+1ϕ2L4ϕ(tn+1)L+en+1ϕ3L6)C(en+1ϕϕ(tn+1)2L+en+1ϕ2H1ϕ(tn+1)L+en+1ϕ3H1)Cen+1ϕH1, (3.10)

    where we have used the a priori bound ϕnH1C implied by the stability result given by Theorem 2. Using the H2 regularity results for elliptic equations, we conclude that

    en+1ϕH2c(en+1ϕL2+en+1ϕL2);

    from (3.4c), we know that

    en+1ϕ=en+1w+1ε2Gn+1c1ε2enϕtn+1tnϕt(t)dt;

    thus,

    en+1ϕH2C(en+1ϕ+en+1w+Gn+1c+enϕ+τ)C(τ+en+1w+en+1ϕH1+enϕ).

    The error estimate for the discrete density ρn+1 is derived in the following lemma.

    Lemma 4. Suppose that the solution to (1.1) satisfies the regularity assumptions given by (3.3), and suppose that (3.1)–(3.2) hold. Then, we have

    en+1ρ2+2nm=0em+1ρemρ2C(τ2+τnm=0σmemu2) (3.11)

    for all 0nN1.

    Proof. Multiplying (3.4a) by 2τen+1ρ and integrating over Ω, we have

    en+1ρ2enρ2+en+1ρenρ2=2τ(en+1ρu(tn+1),en+1ρ)+2τ(Rn+1ρ,en+1ρ)2τ(ρn+1(u(tn+1)u(tn)),en+1ρ)2τ(ρn+1enu,en+1ρ)=4i=1Ki. (3.12)

    Using u(tn+1)=0 in Ω and u(tn+1)=0 on Ω, we have

    K1=12Ω|en+1ρ|2u(tn+1)nds=0. (3.13)

    By using Young's inequality, the Cauchy-Schwarz inequality and Lemma 1, we have

    K22τRn+1ρen+1ρCτRn+1ρ2+ετen+1ρ2Cτ3+ετen+1ρ2,

    where ε>0 is a sufficiently small constant.

    Then, by the Sobolev inequality and Young inequality, we get

    K3=2τ(ρ(tn+1)(u(tn+1)u(tn)),en+1ρ)+2τ(en+1ρ(u(tn+1)u(tn)),en+1ρ)=2τ(ρ(tn+1)(u(tn+1)u(tn)),en+1ρ)2τρ(tn+1)Ltn+1tnut(t)dten+1ρετen+1ρ2+Cτ2tn+1tnut(t)2dtCτ3+ετen+1ρ2. (3.14)

    Similarly, the last term can be estimated as follows:

    K42τρ(tn+1)L1σnLσnenuen+1ρετen+1ρ2+Cτσnenu2. (3.15)

    If ε is sufficiently small such that ετ16, substituting the estimates of Ki (1i4) into (3.12), we have

    en+1ρ2enρ2+2en+1ρenρ2Cτ3+Cτenρ2+Cτσnenu2. (3.16)

    Using the discrete Gronwall inequality, we obtain the desired result.

    Lemma 5. Suppose that the solution to (1.1) satisfies the regularity assumptions given by (3.3), and suppose that (3.1)–(3.2) are valid. For sufficiently small τ, there are the following error estimates:

    τγN1n=0(λen+1w2+en+1w2)+eNϕ2+λeNϕ2+λ2ε2eNϕ4L4+σNeNu2+N1n=0(12σn(˜en+1uenu)2+σn+1(en+1u˜en+1u)2)+τηN1n=0(en+1u2+12˜en+1u2+(en+1u˜en+1u)2)Cτ. (3.17)

    Proof. Let us multiply (3.4a) by τ|˜en+1u|2, (3.4b) by 2τen+1ϕ and 2τλen+1w, (3.4c) by 2τγen+1w and 2λ(en+1ϕenϕ), (3.4d) by 2τ˜en+1u and (3.4e) by 2τen+1u. Summing up all of the above equations, we have

    2τγλen+1w2+en+1ϕ2enϕ2+en+1ϕenϕ2+2τγen+1w2+λen+1ϕ2λenϕ2+λen+1ϕenϕ2σn+1en+1u2σnenu2+σn(˜en+1uenu)2+σn+1(en+1u˜en+1u)2+τηen+1u2+τη˜en+1u2+τη(en+1u˜en+1u)2=2τ(ρ(tn+1)(u(tn+1))u(tn+1),˜en+1u)+2τ(ρn+1(un)˜un+1,˜en+1u)τ(ρn+1un,|en+1u|2)2τ(p(tn+1),˜en+1u)+2τ(Rn+1u,˜en+1u)+2τλ(w(tn+1)ϕ(tn+1)wn+1ϕn,˜en+1u)2τλ(u(tn+1)ϕ(tn+1)˜un+1ϕn,en+1w)+2τλ(Rn+1ϕ,en+1w)2τ(u(tn+1)ϕ(tn+1)˜un+1ϕn,en+1ϕ)+2τ(Rn+1ϕ,en+1ϕ)+2τγε2(ϕ(tn+1)3(ϕn+1)3,en+1w)2τγε2(ϕ(tn+1)ϕn,en+1w)2λε2(ϕ(tn+1)3(ϕn+1)3,en+1ϕenϕ)+2λε2(ϕ(tn+1)ϕn,en+1ϕenϕ)=i=14i=1Ai. (3.18)

    Thanks to un=0 in Ω, we have

    A2+A3=2τ(ρn+1(un)˜un+1,˜en+1u)τ(ρn+1un,|˜en+1u|2)=2τ(ρn+1(un)u(tn+1),˜en+1u)τ((unρn+1|˜en+1u|2),1)=2τ(ρn+1(un)u(tn+1),˜en+1u). (3.19)

    Therefore, we get

    A1+A2+A3 (3.20)
    =2τ(ρn+1(un)u(tn+1)ρ(tn+1)(u(tn+1))u(tn+1),˜en+1u)=2τ(ρn+1(enu)u(tn+1),˜en+1u)2τ(en+1ρ(u(tn))u(tn+1),˜en+1u)2τ(ρ(tn+1)((u(tn+1)u(tn)))u(tn+1),˜en+1u)Cτρn+1L1σnLσnenuu(tn+1)L3˜en+1uL6 (3.21)
    +Cτen+1ρu(tn)Lu(tn+1)L3˜en+1uL6+Cτρ(tn+1)Ltn+1tnut(t)dtu(tn+1)L3˜en+1uL6Cτ(σnenu+en+1ρ+τtn+1tnut(t)dt)˜en+1uητ12˜en+1u2+Cτ(σnenu2+en+1ρ2+τ2)Cτ3+ητ12˜en+1u2+Cτσnenu2+Cτ2nm=0σmemu2, (3.22)

    where the Sobolev embedding inequality vL6CvL2 for any vV is used.

    For A4, we have

    A42τp(tn+1)1σnLσn(˜en+1uenu)Cτ2+12σn(˜en+1uenu)2. (3.23)

    Using the Poincarˊe inequality and Cauchy-Schwarz inequality, we can deal with A6, A7 and A9:

    A6=2τλ(w(tn+1)ϕ(tn+1)wn+1ϕn,˜en+1u)=2τλ(w(tn+1)(ϕ(tn+1)ϕ(tn)),˜en+1u)+2τλ(w(tn+1)enϕ,˜en+1u)+2τλ(ϕnen+1w,˜en+1u)2τλw(tn+1)Ltn+1tnϕtdt˜en+1u+2τλw(tn+1)Lenϕ˜en+1u+2τλ(ϕnen+1w,˜en+1u)Cτ3+ητ12˜en+1u2+Cτenϕ2+2τλ(ϕnen+1w,˜en+1u),A7=2τλ(u(tn+1)ϕ(tn+1)˜un+1ϕn,en+1w)=2τλ(u(tn+1)(ϕ(tn+1)ϕ(tn))+u(tn+1)enϕ+ϕn˜en+1u,en+1w)2τλu(tn+1)Ltn+1tnϕtdten+1w+2τλu(tn+1)Lenϕen+1w2τλ(ϕn˜en+1u,en+1w)Cτ3+τγ3en+1w2+Cτenϕ22τλ(ϕn˜en+1u,en+1w), (3.24)

    and

    A9=2τ(u(tn+1)ϕ(tn+1)˜un+1ϕn,en+1ϕ)2τu(tn+1)Ltn+1tnϕtdten+1ϕ+2τu(tn+1)Lenϕen+1ϕ+2τϕn˜en+1uL6en+1ϕL6Cτ3+Cτen+1ϕ2+Cτenϕ2+ητ12˜en+1u2. (3.25)

    From Lemma 1, we have

    A5+A8+A102τRn+1u˜en+1u+2τλRn+1ϕen+1w+2τRn+1ϕen+1ϕCτ3+Cτ2nm=0σmemu2+ητ12˜en+1u2+τγ3en+1w2+12en+1ϕ2. (3.26)

    From Lemma 3, we obtain

    A11+A12=2τγε2(Gn+1cenϕtn+1tnϕt(t)dt,en+1w)2τγε2(enϕ+Gn+1c+tn+1tnϕt(t)dt)en+1wCτ3+Cτ(enϕ2+en+1ϕ2+en+1ϕ2)+τγ3en+1w2, (3.27)

    and

    A13+A14=2λε2(Gn+1cenϕtn+1tnϕt(t)dt,en+1ϕenϕ)=λ2ε2(en+1ϕ4L4enϕ4L4+(en+1ϕ)2(enϕ)22+2en+1ϕ(en+1ϕenϕ)2)2λε2en+1ϕenϕ22λε2(3(ϕ(tn+1))2en+1ϕ3ϕ(tn+1)(en+1ϕ)2en+1ϕ,en+1ϕenϕ)+2λε2(tn+1tnϕt(t)dt,en+1ϕenϕ), (3.28)

    where we use this identity

    a3(ab)=14(a4b4+(a2b2)2+2a2(ab)2). (3.29)

    We denote

    ˜Gn+1=2λε2(3(ϕ(tn+1))2en+1ϕ3ϕ(tn+1)(en+1ϕ)2en+1ϕ) (3.30)

    Similar to the method estimated from Lemma 3, we can obtain

    ˜Gn+1Cen+1ϕH1. (3.31)

    Taking the gradient of ˜Gn+1, we get

    ˜Gn+1=2λε2[(3(ϕ(tn+1))21)en+1ϕ+6ϕ(tn+1)en+1ϕϕ(tn+1)3(en+1ϕ)2ϕ(tn+1)6ϕ(tn+1)en+1ϕen+1ϕ]. (3.32)

    Since H2(Ω)L(Ω) and by utilizing the bound of en+1ϕL2 implied by Theorem 2, we conclude that

    en+1ϕen+1ϕL2en+1ϕLen+1ϕL2Cen+1ϕH2. (3.33)

    In view of (3.10) and the bound ϕnH1<C, we have

    ˜Gn+1C[(ϕ(tn+1)2L+1)en+1ϕL2+ϕ(tn+1)Lϕ(tn+1)L3en+1ϕL6+ϕ(tn+1)L6en+1ϕ2L6+ϕ(tn+1)Len+1ϕen+1ϕL2]C(en+1ϕL2+en+1ϕH1+en+1ϕ2H1+en+1ϕen+1ϕL2)C(τ+en+1w+en+1ϕ+en+1ϕ+enϕ). (3.34)

    Dealing with the penultimate term of (3.28) and in view of (3.31) and (3.33), we get

    (˜Gn+1,en+1ϕenϕ)=τ(˜Gn+1,γΔen+1wu(tn+1)enϕ˜en+1uϕn)+τ(˜Gn+1,u(tn+1)tn+1tnϕt(t)dt+Rn+1ϕ)γτen+1w˜Gn+1+τϕn˜en+1uH1˜Gn+1H1+τ˜Gn+1(Rn+1ϕ+u(tn+1)Lenϕ+u(tn+1)Ltn+1tnϕt(t)dt)Cτ3+γλτ2en+1w2+ητ12˜en+1u2+Cτ(en+1w2+en+1ϕ2+en+1ϕ2+enϕ2+enϕ2). (3.35)

    Then, we estimate the last term of (3.28),

    2λε2(tn+1tnϕt(t)dt,en+1ϕenϕ)=2λτε2(tn+1tnϕt(t)dt,γΔen+1wu(tn+1)enϕ˜en+1uϕn)+2λτε2(tn+1tnϕt(t)dt,u(tn+1)tn+1tnϕt(t)dt+Rn+1ϕ)2λγτε2tn+1tnϕt(t)dten+1w+2λτε2ϕn˜en+1uH1tn+1tnϕt(t)dtH1+2λτε2tn+1tnϕt(t)dt(Rn+1ϕ+u(tn+1)Lenϕ)+2λτε2tn+1tnϕt(t)dtu(tn+1)Ltn+1tnϕt(t)dtCτ3+γλτ2en+1w2+ητ12˜en+1u2+Cτenϕ2. (3.36)

    Combining (3.35) and (3.36), we can obtain

    A13+A14λ2ε2(en+1ϕ4L4enϕ4L4+(en+1ϕ)2(enϕ)22+2en+1ϕ(en+1ϕenϕ)2)2λε2en+1ϕenϕ2+Cτ3+γλτen+1w2+ητ6˜en+1u2+Cτ(enϕ2+en+1w2+en+1ϕ2+en+1ϕ2+enϕ2+enϕ2). (3.37)

    Substituting the above estimates into (3.18), we have

    τγλen+1w2+12en+1ϕ2enϕ2+(1+2λε2)en+1ϕenϕ2+τγen+1w2+λen+1ϕ2λenϕ2+λen+1ϕenϕ2+σn+1en+1u2σnenu2+12σn(˜en+1uenu)2+σn+1(en+1u˜en+1u)2+τηen+1u2+τη2˜en+1u2+τη(en+1u˜en+1u)2+λ2ε2(en+1ϕ4L4enϕ4L4+(en+1ϕ)2(enϕ)22+2en+1ϕ(en+1ϕenϕ)2)Cτ2+Cτσnenu2+Cτ2nm=0σmemu2+Cτ(enϕ2+en+1w2+en+1ϕ2+en+1ϕ2+enϕ2+enϕ2). (3.38)

    Adding up from 0 to N1, and applying Gronwall's inequality, we infer that

    τγN1n=0(λen+1w2+en+1w2)+eNϕ2+λeNϕ2+λ2ε2eNϕ4L4+σNeNu2+N1n=0(12σn(˜en+1uenu)2+σn+1(en+1u˜en+1u)2)+τηN1n=0(en+1u2+12˜en+1u2+(en+1u˜en+1u)2)Cτ.

    Lemma 5 shows that the fractional scheme converges at a rate of O(τ12). However, since we use the first-order backward Euler method of time discretization, it is not optimal from the perspective of theoretical analysis. Next, we will improve the convergence speed to the first order.

    Lemma 6. Suppose that the solution to (1.1) satisfies the regularity assumptions given by (3.3), and suppose that (3.1)–(3.2) are valid. For sufficiently small τ, there are the following error estimates:

    τγN1n=0(λen+1w2+en+1w2)+eNϕ2+λeNϕ2+λ2ε2eNϕ4L4+σNeNu2+τηN1n=0en+1u2Cτ2. (3.39)

    Proof. Taking the sum of (3.4d) and (3.4e), we get

    ρnen+1uenuτ+ρn+1ρnτ(en+1u˜en+1n)ηen+1u+ρ(tn+1)(u(tn+1))u(tn+1)ρn+1(un)˜un+1+Δen+1pλw(tn+1)ϕ(tn+1)+λwn+1ϕn=Rn+1u. (3.40)

    Let us multiply (3.4a) by τ˜en+1u, (3.4b) by 2τen+1ϕ and 2τλen+1w, (3.4c) by 2τγen+1w and 2λ(en+1ϕenϕ) and (3.40) by 2τen+1u. Summing up all of the above equations, we obtain

    σn+1en+1u2σnenu2+σn(en+1uenu)2+2ητen+1u2+2τγλen+1w2+en+1ϕ2enϕ2+en+1ϕenϕ2+2τγen+1w2+λen+1ϕ2λenϕ2+λen+1ϕenϕ2=2τ(ρn+1un,˜en+1uen+1u)+τ(ρn+1un,|en+1u|2)2τ(ρ(tn+1)(u(tn+1))u(tn+1)ρn+1(un)˜un+1,en+1u)+2τλ(w(tn+1)ϕ(tn+1)wn+1ϕn,en+1u)+2τ(Rn+1u,en+1u)2τλ(u(tn+1)ϕ(tn+1)˜un+1ϕn,en+1w)+2τλ(Rn+1ϕ,en+1w)2τ(u(tn+1)ϕ(tn+1)˜un+1ϕn,en+1ϕ)+2τ(Rn+1ϕ,en+1ϕ)+2τrε2(ϕ(tn+1)3(ϕn+1)3,en+1w)2τrε2(ϕ(tn+1)ϕn,en+1w)2λε2(ϕ(tn+1)3(ϕn+1)3,en+1ϕenϕ)+2λε2(ϕ(tn+1)ϕn,en+1ϕenϕ)=13i=1Li. (3.41)

    Using integration by parts, we have

    L1+L2=2τ(ρn+1(un)˜en+1u,en+1u)+2τ(ρn+1(enu)en+1u,en+1u˜en+1u)2τ(ρn+1(u(tn))en+1u,en+1u˜en+1u); (3.42)

    we rewrite the L3 as

    L3=2τ(ρn+1(un)˜en+1u,en+1u)2τ(ρn+1((u(tn+1)u(tn)))u(tn+1),en+1u)2τ(ρn+1(enu)u(tn+1),en+1u)2τ(en+1ρ(u(tn+1))u(tn+1),en+1u); (3.43)

    then,

    L1+L2+L3=2τ(ρn+1(enu)en+1u,en+1u˜en+1u)2τ(en+1ρ(u(tn+1))u(tn+1),en+1u)2τ(ρn+1(u(tn))en+1u,en+1u˜en+1u)2τ(ρn+1((u(tn+1)u(tn)))u(tn+1),en+1u)2τ(ρn+1(enu)u(tn+1),en+1u)=5i=1Ji. (3.44)

    Lemma 5 shows that

    (en+1u˜en+1u)C. (3.45)

    According to Young's inequality and the Cauchy-Schwarz inequality, we can deduce that

    J1ητ10en+1u2+Cτenu2(en+1u˜en+1u)σn+1(en+1u˜en+1u)ητ10en+1u2+Cτ32enu2,J22τen+1ρu(tn+1)Lu(tn+1)L3en+1uL6ητ10en+1u2+Cτen+1ρ2Cτ3+ητ10en+1u2+Cτ2nm=0σmemu2,J3Cτen+1uu(tn)Lσn+1(en+1u˜en+1u)Cτ3+ητ10en+1u2+Cτσn+1(en+1u˜en+1u)2,J42τρn+1Lu(tn+1)Ltn+1tnutdten+1uCτ3+ητ10en+1u2,J52τρn+1Lu(tn+1)L1σnLσnenuen+1uCτσnenu2+ητ10en+1u2. (3.46)

    Therefore, we have

    M1+M2+M3Cτ3+Cτ32enu2+ητ2en+1u2+Cτσnenu2+Cτ2nm=0σmemu2+Cτσn+1(en+1u˜en+1u)2. (3.47)

    Using the embedding inequality and Cauchy-Schwarz inequality, we can deal with M4M9

    M4=2τλ(w(tn+1)ϕ(tn+1)wn+1ϕn,en+1u)=2τλ(w(tn+1)(ϕ(tn+1)ϕ(tn))+w(tn+1)enϕ+ϕnen+1w,en+1u)2τλw(tn+1)Ltn+1tnϕt(t)dten+1u+2τλw(tn+1)Lenϕen+1u+2τλϕnL2en+1wL3en+1uL6Cτ3+τηen+1u2+Cτenϕ2+Cτen+1w2, (3.48)
    M5+M7+M9=2τ(Rn+1u,en+1u)+2τλ(Rn+1ϕ,en+1w)+2τ(Rn+1ϕ,en+1ϕ)2τRn+1uen+1u+2τλRn+1ϕen+1w+2τRn+1ϕen+1ϕCτ3+Cτ2nm=0σmemu2+τλγ4en+1w2+14en+1ϕ2, (3.49)
    M6=2τλ(u(tn+1)ϕ(tn+1)˜un+1ϕn,en+1w)=2τλ((˜en+1uen+1u)ϕ(tn+1),en+1w)2τλ(en+1uϕ(tn+1),en+1w)2τλ(˜un+1tn+1tnϕt(t)dt,en+1w)2τλ(˜un+1enϕ,en+1w)2τλ1σn+1Lσn+1(˜en+1uen+1u)ϕ(tn+1)Len+1w+2τλ1σn+1Lσn+1en+1uϕ(tn+1)Len+1w+2τλtn+1tnϕt(t)dtL2˜un+1L3en+1wL6+2τλenϕ˜un+1L3en+1wL6Cτ3+τλ2en+1w2+Cτσn+1en+1u2+τλγ4en+1w2+Cτenϕ2+Cτσn+1(en+1u˜en+1u)2, (3.50)

    and

    M8=2τ(u(tn+1)ϕ(tn+1)˜un+1ϕn,en+1ϕ)Cτ3+14en+1ϕ2+Cτσn+1en+1u2+λ2en+1ϕ2+Cτenϕ2. (3.51)

    Next, we estimate M10+M11 and M12+M13 as follows:

    M10+M11Cτ3+Cτ(enϕ2+en+1ϕ2+en+1ϕ2)+τγ2en+1w2. (3.52)
    \begin{align} &\left(\tilde{G}^{n+1}, e_{\phi}^{n+1}-e_{\phi}^{n}\right)\\ = &\tau\left(\tilde{G}^{n+1}, \gamma \triangle e_w^{n+1}-u(t_{n+1} ) \nabla \phi(t _{n+1})+\tilde{\boldsymbol{u}}^{n+1} \nabla \phi^{n}+R_{\phi}^{n+1}\right)\\ = &\tau\left(\tilde{G}^{n+1}, \gamma \triangle e_w^{n+1}-\left(\tilde{e}_{u}^{n+1}-e_{u}^{n+1}\right) \nabla \phi\left(t_{n+1}\right)-e_{u}^{n+1} \nabla \phi(t _{n+1})-\tilde{\boldsymbol{u}}^{n+1} \int_{t _n}^{t _{n+1}} \nabla \phi_t(t) d t\right)\\ &-\tau\left(\tilde{G}^{n+1}, \tilde{\boldsymbol{u}} \nabla e_{\phi}^{n}-R_{\phi}^{n+1}\right) \\ \leq &C \tau^{3}+C\tau\left\|\tilde{G}^{n+1}\right\|^{2}+\frac{\tau\gamma \lambda}{4}\left\|\nabla e_{w}^{n+1}\right\|^{2}+C\tau\left\|\nabla \tilde{G}^{n+1}\right\|^{2}+\frac{1}{4}\left\|\sigma^{n+1} e_u^{n+1}\right\|^{2}+C \tau\left\|\nabla e_{\phi}^{n}\right\|^{2}\\ &+C\tau\left\|\sigma^{n+1}\left(e_{u}^{n+1}- \tilde{e}_{u}^{n+1} \right)\right\|^2\\ \leq& C\tau^{3}+\frac{\tau \gamma \lambda}{4}\left\|\nabla e_{w}^{n+1}\right\|^{2}+\frac{1}{4}\left\|\sigma^{n+1} e_{u}^{n+1}\right\|^{2}+C\tau\left\|\sigma^{n+1}\left(e_{u}^{n+1}- \tilde{e}_{u}^{n+1} \right)\right\|^2\\ &+C\tau\left(\left\|e_{w}^{n+1}\right\|^{2}+\left\|e_{\phi}^{n+1}\right\|^{2}+\left\|\nabla e_{\phi}^{n+1}\right\|^{2}+\left\|e_{\phi}^{n}\right\|^{2}+\left\|\nabla e_{\phi}^{n}\right\|^{2}\right), \end{align} (3.53)

    and

    \begin{align} \left(\int_{t_{n}}^{t_{n+1}} \phi_{t}(t) d t, e_{\phi}^{n+1}-e_{\phi}^{n}\right)\leq C \tau^{3}+\frac{\tau \gamma \lambda}{4}\left\|\nabla e_{w}^{n+1}\right\|^{2}+\frac{1}{4}\left\|\sigma^{n+1} e^{n+1}\right\|^{2}+C \tau\|\nabla e _{\phi}^{n}\|^{2}. \end{align}

    From (3.29), combining the two inequalities above, we deduce that

    \begin{equation} \begin{split} M_{12}+M_{13} \leq&-\frac{\lambda}{2 \varepsilon^{2}}\left(\left\|e_{\phi}^{n+1}\right\|_{L^{4}}^{4}-\left\|e_{\phi}^{n}\right\|_{L^{4}}^{4}+\left\|(e_{\phi}^{n+1})^{2}-\left(e_{\phi}^{n}\right)^{2}\right\|^{2}\right.\\ &\left.+2\left\|e_{\phi}^{n+1}\left(e_{\phi}^{n+1}-e _{\phi}^{n}\right)\right\|^{2}\right)-\frac{2 \lambda}{\varepsilon^{2}}\left\|e_{\phi}^{n+1}-e _{\phi}^{n}\right\|^{2}\\ &+C \tau^{3}+\frac{\tau\gamma \lambda}{2}\left\|\nabla e_{w}^{n+1}\right\|^{2}+\frac{1}{2}\left\|\sigma^{n+1} e_u^{n+1}\right\|^{2} \\ &+C\tau\left(\left\|e_{w}^{n+1}\right\|^{2}+\left\|e_{\phi}^{n+1}\right\|^{2}+\left\|\nabla e_{\phi}^{n+1}\right\|^{2}+\left\|e_{\phi}^{n}\right\|^{2}+\left\|\nabla e_{\phi}^{n}\right\|^{2}\right).\\ \end{split} \end{equation} (3.54)

    Plugging the above inequality into (3.41) gives

    \begin{align} &\frac{1}{2}\left(\left\|\sigma^{n+1} e_{u}^{n+1}\right\|^{2}-\left\|\sigma^{n} e_{u}^{n}\right\|^{2}+\left\|\sigma^{n}\left(e_{u}^{n+1}-e_{u}^{n}\right)\right\|^{2}\right)+ \eta\tau\left\|\nabla e_{u}^{n+1}\right\|^{2}+ \tau \gamma \lambda\left\|\nabla e_{w}^{n+1}\right\|^{2} \\ &+\frac{1}{2}\left(\left\|e_{\phi}^{n+1}\right\|^{2}-\left\|e_{\phi}^{n}\right\|^{2}+\left\|e_{\phi}^{n+1}-e_{\phi}^{n}\right\|^{2}\right)+ \tau \gamma\left\|e_{w}^{n+1}\right\|^{2}+\frac{\lambda}{2}\left\|\nabla e_{\phi}^{n+1}\right\|^{2}-\lambda\left\|\nabla e_{\phi}^{n}\right\|^{2}\\ &+\lambda\left\|\nabla e _{\phi}^{n+1}-\nabla e_{\phi}^{n} \right\|^{2}+\frac{\lambda}{2 \varepsilon^{2}}\left(\left\|e_{\phi}^{n+1}\right\|_{L^{4}}^{4}-\left\|e_{\phi}^{n}\right\|_{L^{4}}^{4}+\left\|(e_{\phi}^{n+1})^{2}-\left(e_{\phi}^{n}\right)^{2}\right\|^{2}\right.\\ &\left.+2\left\|e_{\phi}^{n+1}\left(e_{\phi}^{n+1}-e _{\phi}^{n}\right)\right\|^{2}\right)+\frac{2 \lambda}{\varepsilon^{2}}\left\|e_{\phi}^{n+1}-e _{\phi}^{n}\right\|^{2}\\ \leq& C \tau^{3}+C \tau^{2} \sum\limits_{m = 0}^{n}\left\|\sigma^{m} e_{u}^{m}\right\|^{2}+C\tau\left\|\sigma^{n+1}\left(e_{u}^{n+1}- \tilde{e}_{u}^{n+1} \right)\right\|^2\\ &+C\tau\left(\left\|\sigma^{n} e_{u}^{n}\right\|^{2}+\left\|e_{\phi}^{n}\right\|^{2}+\left\|\nabla e_{\phi}^{n}\right\|^{2}+\left\|\nabla e_{w}^{n+1}\right\|^{2} +\left\|e_{w}^{n+1}\right\|^{2}+\left\|e_{\phi}^{n+1}\right\|^{2}+\left\|\nabla e_{\phi}^{n+1}\right\|^{2}\right){;} \end{align} (3.55)

    for sufficiently small \tau , taking the sum of (3.55) from 0 to N -1 and using the discrete Gronwall inequality, we can obtain Lemma 6.

    Theorem 4. Suppose that the solution to (1.1) satisfies the regularity assumptions given by (3.3), and suppose that (3.1)–(3.2) are valid. For sufficiently small \tau , there are the following error estimates:

    \begin{equation} \begin{aligned} &\left\|\sigma\left(t_{n}\right)\boldsymbol{u}\left(t_{n}\right)-\sigma^{n}\boldsymbol{u}^{n}\right\|^{2}+\left\|\rho\left(t_{n}\right)-\rho^{n}\right\|^{2}+\left\|\nabla\left(w\left(t_{n}\right)-w^{n}\right)\right\|^{2} \\ &+\left\|\nabla\left(\phi\left(t_{n}\right)-\phi^{n}\right)\right\|^{2}+\eta \tau \sum\limits_{m = 1}^{n}\left\|\nabla\left(\boldsymbol{u}\left(t_{m}\right)-\boldsymbol{u}^{m}\right)\right\|^{2} \leq C \tau^{2} \end{aligned} \end{equation} (3.56)

    for all 1 \leq n \leq N .

    Proof. From Lemmas 5 and 6, we obtain

    \begin{equation} \left\|\rho\left(t_{n}\right)-\rho^{n}\right\|^{2}+\left\|\nabla\left(w\left(t_{n}\right)-w^{n}\right)\right\|^{2}+\left\|\nabla\left(\phi\left(t_{n}\right)-\phi^{n}\right)\right\|^{2} +\eta \tau \sum\limits_{m = 1}^{n}\left\|\nabla\left(\boldsymbol{u}\left(t_{m}\right)-\boldsymbol{u}^{m}\right)\right\|^{2} \leq C \tau^{2}. \end{equation} (3.57)

    Thus, we only prove that

    \begin{equation} \left\|\sigma\left(t_{n}\right)\boldsymbol{u}\left(t_{n}\right)-\sigma^{n}\boldsymbol{u}^{n}\right\|^{2} \leq C \tau^{2} . \end{equation} (3.58)

    In fact, we have

    \begin{equation} \begin{aligned} \left\|\sigma\left(t_{n}\right)\boldsymbol{u}\left(t_{n}\right)-\sigma^{n}\boldsymbol{u}^{n}\right\|^{2} & \leq C\left\|\left(\sigma\left(t_{n}\right)-\sigma^{n}\right)\boldsymbol{u}\left(t_{n}\right)\right\|^{2}+\left\|\sigma^{n} e_{u}^{n}\right\|^{2} \\ & \leq C\left(\left\|e_{\rho}^{n}\right\|^{2}+\left\|\sigma^{n} e_{u}^{n}\right\|^{2}\right) \leq C \tau^{2}. \end{aligned} \end{equation} (3.59)

    Theorem 3.6 states that both \sigma^{n}\boldsymbol{u}^{n}, \; \rho^{n}, \; \boldsymbol{u}^{n} , \phi^{n} and w^{n} are order 1 approximations to \sigma\boldsymbol{u}, \; \rho, \; \boldsymbol{u} , \phi and w in l^{\infty}\left(L^{2}(\Omega)\right) , l^{\infty}\left(L^{2}(\Omega)\right) , l^{2}\left(H_{0}^{1}(\Omega)\right) , l^{\infty}\left(H_0^{1}(\Omega)\right) and l^{\infty}\left(H_0^{1}(\Omega)\right) , respectively. Finally, we can obtain order \frac{1}{2} error estimates for p approximation in l^{\infty}\left(L^{2}(\Omega)\right) .

    Theorem 5. Under the assumptions in Theorem 4, the following holds true:

    \begin{equation} \tau\sum\limits_{m = 1}^{N}\left\|p\left(t_{m}\right)-p^{m}\right\| \leq C \tau. \end{equation} (3.60)

    Proof. Let us rewrite (3.40) as

    \begin{equation} \begin{split} -\nabla e_{p}^{n+1} = &\rho^{n} \frac{e_{u}^{n+1}-e_{u}^{n}}{\tau}+\frac{\rho^{n+1}-\rho^{n}}{\tau}\left(e_{u}^{n+1}-\widetilde{e}_{u}^{n+1}\right)-\eta \triangle e_{u}^{n+1}-R_{u}^{n+1}\\ &+\rho\left(t_{n+1}\right)\left(\boldsymbol{u}\left(t_{n+1}\right) \cdot \nabla\right) \boldsymbol{u}\left(t_{n+1}\right)-\rho^{n+1}\left(\boldsymbol{u}^{n} \cdot \nabla\right) \tilde{\boldsymbol{u}}^{n+1}\\ &- \lambda w\left(t_{n+1}\right) \nabla \phi\left(t_{n+1}\right)+ \lambda w^{n+1} \nabla \phi^{n}.\\ \end{split} \end{equation} (3.61)

    To prove the theorem, we introduce the discrete inf-sup condition, i.e.,

    \begin{equation} \beta\left\|e_{p}^{n+1}\right\| \leq \frac{\left(\nabla e_{p}^{n+1}, v\right)}{\|\nabla v\|}. \end{equation} (3.62)

    Then, we can constrain the products of the right-hand side of (3.61) with an arbitrary v \in V as follows:

    \begin{equation*} \begin{aligned} \frac{1}{\tau}\left(\rho^{n}\left(e_{u}^{n+1}-e_{u}^{n}\right), v\right) &\leq C \tau^{-1}\left\|\sigma^{n}\left(e_{u}^{n+1}-e_{u}^{n}\right)\right\| \| \nabla v \|, \\ -\eta\left(\Delta e_{u}^{n+1}, v\right)-\left(R_{u}^{n+1}, v\right) &\leq C\left(\tau^{2}+\left\|\nabla e_{u}^{n+1}\right\|\right) \| \nabla v \|, \end{aligned} \end{equation*}

    and

    \begin{equation} \begin{split} &- \lambda\left(w\left(t_{n+1}\right) \nabla \phi\left(t_{n+1}\right)-w^{n+1} \nabla \phi^{n}, v\right)\\ = & -\lambda\left(w\left(t_{n+1}\right)\left(\nabla \phi\left(t_{n+1}\right)-\nabla \phi\left(t_{n}\right)\right)+w\left(t_{n+1}\right) \nabla e_{\phi}^{n}+\nabla \phi^{n} e_{w}^{n+1}, v\right)\\ \leq& \lambda \left\| w\left(t_{n+1}\right)\right\|_{L^{\infty}}\left\| \int_{t_{n}}^{t_{n+1}} \nabla \phi_{t}(t) d t\right\|\| v\|+ \lambda\left\| w\left(t_{n+1}\right)\right\|_{L^{\infty}}\left\|\nabla e_{\phi}^{n}\right\|\left\|v\right\|\\ &+ \lambda\left\|\nabla \phi^{n}\right\|_{L^2}\left\|e_w^{n+1}\right\|_{L^{3}}\left\|v\right\|_{L^{6}}\\ \leq& C\left( \tau+\left\|\nabla e_{\phi}^{n}\right\|+\left\|\nabla e_{w}^{n+1}\right\|\right)\left\| \nabla v \right\|.\\ \end{split} \end{equation} (3.63)

    For the second term on the right-hand side, using \|v\|_{L^{3}} \leq\|v\|^{\frac{1}{2}}\|\nabla v\|^{\frac{1}{2}} , we have

    \begin{equation} \begin{aligned} &\left(\frac{\rho^{n+1}-\rho^{n}}{\tau}\left(e_{u}^{n+1}-\widetilde{e}_{u}^{n+1}\right), v\right) \\ &\leq C \tau^{-1}\left\|\rho^{n+1}-\rho^{n}\right\|\left\|e_{u}^{n+1}-\widetilde{e}_{u}^{n+1}\right\|_{L^{3}}\|v\|_{L^{6}} \\ &\leq C \tau^{-1}\left\|\rho^{n+1}-\rho^{n}\right\|\left\|e_{u}^{n+1}-\widetilde{e}_{u}^{n+1}\right\|^{\frac{1}{2}}\left\|\nabla\left(e_{u}^{n+1}-\widetilde{e}_{u}^{n+1}\right)\right\|^{\frac{1}{2}}\|\nabla v\| \\ &\leq C\left\|\sigma^{n+1}\left(e_{u}^{n+1}-\widetilde{e}_{u}^{n+1}\right)\right\|^{\frac{1}{2}}\left\|\nabla\left(e_{u}^{n+1}-\widetilde{e}_{u}^{n+1}\right)\right\|^{\frac{1}{2}}\|\nabla v\|. \end{aligned} \end{equation} (3.64)

    By the split method, we arrive at

    \begin{equation} \begin{aligned} &\left(\rho\left(t_{n+1}\right)\left(\boldsymbol{u}\left(t_{n+1}\right) \cdot \nabla\right) \boldsymbol{u}\left(t_{n+1}\right)-\rho^{n+1}\left(\boldsymbol{u}^{n} \cdot \nabla\right) \widetilde{\boldsymbol{u}}^{n+1}, v\right) \\ & = \left(\rho^{n+1}\left(\left(\boldsymbol{u}\left(t_{n+1}\right)-\boldsymbol{u}\left(t_{n}\right)\right) \cdot \nabla\right) \boldsymbol{u}\left(t_{n+1}\right), v\right)+\left(\rho^{n+1}\left(\boldsymbol{u}^{n} \cdot \nabla\right) \tilde{e}_{u}^{n+1}, v\right) \\ &\quad+\left(\rho^{n+1}\left(e_{u}^{n} \cdot \nabla\right) \boldsymbol{u}\left(t_{n+1}\right), v\right)+\left(e_{\rho}^{n+1}\left(\boldsymbol{u}\left(t_{n+1}\right) \nabla\right) \boldsymbol{u}\left(t_{n+1}\right), v\right). \end{aligned} \end{equation} (3.65)

    Utilizing H \ddot{o} lder's inequality and Young's inequality, we derive

    \begin{equation} \begin{split} &\left(\rho\left(t_{n+1}\right)\left(\boldsymbol{u}\left(t_{n+1}\right) \cdot \nabla\right) \boldsymbol{u}\left(t_{n+1}\right)-\rho^{n+1}\left(\boldsymbol{u}^{n} \cdot \nabla\right) \widetilde{\boldsymbol{u}}^{n+1}, v\right) \\ \leq& C\left(\tau+\left\| e_{\rho}^{n+1}\right\|+\left\|\nabla e_{u}^{n}\right\|+\left\|\nabla \tilde{e}_{u}^{n+1}\right\|\right) \| \nabla v\|.\\ \end{split} \end{equation} (3.66)

    By adding up the inequalities and incorporating (3.62), we get the desired result.

    In this section, we will present some numerical experiments to prove the validity and accuracy of our method. In the following simulation, for phase field \phi , chemical potential w and pressure p , we take the P1 finite element space (continuous piecewise linear), and for fluid velocity \boldsymbol{u} , we take the P2 finite element space. All experiments were conducted in Freefem++. We fixed \eta = 0.8, \lambda = 0.7, \gamma = 0.0006, \varepsilon = 0.1, T = 0.1, \rho_1 = 1 and \rho_2 = 3. The computational domain and the initial conditions were taken as

    \begin{equation*} \begin{split} &\Omega = \{(x, y)\in R^2:x^2+y^2 < 1\}, \\ &\phi_0 = cos(\pi x)cos(\pi y), \\ &\boldsymbol{u}_0 = ( \pi cos(\pi x)sin(\pi y), -\pi sin(\pi x)cos(\pi y)).\\ \end{split} \end{equation*}

    Tables 1 and 2 verify that (3.4) is the first-order convergence rate O(\tau) of (\phi, \sigma\boldsymbol{u}, \boldsymbol{u}, \rho) , which is consistent with the conclusion obtained from theoretical analysis. It is only proved in Theorem 3.2 that the pressure is the half-order convergence rate O(\tau^{\frac{1}{2}}) because of technical reasons. However, the numerical results on p in Table 1 still reach the first-order optimal convergence rate O(\tau) . Tables 3 and 4 show the convergence rate with another set of parameters.

    Table 1.  The order of temporal convergence with \eta = 0.8, \lambda = 0.7, \gamma = 0.0006, \varepsilon = 0.1 .
    \tau \Vert \phi\Vert_{H^1} Rate \Vert p\Vert_{L^2} Rate
    0.007812 0.102393 0.469438
    0.003906 0.0653769 0.647262 0.282333 0.733534
    0.001953 0.0367805 0.829842 0.143976 0.971574
    0.000976 0.0196276 0.906053 0.0725853 0.988076

     | Show Table
    DownLoad: CSV
    Table 2.  The order of temporal convergence with \eta = 0.8, \lambda = 0.7, \gamma = 0.0006, \varepsilon = 0.1 .
    \tau \Vert \sigma \boldsymbol{u}\Vert_{L^2} Rate \Vert \boldsymbol{u}\Vert_{H^1} Rate \Vert \rho\Vert_{L^2} Rate
    0.007812 0.114448 0.554732 0.0779992
    0.003906 0.0610184 0.90737 0.2956 0.90814 0.0463154 0.751968
    0.001953 0.02533 1.2684 0.132451 1.15819 0.022386 1.04889
    0.000976 0.0127883 0.98602 0.071371 0.89204 0.0107557 1.05749

     | Show Table
    DownLoad: CSV
    Table 3.  The order of temporal convergence with \eta = 0.1, \lambda = 0.2, \gamma = 0.0003, \varepsilon = 0.08 .
    \tau \Vert \phi\Vert_{H^1} Rate \Vert p\Vert_{L^2} Rate
    0.007812 0.041787 0.0794722
    0.003906 0.0215011 0.958642 0.0435045 0.869285
    0.001953 0.0106192 1.01774 0.0210772 1.04548
    0.000976 0.0054229 0.96953 0.0106356 0.986787

     | Show Table
    DownLoad: CSV
    Table 4.  The order of temporal convergence with \eta = 0.8, \lambda = 0.7, \gamma = 0.0006, \varepsilon = 0.1 .
    \tau \Vert \sigma \boldsymbol{u}\Vert_{L^2} Rate \Vert \boldsymbol{u}\Vert_{H^1} Rate \Vert \rho\Vert_{L^2} Rate
    0.007812 0.0308723 0.387768 0.0585448
    0.003906 0.0161585 0.934018 0.201806 0.942225 0.0300705 0.961192
    0.001953 0.00587585 1.45943 0.0765909 1.39772 0.00932566 1.68907
    0.000976 0.00302534 0.957701 0.0398452 0.942768 0.00468607 0.992826

     | Show Table
    DownLoad: CSV

    Figure 1 shows the evolution of the total energy at \tau = 0.02 . The downward trend of the energy curve confirms that our scheme is unconditionally energy stable. We also see a downward trend in energy when using different parameters. The energy curves for different time steps are shown in Figure 2 as a result of keeping the other parameters unchanged. It can been found that the curves are very similar, which means that the scheme is robust against different time steps.

    Figure 1.  Energy evolution with different parameters.
    Figure 2.  Energy evolution with different time step sizes.

    To solve the Cahn-Hilliard phase field model for two-phase incompressible flows with variable density, we have designed a novel time marching scheme, which can significantly improve the calculation efficiency. The method is efficient because we decoupled the pressure from the velocity and phase field. We have also proved the unconditional energy stability, presented the error analysis and provided various numerical examples to demonstrate the stability and accuracy of the scheme. In addition, the decoupling method developed in this paper is universally applicable, and this method is always applicable for the generation of an effective fully decoupling scheme.

    The authors declare they have not used Artificial Intelligence tools (AI) in the creation of this article.

    This work was supported by the Research Project Supported by Shanxi Scholarship Council of China (No. 2021-029), Shanxi Provincial International Cooperation Base and Platform Project (202104041101019), Shanxi Province Natural Science Research (202203021211129), Shanxi Province Natural Science Research (No. 202203021212249) and Special/Youth Foundation of Taiyuan University of Technology (No. 2022QN101).

    All authors declare that they have no competing interests in this paper.



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