Processing math: 41%
Research article

Convergence analysis of Euler and BDF2 grad-div stabilization methods for the time-dependent penetrative convection model

  • Received: 31 October 2023 Revised: 20 November 2023 Accepted: 20 November 2023 Published: 30 November 2023
  • MSC : 36Q30, 65M60, 76M10

  • Based on the grad-div stabilization method, the first-order backward Euler and second-order BDF2 finite element schemes were studied for the approximations of the time-dependent penetrative convection equations. The proposed schemes are both unconditionally stable. We proved the error bounds of the velocity and temperature in which the constants are independent of inverse powers of the viscosity and thermal conductivity coefficients when the Taylor-Hood element and P2 element are used in finite element discretizations. Finally, numerical experiments with high Reynolds numbers were shown to confirm the theoretical results.

    Citation: Weiwen Wan, Rong An. Convergence analysis of Euler and BDF2 grad-div stabilization methods for the time-dependent penetrative convection model[J]. AIMS Mathematics, 2024, 9(1): 453-480. doi: 10.3934/math.2024025

    Related Papers:

    [1] Yunzhang Zhang, Xinghui Yong . Analysis of the linearly extrapolated BDF2 fully discrete Modular Grad-div stabilization method for the micropolar Navier-Stokes equations. AIMS Mathematics, 2024, 9(6): 15724-15747. doi: 10.3934/math.2024759
    [2] Eunjung Lee, Dojin Kim . Stability analysis of the implicit finite difference schemes for nonlinear Schrödinger equation. AIMS Mathematics, 2022, 7(9): 16349-16365. doi: 10.3934/math.2022893
    [3] Zuliang Lu, Ruixiang Xu, Chunjuan Hou, Lu Xing . A priori error estimates of finite volume element method for bilinear parabolic optimal control problem. AIMS Mathematics, 2023, 8(8): 19374-19390. doi: 10.3934/math.2023988
    [4] Danuruj Songsanga, Parinya Sa Ngiamsunthorn . Single-step and multi-step methods for Caputo fractional-order differential equations with arbitrary kernels. AIMS Mathematics, 2022, 7(8): 15002-15028. doi: 10.3934/math.2022822
    [5] Rubayyi T. Alqahtani, Jean C. Ntonga, Eric Ngondiep . Stability analysis and convergence rate of a two-step predictor-corrector approach for shallow water equations with source terms. AIMS Mathematics, 2023, 8(4): 9265-9289. doi: 10.3934/math.2023465
    [6] Philsu Kim, Seongook Heo, Dojin Kim . A high-order convergence analysis for semi-Lagrangian scheme of the Burgers' equation. AIMS Mathematics, 2023, 8(5): 11270-11296. doi: 10.3934/math.2023571
    [7] Danxia Wang, Ni Miao, Jing Liu . A second-order numerical scheme for the Ericksen-Leslie equation. AIMS Mathematics, 2022, 7(9): 15834-15853. doi: 10.3934/math.2022867
    [8] Narcisse Batangouna . A robust family of exponential attractors for a time semi-discretization of the Ginzburg-Landau equation. AIMS Mathematics, 2022, 7(1): 1399-1415. doi: 10.3934/math.2022082
    [9] Ruby, Vembu Shanthi, Higinio Ramos . A numerical approach to approximate the solution of a quasilinear singularly perturbed parabolic convection diffusion problem having a non-smooth source term. AIMS Mathematics, 2025, 10(3): 6827-6852. doi: 10.3934/math.2025313
    [10] Sara S. Alzaid, Pawan Kumar Shaw, Sunil Kumar . A study of Ralston's cubic convergence with the application of population growth model. AIMS Mathematics, 2022, 7(6): 11320-11344. doi: 10.3934/math.2022632
  • Based on the grad-div stabilization method, the first-order backward Euler and second-order BDF2 finite element schemes were studied for the approximations of the time-dependent penetrative convection equations. The proposed schemes are both unconditionally stable. We proved the error bounds of the velocity and temperature in which the constants are independent of inverse powers of the viscosity and thermal conductivity coefficients when the Taylor-Hood element and P2 element are used in finite element discretizations. Finally, numerical experiments with high Reynolds numbers were shown to confirm the theoretical results.



    The grad-div stabilization method was first introduced for the finite element approximation of the incompressible Navier-Stokes equations by adding the grad-div stabilization term to the momentum equation in [12]. For the most of Lagrange finite element pairs, such as the Taylor-Hood element and MINI element, the grad-div stabilization term is nonzero, and there is no point-wise mass conservation. Thus, this stabilization term is used to improve the mass conservation and to reduce the velocity error caused by the pressure error. The influence of the grad-div stabilization term on the accuracy of the numerical solution was studied in [40] for the steady Stokes problem.

    It is well known that the standard Galerkin finite element methods are not suitable for the numerical simulation of the incompressible flow problems in the case of the high Reynolds number or the small viscosity. Different stabilized methods have been developed to overcome the instabilities of numerical simulation, such as Galerkin least square methods in [13,25], the residual-free bubbles methods in [14,15], the large eddy simulation methods in [45], the sub-grid scale methods in [23,31], the variational multiscale methods in [26,27,34,49] and the streamline upwind Petrov-Galerkin (SUPG) method [5]. We note that the grad-div stabilization method is also a powerful tool for the high Reynolds flow problems. There have an amount of works for this issue. For example, the SUPG and grad-div stabilization methods were studied for the generalized Oseen problem in [37], where it was concluded that the SUPG method is less important for the inf-sup stable pair of velocity and pressure due to the choice of stabilization parameter. Moreover, if one considers only the grad-div stabilization method, it was acknowledged that the grad-div method can lead to stabilized results. The grad-div method for the rotation form of the Navier-Stokes equations can be found in [32], where numerical results showed that the difference between the skew-symmetric and the rotation form of the nonlinearity is from the increased error in the Bernoulli pressure, which results in the increasing of velocity error. The use of the grad-div stabilization term ameliorates the effect and reduces the velocity error. For the time-dependent Oseen problem, the semi-discrete and fully discrete schemes based on the grad-div stabilization method were analyzed in [17]. Under the assumption of sufficiently smooth solutions and based on the specific Stokes projection, the authors proved the optimal error estimates for velocity and pressure with constants that are independent of the viscosity. Subsequently, this observation was extended to the time-dependent Navier-Stokes equations in [18], where the constants in the optimal error bounds do not depend on the inverse power of viscosity. Thus, the results in [17] and [18] gave a theoretical confirmation about stable simulations of the high Reynolds number flows by using the grad-div stabilization method. Recently, the analysis technique in [18] was extended to the second-order backward differentiation formula (BDF2) scheme with variable time-step size of the Navier-Stokes equations in [21]. A review study about error analysis of the grad-div stabilization methods can be found in [20].

    Since there are some advantages of the grad-div stabilization method in numerical simulations of the incompressible flows, it has been applied to some related and coupled systems, such as the magnetohydrodynamic (MHD) equations [11] and the time-dependent dual-porosity-Stokes equations [33]. In this paper, we will consider the following time-dependent penetrative convection equations in the nondimensional form:

    utνΔu+(u)u+p(γ1θ+γ2θ2)i3=f,for(x,t) Ω×(0,T], (1.1)
    u=0,for(x,t) Ω×(0,T], (1.2)
    θtκΔθ+(u)θ=g,for(x,t) Ω×(0,T], (1.3)
    u=0,θ=0,for(x,t) Ω×(0,T], (1.4)
    u(x,0)=u0(x),θ(x,0)=θ0(x),forxΩ, (1.5)

    where Ω is a bounded and convex polyhedral domain in R3 with boundary Ω, and (0,T] is a finite time interval. In the above system, the unknown (u,p,θ) denotes the velocity of the fluid, the pressure and temperature, respectively. ν>0 and κ>0 represent the viscosity coefficient and thermal conductivity parameter. f and g are two given functions. i3 is the unit basis vector given by i3=(0,0,1)T, and (γ1θ+γ2θ2)i3 in (1.1) denotes the buoyancy term. The constraint u=0 represents the incompressibility of the fluid.

    The penetrative convection equations (1.1)–(1.3) used to describe the motion in which convection in a thermally unstable region extends into an adjacent stable region. It has a wide range of applications in the fields of atmospheric dynamics, atmospheric fronts, katabatic winds, dense gas dispersion, natural ventilation, cooling of electronic equipment [29,36,39,42]. The stabilities of penetrative convection equations have been studied in [7,8,16,41,46], and numerical simulations have been reported in [3,38]. Convergence and error estimates of finite element fully discrete schemes were studied in [44] and [6], where the BDF2 and Crank-Nicolson schemes were used, respectively. However, the high Reynolds number was not considered in [6,44]. By considering a linear Boussinesq approximation, i.e., γ2=0 in (1.1), a grad-div stabilization finite element method was studied in [19]. In addition, a rigorous error analysis was given and error estimates were derived in [19], where the constants in error bounds are independent of inverse powers of the viscosity and thermal conductivity coefficients.

    In this paper, based on the grad-div stabilization method, we consider and study two finite element schemes by using the first-order backward Euler and BDF2 formulas to discrete the time derivatives, respectively. The proposed schemes are both linearized semi-implicit schemes where we use the implicit-explicit method and the standard extrapolation formula to deal with nonlinear terms. Moreover, these schemes are unconditionally stable without any condition of the time step size and mesh size. Following the analysis techniques developed in [18,19], we derive error bounds of the velocity and temperature in which the constants are independent of negative powers of the viscosity and thermal conductivity coefficients.

    The rest of this paper is organized as follows. In next section, we state some notations and recall some known results in finite element theory. In sections three and four, the first-order Euler and second-order BDF2 schemes with the grad-div stabilization are proposed, respectively. Unconditional stability of numerical schemes and error analysis are also presented in these sections. In section five, we give numerical results to support the theory analysis and numerical stability of the grad-div stabilization method for the high Reynolds flows.

    For kN+ and 1p+, let Lp(Ω) and Wk,p(Ω) denote the standard Lebesgue space and Sobolev space, respectively. When p=2, Wk,2(Ω) is the Hilber space Hk(Ω). The norms in Lp(Ω), Wk,p(Ω) and Hk(Ω) are defined as the classical senses (cf.[1]) and are denoted by Lp, Wk,p and Hk. We define Hk0(Ω) to be the subspace of Hk(Ω) of functions with zero trace on Ω. The dual space of H10(Ω) is denoted by H1(Ω). The boldface notations Lp(Ω), Wk,p(Ω) and Hk(Ω) are used to denote the vector-value spaces Lp(Ω)3, Wk,p(Ω)3 and Hk(Ω)3, respectively. The corresponding norms are denoted by Lp, Wk,p and Hk.

    For simplicity, we denote V=H10(Ω), Y=H10(Ω),

    V0={uV,u=0 in Ω},Q=L20(Ω)={q L2(Ω),Ωqdx=0}.

    Introduce the trilinear forms c1(,,) and c2(,,), which are given by

    c1(u1,u2,u3)=Ω((u1)u2u3+12(u1)u2u3)dxc2(u1,θ1,θ2)=Ω((u1θ1)θ2+12(u1)θ1θ2)dx

    for uiV with i=1,2,3 and θjY with j=1,2. By the integration by parts, it is easy to check that the trilinear forms c1(,,) and c2(,,) satisfy the skew-symmetric properties, i.e.,

    c1(u1,u2,u3)=c1(u1,u3,u2)andc2(u1,θ1,θ2)=c2(u1,θ2,θ1),

    which give

    c1(u1,u2,u2)=0andc2(u1,θ1,θ1)=0. (2.1)

    For given fL2(Ω),gL2(Ω), u0L2(Ω) and θ0L2(Ω), in terms of ther skew-symmetric properties (2.1), there {holds} the following energy inequalities:

    θ(t)2L2+κt0θ(s)2L2dsθ02L2+Ct0g(s)2L2ds, (2.2)

    and

    u(t)2L2+νt0u(s)2L2dsu02L2+Ct0(f(s)2L2+θ(s)2L2+θ(s)2L2θ(s)2L3)dsu02L2+Ct0(f(s)2L2+g(s)2L2)ds+C(t0g(s)2L2ds)2 (2.3)

    for 0tT.

    Let 0=t0<t1<<tN=T be a uniform partition of the time interval [0,T] with time step τ=T/N and tn=nτ with 0nN. Denote un=u(tn), pn=p(tn), θn=θ(tn), fn=f(tn) and gn=g(tn). For any sequence of functions {fn}Nn=0, denote the backward Euler discretization and BDF2 discretization by

    D1,τfn+1=(fn+1fn)/τfor  0nN1,

    and

    D2,τfn+1=(3fn+14fn+fn1)/2τ,ˆfn=2fnfn1for  1nN1,

    where ˆfn is the standard extrapolation formula. For the discrete time derivative D2,τ, there holds the telescope formula [35]:

    (D2,τfn+1,fn+1)=14τ(fn+12L2fn2L2+ˆfn+12L2ˆfn2L2)+14τfn+12fn+fn12L2. (2.4)

    We define the finite element spaces as follows. Let Th={Kj}Lj=1 be a quasi-uniform tetrahedron partition of Ω with the mesh size h=max{diamKj, j=1,,L}. In finite element discretization of (1.1)–(1.3), we use the Taylor-Hood (P2P1) finite element for the velocity u and pressure p, and the piece-wise quadratic finite element for the temperature θ. The corresponding finite element subspaces of V,Q and Y are denoted by Vh, Qh and Yh, respectively, i.e.,

    Vh={vhC(¯Ω)V, vh|K(P2(K))3,  KTh},Qh={qhC(¯Ω)H1(Ω), qh|KP1(K),  KTh, Ωqhdx=0},Yh={rhC(¯Ω)Y, rh|KP2(K),  KTh}.

    It is {well known} that the discrete inf-sup condition holds for the inf-sup stable element such that

    β0qhL2supvhVh(vh,qh)vhL2, qhQh, (2.5)

    where β0>0 is independent of the mesh size h.

    Denote the space of discrete divergence-free functions by

    V0h={vhVh,(vh,qh)=0 qhQh}.

    In the sequel, let πh and Rh be the L2 orthogonal and Ritz orthogonal projections onto Qh and Yh, respectively. Then the following error bounds and stability hold [4,43]:

    pπhpL2Ch2pH2 pH2(Ω), (2.6)
    ψRhψL2+h(ψRhψ)L2Ch3ψH3 ψH3(Ω), (2.7)
    RhψW1,CψH3 ψH3(Ω). (2.8)

    Let Ih be the Lagrange interpolation operator onto Vh. The following bound can be found in [4]:

    uIhuWm,pChnmuWn,p0mn3, (2.9)

    where n>3/p when 1<p and n3 when p=1.

    Next, we recall some known results about finite element approximations of Stokes problem in [17,18]. Consider the Stokes problem:

    νΔu+p=Fin Ω,u=0in Ω,u=0on Ω. (2.10)

    Let us denote (uh,ph)Vh×Qh, the mixed finite element approximation solution to the Stokes problem (2.10). Then one has error estimates [22,30]

    (uuh)L2C(infvhVhuvhH1+ν1infqhQhpqhL2), (2.11)
    pphL2C(νinfvhVhuvhH1+infqhQhpqhL2), (2.12)
    uuhL2Ch(infvhVhuvhH1+ν1infqhQhpqhL2), (2.13)

    where C>0 is independent of h,ν and κ. It is clear that the error estimates of the velocity depend on the negative power of ν. To prove error bounds with constants being independent of the negative power of ν, we recall the specific projection of (u,p) introduced in [17] that we will denote by sh defined by

    (sh,vh)=(u,vh),vhV0h.

    Let (u,p,θ) be the solution to the system (1.1)–(1.5) with uL((0,T];VH3(Ω)), pL((0,T];QH2(Ω)), θL((0,T];YH3(Ω)) and utL((0,T];H1(Ω)). We note that (u,0) is the solution to the Stokes problem (2.10) with the right-hand side F given by

    F=fut(u)up+(γ1θ+γ2θ2)i3. (2.14)

    Let (sh,lh)Vh×Qh be the corresponding mixed finite element approximation of (u,0). Then, from (2.11)–(2.13), we have

    ushL2+hushH1Ch3uH3, (2.15)
    lhL2Cνh2uH3, (2.16)

    where C>0 is independent of h and ν. In terms of the inverse inequality in the theory of the finite element method and IhuLCuL, we have

    shLshIhuL+IhuLC(h3/2shIhuL2+uL)C(h3/2shuL2+h3/2uIhuL2+uH3)CuH3, (2.17)

    where C>0 is independent of h,ν and κ. Furthermore, there holds

    (ush)LCuLCuH3andshLCuLCuH3, (2.18)

    where C>0 is independent of h,ν and κ.

    Finally, we recall the discrete Gronwall inequality established in [24].

    Lemma 2.1. Let ak,bk,ck and γk, for integers k0, be the nonnegative numbers such that

    an+τnk=0bkτnk=0γkak+τnk=0ck+Bfor n0. (2.19)

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

    an+τnk=0bkexp(τnk=0γkσk)(τnk=0ck+B)for n0. (2.20)

    Remark 2.1. If the first sum on the right in (2.19) extends only up to n1, then the estimate (2.20) holds for all τ>0 with σk=1.

    In this section, we consider the first-order Euler fully discrete scheme for the problems (1.1)–(1.5). This Euler scheme is a semi-implicit scheme, i.e., one only solves two linear systems at each time discrete level. Based on the grad-div stabilization method, we propose the following first-order Euler finite element scheme for 0nN1.

    Step Ⅰ: Find θn+1hYh by

    (D1,τθn+1h,ψh)+κ(θn+1h,ψh)+c2(unh,θn+1h,ψh)=(gn+1,ψh) (3.1)

    with the initial iteration values θ0h=Rhθ0, u0h=Ihu0 and for any ψhYh.

    Step Ⅱ: Find (un+1h,pn+1h)Vh×Qh by

    (D1,τun+1h,vh)+ν(un+1h,vh)+c1(unh,un+1h,vh)(vh,pn+1h)+(un+1h,qh)+β(un+1h,vh)γ1(θnhi3,vh)γ2((θnh)2i3,vh)=(fn+1,vh) (3.2)

    for any (vh,qh)Vh×Qh, where β>0 is the stabilization parameter. The detailed discussion about the choice of β can be found in [28].

    Next lemma gives the stability of the scheme (3.1)–(3.2), where the discrete energy inequalities can be viewed as the discrete version of the energy estimates (2.2)–(2.3). Moreover, since (3.1)–(3.2) are both linearized problems, the discrete energy inequalities imply the existence and uniqueness of numerical solution (un+1h,pn+1h,θn+1h) to the Euler scheme (3.1)–(3.2).

    Lemma 3.1. For 0nN1 and all τ>0, h>0, the finite element scheme (3.1) and (3.2) has a unique solution θn+1hYh and (un+1h,pn+1h)Vh×Qh. Moreover, the discrete energy inequalities hold:

    θm+1h2L2+κτmn=0θn+1h2L2CτN1n=0gn+12L2+θ0h2L2, (3.3)

    and

    um+1h2L2+τνmn=0um+1h2L2+2τβmn=0um+1h2L2CτN1n=0(fn+12L2+gn+12L2)+C(u0h2L2+θ0h2L2)+C(τN1n=0gn+12L2+θ0h2L2)2 (3.4)

    for all 0mN1, where C>0 is independent of τ and h.

    Proof. Setting ψh=2τθn+1h in (3.1), using the skew-symmetric property (2.1), the Hölder inequality and Young inequality, we have

    θn+1h2L2θnh2L2+θn+1hθnh2L2+2τκθn+1h2L22τgn+1L2θn+1hL2Cτgn+12L2+τκθn+1h2L2.

    Summing up the above inequality from n=0 to n=m gives (3.3).

    Similarly, taking (vh,qh)=2τ(un+1h,pn+1h) in (3.2), we have

    un+1h2L2unh2L2+un+1hunh2L2+2τνun+1h2L2+2τβun+1h2L2Cτfn+1L2un+1hL2+CτθnhL2un+1hL2+CτθnhL2θnhL3un+1hL6τνun+1h2L2+Cτ(fn+12L2+θnh2L2+θnh2L2θnh2L3).

    Summing up the above inequality from n=0 to n=m and noticing (3.3), we get (3.4).

    Denote

    enh=snhunh,θnθnh=(θnRhθn)+(Rhθnθnh):=ηnθ+enθ.

    The sh(t) is the finite element solution to the Stokes problem (2.10) with the right-hand side (2.14) and snh=sh(tn).

    Thanks to the discrete inf-sup condition (2.5), the finite element scheme (3.2) is equivalent to: Find un+1hV0h by

    (D1,τun+1h,vh)+ν(un+1h,vh)+c1(unh,un+1h,vh)+β(un+1h,vh)γ1(θnhi3,vh)γ2((θnh)2i3,vh)=(fn+1,vh) vhV0h. (3.5)

    For simplicity, we take s0h=u0h. For 0nN1, sn+1h and θn+1 satisfy

    (D1,τsn+1h,vh)+ν(sn+1h,vh)+c1(snh,sn+1h,vh)+β(sn+1h,vh)γ1(θni3,vh)γ2((θn)2i3,vh)=(fn+1,vh)+(vh,pn+1πhpn+1)+β((sn+1hun+1),vh)+(D1,τsn+1hut(tn+1),vh)+c1(snh,sn+1h,vh)c1(un+1,un+1,vh)+γ1(θn+1θn,i3vh)+γ2((θn+1)2(θn)2,i3vh) vhV0h, (3.6)

    and

    (D1,τθn+1,ψh)+κ(θn+1,ψh)+c2(un+1,θn+1,ψh)=(D1,τθn+1θt(tn+1),ψh)+(gn+1,ψh) ψhYh. (3.7)

    In (3.6), we use un+1=0 in Ω and (vh,πhpn+1)=0 due to vhV0h.

    Next, we give error equations. Subtracting (3.5) from (3.6) leads to

    (D1,τen+1h,vh)+ν(en+1h,vh)+β(en+1h,vh)=6j=1(Jn+1j,vh) vhV0h, (3.8)

    where

    (Jn+11,vh)=(D1,τsn+1hut(tn+1),vh),(Jn+12,vh)=(vh,(pn+1πhpn+1)+β(sn+1hun+1)),(Jn+13,vh)=c1(snh,sn+1h,vh)c1(un+1,un+1,vh),(Jn+14,vh)=c1(unh,un+1h,vh)c1(snh,sn+1h,vh),(Jn+15,vh)=γ1(θn+1θn,i3vh)+γ2((θn+1)2(θn)2,i3vh),(Jn+16,vh)=γ1((θnθnh),i3vh)+γ2(((θn)2(θnh)2),i3vh).

    Subtracting (3.1) from (3.7), we have

    (D1,τen+1θ,ψh)+κ(en+1θ,ψh)+c2(unh,en+1θ,ψh)=3j=1(λn+1j,ψh) ψhYh, (3.9)

    where

    (λn+11,ψh)=(D1,τθn+1θt(tn+1),ψh),(λn+12,ψh)=(D1,τηn+1θ,ψh),(λn+13,ψh)=c2(unh,Rhθn+1,ψh)c2(un+1,θn+1,ψh).

    The main result in this section is presented in the following theorem.

    Theorem 3.1. Let (u,p,θ) be the unique solution to the time-dependent penetrative convection problems (1.1)–(1.5) and satisfy the following regularity assumptions:

    {u0H3(Ω), θ0H3(Ω), fL((0,T];L2(Ω)),gL((0,T];L2(Ω)),uL((0,T];H3(Ω)),pL((0,T];H2(Ω)),θL((0,T];H3(Ω)),utL((0,T];H1(Ω))L2((0,T];H2(Ω)),θtL((0,T];H1(Ω))L2((0,T];H2(Ω)),uttL2((0,T];L2(Ω)),θttL2((0,T];L2(Ω)). (3.10)

    Then, for the first-order Euler grad-div scheme (3.1)–(3.2) under the time step condition τCh2, when h and τ are sufficiently small, we have the following error estimate:

    em+1h2L2+em+1θ2L2+τmn=0(νem+1h2L2+κem+1θ2L2)C20(h4+τ2) (3.11)

    with 0mN1, where C0>0 is independent of h,τ, ν and κ.

    Proof. Taking vh=2τen+1h in (3.8) and summing up the resulting equation from n=0 to n=m, we have

    em+1h2L2+2τmn=0(νen+1h2L2+βen+1h2L2)=2τmn=06j=1(Jn+1j,en+1h). (3.12)

    We estimate the right-hand side of (3.12) as follows. For Jn+11, we split it as

    Jn+11=(D1,τsn+1hD1,τun+1)+(D1,τun+1ut(tn+1)).

    According to the Taylor's formula and the regularity assumption (3.10), one has

    τmn=0D1,τun+1ut(tn+1)2L2Cτ2.

    Notice that

    D1,τsn+1hD1,τun+1=1τtn+1tnt(sh(t)u(t))dt.

    From the Hölder inequality, we have

    D1,τsn+1hD1,τun+1L21τtn+1tnt(sh(t)u(t))L2dtCh2τtn+1tntu(t)H2dtCh2τ1/2(tn+1tntu(t)2H2dt)1/2.

    Thus,

    2τmn=0(Jn+11,en+1h)Cτmn=0en+1h2L2+C(τ2+h4), (3.13)

    where C>0 is independent of h,τ,ν and κ.

    For Jn+12, it follows from the Hölder inequality and (2.6) that

    (Jn+12,en+1h)(pn+1πhpn+1L2+βsn+1hun+1H1)en+1hL2Ch2en+1hL2,

    then

    2τmn=0(Jn+12,en+1h)Ch4+ϵ1βτmn=0en+1h2L2, (3.14)

    where C>0 is independent of h,τ,ν,κ and ϵ1>0 is some small constant determined later.

    For Jn+13, in terms of the skew-symmetric property (2.1), (2.18), the Hölder inequality and the regularity assumption (3.10), there holds

    (Jn+13,en+1h)=c1(snhun,sn+1h,en+1h)+c1(unun+1,sn+1h,en+1h)+c1(un+1,sn+1hun+1,en+1h)(snhunH1+un+1unH1)sn+1hW1,en+1hL2+un+1Lsn+1hun+1H1en+1hL2C(h2+τ)en+1hL2,

    where C>0 is independent of h,τ,ν and κ. Thus, the third term can be estimated by

    2τmn=0(Jn+13,en+1h)C(h4+τ2)+Cτmn=0en+1h2L2, (3.15)

    where C>0 is independent of h,τ,ν and κ.

    A similar method to bound Jn+14 leads to

    (Jn+14,en+1h)=c1(enh,sn+1h,en+1h)enhL2sn+1hLen+1hL2+sn+1hLenhL2en+1hL2,

    and

    2τmn=0(Jn+14,en+1h)ϵ1βτmn=0en+1h2L2+Cτmn=0(en+1h2L2+enh2L2), (3.16)

    where C>0 is independent of h,τ,ν and κ.

    For Jn+15, using the Hölder inequality, Young inequality and the regularity assumption (3.10), we have

    |(Jn+15,en+1h)|C(1+θn+θn+1L)tn+1tnθt(t)L2dten+1hL2Cen+1h2L2+Cτ2,

    then

    2τmn=0(Jn+15,en+1h)Cτmn=0en+1h2L2+Cτ2, (3.17)

    where C>0 is independent of h,τ,ν and κ.

    By a similar method, we estimate the last term by

    (Jn+16,en+1h)ηnθ+enθL2en+1hL2+(ηnθ+enθ)L2θn+θnhLen+1hL2C(1+θnh2L)(h4+enθ2L2)+C(enθ2L2+en+1h2L2),

    where we use the Sobolev imbedding inequality and C>0 is independent of h,τ,ν and κ. Taking the sum gives

    2τmn=0(Jn+16,en+1h)Ch4τmn=0(1+θnh2L)+Cτmn=0(enθ2L2+en+1h2L2)+Cτmn=0(1+θnh2L)enθ2L2, (3.18)

    where C>0 is independent of h,τ,ν and κ.

    Substituting the estimates (3.13)–(3.18) into (3.12), we obtain

    em+1h2L2+2τmn=0(νen+1h2L2+βen+1h2L2)C(h4+τ2)+Ch4τmn=0(1+θnh2L)+Cτmn=0(enθ2L2+en+1h2L2+enh2L2)+2ϵ1βτmn=0en+1hL2+Cτmn=0(1+θnh2L)enθ2L2, (3.19)

    where C > 0 is independent of h, \tau, \nu and \kappa .

    Next, we estimate e_\theta^{n+1} . Taking \psi_h = 2\tau e_\theta^{n+1} in (3.9), summing up the resulting equation from n = 0 to n = m and noticing the skew-symmetric property (2.1), we have

    \begin{align} \|e_{\theta}^{m+1}\|_{L^2}^{2}+2\kappa \tau\sum\limits_{n = 0}^{m} \|\nabla e_\theta^{n+1}\|_{{{\textbf{L}}}^2}^{2} = 2\tau\sum\limits_{n = 0}^{m} \sum_{j = 1}^3 (\lambda_{j}^{n+1}, e_\theta^{n+1}) , \end{align} (3.20)

    where we noted e_{\theta}^0 = 0 . By the Taylor's formula and the Hölder inequality, it is easy to prove that

    \begin{align} \begin{split} 2\tau\sum\limits_{n = 0}^{m} (\lambda_{1}^{n+1}, e_\theta^{n+1}) \leq C \tau \sum_{n = 0}^{m} \| e_\theta^{n+1}\|_{L^2}^2 + C\tau^2 , \\ \end{split} \end{align} (3.21)
    \begin{align} \begin{split} 2\tau\sum\limits_{n = 0}^{m} (\lambda_{2}^{n+1}, e_\theta^{n+1}) &\leq C \tau \sum_{n = 0}^{m} \| e_{\theta}^{n+1}\|_{L^2}^2 + Ch^{4} \tau \sum_{n = 0}^{m} \|D_{1,\tau}\theta^{n+1}\|_{H^1}^2 \\ &\leq C \tau \sum_{n = 0}^{m} \| e_{\theta}^{n+1}\|_{L^2}^2 + Ch^{4} \end{split} \end{align} (3.22)

    by noticing

    \begin{align*} \tau \sum_{n = 0}^{m} \|D_{1,\tau}\theta^{n+1}\|_{H^1}^2 \leq \frac 1 \tau \sum_{n = 0}^{m} \left( { \int}_{t_n}^{t_{n+1}} \|\theta_t(t)\|_{H^1}dt \right)^2 \leq C, \end{align*}

    where C > 0 is independent of h, \tau , \nu and \kappa .

    For last term \lambda_3^{n+1} , in terms of the following splitting:

    \begin{align*} (\lambda_{3}^{n+1}, e_{\theta}^{n+1}) & = ({{\textbf{u}}}_h^n\cdot\nabla R_h\theta^{n+1},e_{\theta}^{n+1}) +\frac{1}{2}((\nabla\cdot{{\textbf{u}}}_h^n)R_h\theta^{n+1},e_{\theta}^{n+1}) -({{\textbf{u}}}^{n+1}\cdot\nabla\theta^{n+1},e_{\theta}^{n+1})\\ & = -(e_h^n\cdot\nabla R_h\theta^{n+1},e_{\theta}^{n+1}) -({{\textbf{s}}}_h^n\cdot\nabla\eta_{\theta}^{n+1},e_{\theta}^{n+1}) +(({{\textbf{s}}}_h^n-{{\textbf{u}}}^{n+1})\cdot\nabla\theta^{n+1},e_{\theta}^{n+1})\\ &-\frac{1}{2}((\nabla\cdot e_h^n)R_h\theta^{n+1},e_{\theta}^{n+1}) +\frac{1}{2}(\nabla\cdot({{\textbf{s}}}_{h}^n-{{\textbf{u}}}^n)R_h\theta^{n+1},e_{\theta}^{n+1}){,} \end{align*}

    we estimate it by

    \begin{align*} (\lambda_{3}^{n+1}, e_{\theta}^{n+1}) \leq & ( \|e_h^n\|_{{{\textbf{L}}}^2} \|\nabla R_h\theta^{n+1}\|_{{{\textbf{L}}}^\infty} +\|{{\textbf{s}}}_{h}^{n}\|_{{{\textbf{L}}}^\infty}\|\nabla\eta_{\theta}^{n+1}\|_{{{\textbf{L}}}^2} + \|{{\textbf{s}}}_h^n-{{\textbf{u}}}^{n+1}\|_{{{\textbf{L}}}^3}\|\nabla\theta^{n+1}\|_{{{\textbf{L}}}^6}) \|e_{\theta}^{n+1}\|_{L^2} \\ +&\frac{1}{2}(\|\nabla\cdot e_h^n\|_{L^2}\|R_h\theta^{n+1}\|_{L^{\infty}} +\|{{\textbf{s}}}_h^n-{{\textbf{u}}}^n\|_{{{\textbf{H}}}^1}\|R_h\theta^{n+1}\|_{L^{\infty}})\|e_{\theta}^{n+1}\|_{L^2}\\ \leq & C (h^{4}+\tau^2) + C (\| e_{\theta}^{n+1}\|_{L^2}^2 + \|e_h^n\|_{{{\textbf{L}}}^2}^2) + \frac {\epsilon_1\beta }2 \|\nabla \cdot e_h^{n}\|_{L^2}^2 . \end{align*}

    Furthermore, we get

    \begin{align} \begin{split} 2\tau \sum\limits_{n = 0}^{m} (\lambda_{3}^{n+1}, e_\theta^{n+1}) \leq & C (h^{4}+\tau^2) + C\tau \sum\limits_{n = 0}^{m} (\| e_{\theta}^{n+1}\|_{L^2}^2 + \|e_h^n\|_{{{\textbf{L}}}^2}^2) + \epsilon_1\beta \tau \sum_{n = 0}^{m} \|\nabla \cdot e_h^{n}\|_{L^2}^2, \end{split} \end{align} (3.23)

    where C > 0 is independent of h, \tau , \nu and \kappa .

    Substituting the estimates (3.21)–(3.23) into (3.20), we get

    \begin{align} \begin{split} & \|e_{\theta}^{m+1}\|_{L^2}^{2}+2\kappa \tau\sum\limits_{n = 0}^{m} \|\nabla e_\theta^{n+1}\|_{{{\textbf{L}}}^2}^{2} \\ \leq & C(h^{4}+\tau^2) + C\tau \sum\limits_{n = 0}^{m} (\| e_{\theta}^{n+1}\|_{L^2}^2 + \|e_h^n\|_{{{\textbf{L}}}^2}^2) + \epsilon_1\beta \tau \sum_{n = 0}^{m} \|\nabla \cdot e_h^{n}\|_{L^2}^2, \end{split} \end{align} (3.24)

    where C > 0 is independent of h, \tau , \nu and \kappa .

    Taking the sum of (3.19) and (3.24), we have

    \begin{align} \begin{split} & \|e_{h}^{m+1}\|_{{{\textbf{L}}}^2}^{2}+\|e_{\theta}^{m+1}\|_{L^2}^{2}+2\tau\sum\limits_{n = 0}^{m} \left ( \nu\|\nabla e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \kappa \|\nabla e_\theta^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \beta \|\nabla\cdot e_{h}^{n+1}\|_{L^2}^{2} \right ) \\ \leq & C \tau \sum_{n = 0}^{m} (\|e_{\theta}^{n+1}\|_{L^2}^{2}+\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} +\|e_{\theta}^{n}\|_{L^2}^{2}+\|e_{h}^{n}\|_{{{\textbf{L}}}^2}^{2}) + C(h^{4}+\tau^2) \\ & + C h^{4} \tau \sum_{n = 0}^{m} (1+\|\theta_{h}^{n}\|_{L^\infty}^2) + 3\epsilon_1\beta \tau \sum_{n = 0}^{m} \|\nabla\cdot e_h^{n+1}\|_{L^2} + C\tau \sum_{n = 0}^{m} (1+\|\theta_{h}^{n}\|_{L^\infty}^2) \|e_{\theta}^{n}\|_{L^2}^2 , \end{split} \end{align} (3.25)

    where C > 0 is independent of h, \tau , \nu and \kappa .

    To uniformly bound the norm \|\theta_h^n\|_{L^\infty} , we use the method of mathematical induction. Taking m = 0 in (3.25) and noting e_h^0 = 0 , e_\theta^0 = 0 and \|\theta_h^0\|_{L^\infty} = \|R_h\theta_0\|_{L^\infty}\leq \|\theta_0\|_{H^3} , we get

    \begin{align} \begin{split} & \|e_{h}^{1}\|_{{{\textbf{L}}}^2}^{2}+\|e_{\theta}^{1}\|_{L^2}^{2}+2\tau \left ( \nu\|\nabla e_{h}^{1}\|_{{{\textbf{L}}}^2}^{2} + \kappa \|\nabla e_\theta^{1}\|_{{{\textbf{L}}}^2}^{2} + \beta \|\nabla\cdot e_{h}^{1}\|_{L^2}^{2} \right ) \\ \leq & C \tau (\|e_{\theta}^{1}\|_{L^2}^{2}+\|e_{h}^{1}\|_{{{\textbf{L}}}^2}^{2} ) + C(h^4+\tau^2) + 3\epsilon_1\beta \tau \|\nabla\cdot e_h^{1}\|_{L^2}. \end{split} \end{align} (3.26)

    For sufficiently small \tau with C\tau < 1/2 , we select small parameter \epsilon_1\leq 2/3 in (3.26). We then conclude that there exists some C_1 > 0 , which is independent of h, \tau , \nu and \kappa such that

    \begin{align*} \|e_{h}^{1}\|_{{{\textbf{L}}}^2}^{2}+\|e_{\theta}^{1}\|_{L^2}^{2}+\tau \left ( \nu\|\nabla e_{h}^{1}\|_{{{\textbf{L}}}^2}^{2} + \kappa \|\nabla e_\theta^{1}\|_{{{\textbf{L}}}^2}^{2}\right ) \leq C_1^2 (h^4+\tau^2). \end{align*}

    Thus, the error estimate (3.11) holds for m = 0 by taking C_0 > C_1 . Now, we assume that the error estimate (3.11) holds for m\leq k-1 with 1\leq k\leq N-1 . Under the time step condition \tau\leq Ch^2 , we have

    \begin{align} \|e_{\theta}^{n}\|_{L^2} \leq CC_0h^2 \qquad \forall \ 1\leq n\leq k. \end{align} (3.27)

    By the inverse inequality, we get

    \begin{align*} \|e_{\theta}^{n}\|_{L^\infty} \leq Ch^{-3/2}\|e_{\theta}^{n}\|_{L^2} \leq CC_0h^{1/2} \leq C \end{align*}

    for sufficiently small h with C_0h^{1/2} \leq 1 . Thus,

    \begin{align} \|\theta_h^{n}\|_{L^\infty} \leq \|e_\theta^n\|_{L^\infty} + \|\theta^n\|_{L^\infty}\leq C \quad {\mbox{and}}\quad \tau \sum_{n = 1}^{k} \| \theta_{h}^{n}\|_{L^\infty}^2 \leq C. \end{align} (3.28)

    As a result, resetting m by k in (3.25), we have

    \begin{align} \begin{split} & \|e_{h}^{k+1}\|_{{{\textbf{L}}}^2}^{2}+\|e_{\theta}^{k+1}\|_{L^2}^{2}+2\tau\sum\limits_{n = 0}^{k} \left ( \nu\|\nabla e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \kappa \|\nabla e_\theta^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \beta \|\nabla\cdot e_{h}^{n+1}\|_{L^2}^{2} \right ) \\ \leq & C(h^4+\tau^2) + C \tau \sum_{n = 0}^{k} (\|e_{\theta}^{n+1}\|_{L^2}^{2}+\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} +\|e_{\theta}^{n}\|_{L^2}^{2}+\|e_{h}^{n}\|_{{{\textbf{L}}}^2}^{2}) \\ & + 3\epsilon_1\beta \tau \sum_{n = 0}^{k} \|\nabla\cdot e_h^{n+1}\|_{L^2} . \end{split} \end{align} (3.29)

    We select small parameter \epsilon_1\leq 2/3 in (3.29), then

    \begin{align*} & \|e_{h}^{k+1}\|_{{{\textbf{L}}}^2}^{2}+\|e_{\theta}^{k+1}\|_{L^2}^{2}+\tau\sum\limits_{n = 0}^{k} \left ( \nu\|\nabla e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \kappa \|\nabla e_\theta^{n+1}\|_{{{\textbf{L}}}^2}^{2} \right ) \\ \leq & C(h^4+\tau^2) + C \tau \sum_{n = 0}^{k} (\|e_{\theta}^{n+1}\|_{L^2}^{2}+\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} +\|e_{\theta}^{n}\|_{L^2}^{2}+\|e_{h}^{n}\|_{{{\textbf{L}}}^2}^{2}). \end{align*}

    Applying the discrete Gronwall inequality to the above inequality, we conclude that there exists some C_2 > 0 , which is independent of h, \tau , \nu and \kappa such that

    \begin{align*} \|e_{h}^{k+1}\|_{{{\textbf{L}}}^2}^{2}+\|e_{\theta}^{k+1}\|_{L^2}^{2}+\tau\sum\limits_{n = 0}^{k} \left ( \nu\|\nabla e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \kappa \|\nabla e_{\theta}^{n+1}\|_{{{\textbf{L}}}^2}^{2} \right ) \leq C_2^2(h^4+\tau^2) \end{align*}

    for sufficiently small \tau . Thus, we prove that the error estimate (3.11) also holds for m = k by taking C_0\geq C_2 , and we finish the mathematical induction.

    Based on (3.11) in Theorem 3.1, we get the following error estimate for the first-order Euler grad-div finite element scheme (3.1)–(3.2).

    Theorem 3.2 Under the assumptions in Theorem 3.1, we have

    \begin{align} \|{{\textbf{u}}}^{m+1}-{{\textbf{u}}}_h^{m+1}\|_{{{\textbf{L}}}^2}^{2}+\|\theta^{m+1}-\theta_h^{m+1}\|_{L^2}^{2} \leq C(h^{4}+\tau^2) \end{align} (3.30)

    with 0\leq m\leq N-1 , where C > 0 is independent of h, \tau , \nu and \kappa .

    In this section, we consider the second-order BDF2 fully discrete scheme. Based on the extrapolation method and grad-div stabilization method, we propose the following linearized second-order finite element scheme for 1\leq n\leq N-1 :

    Step Ⅰ: Find \theta_h^{n+1}\in Y_h by

    \begin{align} (D_{2,\tau}\theta_{h}^{n+1},\psi_{h})+\kappa(\nabla \theta_{h}^{n+1},\nabla\psi_{h})+c_2(\widehat{{{\textbf{u}}}}_{h}^{n},\theta_{h}^{n+1},\psi_{h}) = (g^{n+1},\psi_{h}) \quad \forall \ \psi_h\in Y_h. \end{align} (4.1)

    Step Ⅱ: Find ({{\textbf{u}}}_{h}^{n+1}, \theta_{h}^{n+1})\in {{\textbf{V}}}_{h}\times Q_{h} by

    \begin{eqnarray} (D_{2,\tau}{{\textbf{u}}}_{h}^{n+1},{{\textbf{v}}}_{h})+\nu(\nabla{{\textbf{u}}}_{h}^{n+1},\nabla {{\textbf{v}}}_{h})+c_1(\widehat {{{\textbf{u}}}}_{h}^{n},{{\textbf{u}}}_{h}^{n+1},{{\textbf{v}}}_{h})-(\nabla\cdot{{\textbf{v}}}_{h},p_{h}^{n+1}) +(\nabla\cdot{{\textbf{u}}}_{h}^{n+1},q_{h})\\ +\beta(\nabla\cdot{{\textbf{u}}}_{h}^{n+1},\nabla\cdot{{\textbf{v}}}_{h})-(\gamma_{1}\widehat {\theta}_{h}^{n} + \gamma_2(\widehat \theta_{h}^{n})^{2} ,{\textbf{i}}_{3} \cdot {{\textbf{v}}}_{h}) = ({{\textbf{f}}}^{n+1},{{\textbf{v}}}_{h}) \quad \forall \ ({{\textbf{v}}}_{h},q_{h}) \in{{\textbf{V}}}_{h} \times Q_{h}. \end{eqnarray} (4.2)

    Remark 4.1. Numerical solution ({{\textbf{u}}}_h^1, \theta_h^1)\in {{\textbf{V}}}_{0h}\times Y_h can be solved from the Euler finite element scheme (3.1)–(3.2) in the above section. Usually, since the one-step error is first higher order, then one has

    \begin{align} \|e_{h}^{1}\|_{{{\textbf{L}}}^2}^{2}+\|e_{\theta}^{1}\|_{L^2}^{2}+\tau \left ( \nu\|\nabla e_{h}^{1}\|_{{{\textbf{L}}}^2}^{2} + \kappa \|\nabla e_{\theta}^{1}\|_{{{\textbf{L}}}^2}^{2} + \beta\|\nabla\cdot e_h^1\|_{L^2}^2 \right ) \leq C_3^2(h^{4}+\tau^4), \end{align} (4.3)

    where C_3 > 0 is independent of h, \tau , \nu and \kappa .

    By a similar proof for Lemma 3.1, we can get the following unconditional stabilities of the numerical scheme (4.1)–(4.2), which imply the existence and uniqueness of numerical solution ({{\textbf{u}}}_h^{n+1}, p_h^{n+1}, \theta_h^{n+1}) to the BDF2 scheme (4.1)–(4.2).

    Lemma 4.1. For 1\leq n\leq N-1 and all \tau > 0 , h > 0 , the BDF2 finite element scheme (4.1) and (4.2) has a unique solution \theta_{h}^{n+1}\in Y_{h} and ({{\textbf{u}}}_{h}^{n+1}, p_{h}^{n+1})\in {{\textbf{V}}}_{h}\times Q_{h} . Moreover, the discrete energy inequalities hold:

    \begin{align} \|\theta_{h}^{m+1}\|_{L^{2}}^{2}+\|\widehat {\theta}_{h}^{m+1}\|_{L^{2}}^{2} +2\kappa\tau\sum\limits_{n = 1}^{m}\|\nabla\theta_{h}^{n+1}\|_{{{\textbf{L}}}^{2}}^{2}\leq C\tau\sum\limits_{n = 1}^{m}\|g^{n+1}\|_{L^{2}}^{2} +C\left( \|\theta_{h}^{1}\|_{L^{2}}^{2} + \| {\theta}_{h}^{0}\|_{L^{2}}^{2} \right), \end{align} (4.4)

    and

    \begin{align} \begin{split} &\|{{\textbf{u}}}_{h}^{m+1}\|_{{{\textbf{L}}}^{2}}^{2}+\|\widehat {{\textbf{u}}}_{h}^{m+1}\|_{{{\textbf{L}}}^{2}}^{2}+2\tau\nu\sum\limits_{n = 1}^{m}\|\nabla {{\textbf{u}}}_{h}^{n+1}\|_{{{\textbf{L}}}^{2}}^{2}+4\tau\beta\sum\limits_{n = 1}^{m}\|\nabla\cdot{{\textbf{u}}}_{h}^{n+1}\|^{2}_{{{\textbf{L}}}^2}\\ \leq & C\tau\sum\limits_{n = 1}^{m}(\|{{\textbf{f}}}^{n+1}\|_{{{\textbf{L}}}^{2}}^{2}+\|g^{n+1}\|_{L^{2}}^{2})+C(\|\theta_{h}^{1}\|_{L^{2}}^{2} +\|{\theta}_{h}^{0}\|_{L^{2}}^{2}) \\ & +C\left (\tau\sum\limits_{n = 1}^{m}\|g^{n+1}\|_{L^{2}}^{2}+\|\theta_{h}^{1}\|_{L^{2}}^{2} +\|{\theta}_{h}^{0}\|_{L^{2}}^{2} \right )^{2} \end{split} \end{align} (4.5)

    for all 1\leq m\leq N-1 , where C > 0 is independent of h and \tau .

    In terms of the discrete inf-sup condition (2.5), we rewrite the discrete variational problem (4.2) as: find {{\textbf{u}}}_{h}^{n+1}\in {{\textbf{V}}}_{0h} by

    \begin{eqnarray} (D_{2,\tau}{{\textbf{u}}}_{h}^{n+1},{{\textbf{v}}}_{h})+\nu(\nabla{{\textbf{u}}}_{h}^{n+1},\nabla {{\textbf{v}}}_{h}) +c_1(\widehat {{{\textbf{u}}}}_{h}^{n},{{\textbf{u}}}_{h}^{n+1},{{\textbf{v}}}_{h})+\beta(\nabla\cdot{{\textbf{u}}}_{h}^{n+1},\nabla\cdot{{\textbf{v}}}_{h}) -(\gamma_{1}\widehat {\theta}_{h}^{n}+\gamma_2 (\widehat \theta_{h}^{n})^2, {\textbf{i}}_{3}\cdot{{\textbf{v}}}_{h}) = ({{\textbf{f}}}^{n+1},{{\textbf{v}}}_{h}) \quad \forall \ {{\textbf{v}}}_h\in{{\textbf{V}}}_{0h}. \end{eqnarray} (4.6)

    On the other hand, let {{\textbf{s}}}_{h}^{0} = {{\textbf{u}}}_{h}^{0} . For 1\leq n\leq N-1 , {{\textbf{s}}}_{h}^{n+1} and \theta^{n+1} satisfy

    \begin{align} \begin{split} &(D_{2,\tau}{{\textbf{s}}}_{h}^{n+1},{{\textbf{v}}}_{h})+\nu(\nabla {{\textbf{s}}}_{h}^{n+1},\nabla {{\textbf{v}}}_{h})+c_1(\widehat{{{\textbf{s}}}}_{h}^{n},{{\textbf{s}}}_{h}^{n+1},{{\textbf{v}}}_{h}) +\beta(\nabla\cdot {{\textbf{s}}}_{h}^{n+1},\nabla\cdot{{\textbf{v}}}_{h})\\ &-\gamma_{1}(\widehat {\theta}^{n},\textbf i_{3}\cdot{{\textbf{v}}}_{h}) -\gamma_{2}((\widehat \theta^{n})^{2},\textbf i_{3}\cdot{{\textbf{v}}}_{h})\\ & = ({{\textbf{f}}}^{n+1},{{\textbf{v}}}_{h})+(\nabla\cdot{{\textbf{v}}}_{h}, p^{n+1}-\pi_{h}p^{n+1})\\ &+\beta(\nabla\cdot({{\textbf{s}}}_{h}^{n+1}-{{\textbf{u}}}^{n+1}),\nabla\cdot{{\textbf{v}}}_{h}) +(D_{2,\tau} {{\textbf{s}}}_{h}^{n+1}-{{\textbf{u}}}_{t}(t_{n+1}),{{\textbf{v}}}_{h})\\ & +c_1(\widehat{{{\textbf{s}}}}_{h}^{n},{{\textbf{s}}}_{h}^{n+1},{{\textbf{v}}}_{h})-c_1({{\textbf{u}}}^{n+1},{{\textbf{u}}}^{n+1},{{\textbf{v}}}_{h})\\ &+\gamma_{1}(\theta^{n+1}-\widehat {\theta}^{n}, \textbf i_{3}\cdot{{\textbf{v}}}_{h}) +\gamma_{2}((\theta^{n+1})^{2}-(\widehat \theta^{n})^{2},\textbf i_{3}\cdot{{\textbf{v}}}_{h}) \quad \forall \ {{\textbf{v}}}_h\in{{\textbf{V}}}_{0h}, \end{split} \end{align} (4.7)

    and

    \begin{eqnarray} (D_{2,\tau}\theta^{n+1},\psi_{h})+\kappa(\nabla \theta^{n+1},\nabla\psi_{h})+c_2({{\textbf{u}}}^{n+1},\theta^{n+1},\psi_{h}) = (g^{n+1},\psi_{h}) +(D_{2,\tau}\theta^{n+1}-\theta_{t}^{n+1},\psi_{h}) \quad \forall \ \psi_h\in Y_h. \end{eqnarray} (4.8)

    In (4.7), we use \nabla\cdot{{\textbf{u}}}^{n+1} = 0 in \Omega and (\nabla\cdot{{\textbf{v}}}_h, \pi_hp^{n+1}) = 0 .

    Next, we give error equations corresponding to the BDF2 scheme. Subtracting (4.6) from (4.7) leads to

    \begin{align} (D_{2,\tau}e_{h}^{n+1},{{\textbf{v}}}_{h})+\nu(\nabla e_{h}^{n+1},\nabla{{\textbf{v}}}_{h}) +\beta(\nabla\cdot e_{h}^{n+1},\nabla\cdot{{\textbf{v}}}_{h}) = \sum_{j = 1}^6 (I_{j}^{n+1},{{\textbf{v}}}_{h}) \quad \forall \ {{\textbf{v}}}_h\in{{\textbf{V}}}_{0h}, \end{align} (4.9)

    where

    \begin{eqnarray*} (I_{1}^{n+1},{{\textbf{v}}}_{h})& = &( D_{2,\tau}{{\textbf{s}}}_{h}^{n+1}-{{\textbf{u}}}_{t}(t_{n+1}),{{\textbf{v}}}_{h}) , \\ (I_{2}^{n+1},{{\textbf{v}}}_{h})& = &(\nabla\cdot{{\textbf{v}}}_{h},(p^{n+1}-\pi_h p^{n+1})+\beta\nabla\cdot({{\textbf{s}}}_h^{n+1}-{{\textbf{u}}}^{n+1})), \\ (I_{3}^{n+1},{{\textbf{v}}}_{h})& = &c_1(\widehat {{{\textbf{s}}}}_{h}^{n},{{\textbf{s}}}_{h}^{n+1},{{\textbf{v}}}_{h})-c_1({{\textbf{u}}}^{n+1},{{\textbf{u}}}^{n+1},{{\textbf{v}}}_{h}), \\ (I_{4}^{n+1},{{\textbf{v}}}_{h})& = &c_1(\widehat{{{\textbf{u}}}}_{h}^{n},{{\textbf{u}}}_{h}^{n+1},{{\textbf{v}}}_{h})-c_1(\widehat {{{\textbf{s}}}}_{h}^{n},{{\textbf{s}}}_{h}^{n+1},{{\textbf{v}}}_{h}),\\ (I_{5}^{n+1},{{\textbf{v}}}_{h})& = &\gamma_{1}(\theta^{n+1}-\widehat {\theta}^{n}, \textbf i_{3}\cdot{{\textbf{v}}}_{h}) +\gamma_{2}( (\theta^{n+1})^{2}- (\widehat \theta^{n})^{2}, \textbf i_{3}\cdot{{\textbf{v}}}_{h}),\\ (I_{6}^{n+1},{{\textbf{v}}}_{h})& = &\gamma_{1}((\widehat \theta^{n}-\widehat \theta_{h}^{n}),\textbf i_{3}\cdot{{\textbf{v}}}_{h})+ \gamma_2(((\widehat\theta^{n})^{2}-(\widehat\theta_{h}^{n})^{2}), \textbf i_{3}\cdot{{\textbf{v}}}_{h}). \end{eqnarray*}

    Subtracting (4.1) from (4.8) leads to

    \begin{align} (D_{2,\tau}e_{\theta}^{n+1},\psi_{h})+\kappa(\nabla e_{\theta}^{n+1},\nabla\psi_{h})+c_2(\widehat{{\textbf{u}}}_h^n,e_{\theta}^{n+1},\psi) = \sum_{j = 1}^{3}(X_{j}^{n+1},\psi_{h}) \quad \forall \ \psi_h\in Y_{h}, \end{align} (4.10)

    where

    \begin{eqnarray*} (X_{1}^{n+1},\psi_{h})& = &( D_{2,\tau}\theta^{n+1}-\theta_t(t_{n+1}),\psi_{h}),\\ (X_{2}^{n+1},\psi_{h})& = &- (D_{2,\tau}\eta_\theta^{n+1},\psi_{h}),\\ (X_{3}^{n+1},\psi_{h})& = &c_2(\widehat {{\textbf{u}}}_{h}^{n},R_h\theta^{n+1},\psi_{h}) -c_2( {{\textbf{u}}}^{n+1},\theta^{n+1},\psi_{h}). \end{eqnarray*}

    The main result in this section is presented in the following theorem.

    Theorem 4.1. Under the assumptions in (3.10), we further assume that

    \begin{align} {{\textbf{u}}}_{ttt} \in {{\textbf{L}}}^2((0, T]; {{\textbf{L}}}^2(\Omega)), \quad \theta_{ttt} \in L^2((0, T]; L^2(\Omega)). \end{align} (4.11)

    Then, for the second-order BDF2 grad-div scheme (4.1)–(4.2), under the time step condition \tau\leq Ch , when h and \tau are sufficiently small, we have the following error estimate:

    \begin{align} \|e_h^{m+1}\|_{{{\textbf{L}}}^2}^{2}+\|e_\theta^{m+1}\|_{L^2}^{2} +\tau\sum\limits_{n = 0}^{m} \left ( \nu\|\nabla e_h^{m+1}\|_{{{\textbf{L}}}^2}^{2} + \kappa \|\nabla e_\theta^{m+1}\|_{{{\textbf{L}}}^2}^{2} \right ) \leq \widehat C_0^2(h^{4}+\tau^4) \end{align} (4.12)

    with 0\leq m\leq N-1 , where \widehat C_0 > 0 is independent of h, \tau , \nu and \kappa .

    Proof. Taking {{\textbf{v}}}_h = 4\tau e_h^{n+1} in (4.9) and summing up the resulting equation from n = 1 to n = m , we have

    \begin{align} \begin{split} & \|e_{h}^{m+1}\|_{{{\textbf{L}}}^2}^{2}+\|\widehat e_{h}^{m+1}\|_{{{\textbf{L}}}^2}^{2}+4\tau\sum\limits_{n = 1}^{m} \left ( \nu\|\nabla e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \beta \|\nabla\cdot e_{h}^{n+1}\|_{L^2}^{2} \right ) \\ \leq & 4\tau\sum\limits_{n = 1}^{m} \sum_{j = 1}^6 (I_{j}^{n+1},e_{h}^{n+1})+C\|e_h^1\|_{{{\textbf{L}}}^2} , \end{split} \end{align} (4.13)

    where C > 0 is independent of h, \tau , \nu, \kappa and we use

    \begin{align*} \|\widehat e_h^1\|_{{{\textbf{L}}}^2} \leq 2 \|e_h^1\|_{{{\textbf{L}}}^2} +\|e_h^0\|_{{{\textbf{L}}}^2} = 2\|e_h^1\|_{{{\textbf{L}}}^2} . \end{align*}

    We estimate the right-hand side of (4.13) as follows. Since

    \begin{align*} I_1^{n+1} = ( D_{2,\tau}{{\textbf{s}}}_{h}^{n+1}-D_{2,\tau}{{\textbf{u}}}^{n+1})+ (D_{2,\tau}{{\textbf{u}}}^{n+1} -{{\textbf{u}}}_{t}(t_{n+1})) , \end{align*}

    by using the Taylor's formula and the regularity assumption (4.11), one has

    \begin{align*} D_{2,\tau}{{\textbf{u}}}^{n+1} -{{\textbf{u}}}_{t}(t_{n+1}) = \frac{1}{2\tau}\int_{t_{n-1}}^{t^{n+1}} \left( 2(t-t_{n})^{2}-\frac{1}{2}(t-t_{n-1})^{2}\right)\partial_{ttt}{{\textbf{u}}}(t) dt, \end{align*}

    which results in

    \begin{align*} \tau\sum\limits_{n = 1}^{m} \|D_{2,\tau}{{\textbf{u}}}^{n+1} -{{\textbf{u}}}_{t}(t_{n+1})\|_{{{\textbf{L}}}^2}^2 \leq C\tau^4, \end{align*}

    where C > 0 is independent of h, \tau , \nu and \kappa . From the Hölder inequality, we have

    \begin{align*} &\quad \|D_{2,\tau}{{\textbf{s}}}_{h}^{n+1}-D_{2,\tau}{{\textbf{u}}}^{n+1}\|_{{{\textbf{L}}}^2} \\ &\leq \frac {3}{2\tau}\int_{t_{n}}^{t^{n+1}} \|\partial_{t}({{\textbf{s}}}_{h}(t)-{{\textbf{u}}}(t))\|_{{{\textbf{L}}}^2}dt+ \frac{1}{2\tau}\int_{t_{n-1}}^{t^{n}} \|\partial_{t}({{\textbf{s}}}_{h}(t)-{{\textbf{u}}}(t))\|_{{{\textbf{L}}}^2}dt\\ &\leq \frac {Ch^2}{\tau} \int_{t_{n-1}}^{t^{n+1}} \|\partial_{t}{{\textbf{u}}}(t)\|_{{{\textbf{H}}}^2}dt\\ &\leq \frac {Ch^2}{\tau^{1/2}} \left( { \int}_{t_n-1}^{t_{n+1}} \| \partial_t{{\textbf{u}}}(t)\|_{{{\textbf{H}}}^2}^2 dt \right)^{1/2}. \end{align*}

    Thus, we can obtain

    \begin{align} 4\tau\sum\limits_{n = 1}^{m} (I_{1}^{n+1},e_{h}^{n+1}) \leq C \tau \sum_{n = 1}^{m} \| e_h^{n+1}\|_{{{\textbf{L}}}^2}^{2} + C(\tau^4+h^{4}), \end{align} (4.14)

    where C > 0 is independent of h, \tau , \nu and \kappa .

    From the Hölder inequality and (2.6), we estimate (I_{2}^{n+1}, e_{h}^{n+1}) by

    \begin{align*} (I_{2}^{n+1},e_{h}^{n+1}) \leq & \left( \|p^{n+1}-\pi_h p^{n+1}\|_{L^2} + \beta \|{{\textbf{s}}}_h^{n+1}-{{\textbf{u}}}^{n+1}\|_{{{\textbf{H}}}^1} \right) \|\nabla\cdot e_h^{n+1}\|_{L^2} \\ \leq & C h^2 \|\nabla\cdot e_h^{n+1}\|_{L^2}. \end{align*}

    Then, by the Young inequality, it is easy to see that

    \begin{align} 4\tau \sum\limits_{n = 1}^{m} (I_{2}^{n+1},e_{h}^{n+1}) \leq Ch^{4} + \epsilon_2\beta \tau \sum_{n = 1}^{m} \|\nabla\cdot e_h^{n+1}\|_{L^2}^{2} , \end{align} (4.15)

    where C > 0 is independent of h, \tau, \nu, \kappa and \epsilon_2 > 0 is a small constant determined later.

    For the term of I_3^{n+1} , according to the Hölder inequality, (2.6), (2.18), the regularity assumption (3.10) and the Taylor formula with integral remainder

    \begin{align} \widehat {{{\textbf{u}}}}^{n}-{{\textbf{u}}}^{n+1} = \int_{t_{n-1}}^{t_{n+1}}\{2(t-t_{n})_+ -(t-t_{n-1})\}\partial_{tt}{{\textbf{u}}}(t) dt, \end{align} (4.16)

    where (t-t_{n})_+ = \max\{t-t_n, 0\} , there holds

    \begin{align*} (I_{3}^{n+1},e_{h}^{n+1}) & = c_1(\widehat {{{\textbf{s}}}}_{h}^{n}-\widehat {{{\textbf{u}}}}^{n},{{\textbf{s}}}_{h}^{n+1},e_{h}^{n+1}) +c_1(\widehat {{{\textbf{u}}}}^{n}-{{\textbf{u}}}^{n+1},{{\textbf{s}}}_{h}^{n+1},e_{h}^{n+1}) \\ &\quad +c_1({{\textbf{u}}}^{n+1},{{\textbf{s}}}_{h}^{n+1}-{{\textbf{u}}}^{n+1},e_{h}^{n+1})\\ &\leq ( \|\widehat {{{\textbf{s}}}}_{h}^{n}-\widehat {{{\textbf{u}}}}^{n}\|_{{{\textbf{H}}}^1} + \|\widehat {{{\textbf{u}}}}^{n}-{{\textbf{u}}}^{n+1}\|_{{{\textbf{L}}}^2} )\|{{\textbf{s}}}_h^{n+1}\|_{{{\textbf{W}}}^{1,\infty}} \|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}\\ & \quad + C \|{{\textbf{u}}}^{n+1}\|_{{{\textbf{L}}}^{\infty}} \| {{\textbf{s}}}_h^{n+1}-{{\textbf{u}}}^{n+1}\|_{{{\textbf{H}}}^1} \|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}\\ &\leq C(h^2+(\tau^{3}\int_{t_{n-1}}^{t^{n+1}}\|\partial_{tt}{{\textbf{u}}}(t)\|_{{{\textbf{L}}}^2}^{2}dt)^{\frac{1}{2}})\|e_{h}^{n+1}\|_{{{\textbf{L}}}^{2}}. \end{align*}

    Thus, we have

    \begin{align} 4\tau \sum\limits_{n = 1}^{m} (I_{3}^{n+1},e_{h}^{n+1}) \leq C (h^{4}+\tau^4) + C \tau \sum_{n = 1}^{m} \| e_h^{n+1}\|_{{{\textbf{L}}}^2}^2, \end{align} (4.17)

    where C > 0 is independent of h, \tau , \nu and \kappa .

    A similar method leads to

    \begin{align*} (I_{4}^{n+1},e_{h}^{n+1}) & = c_1 (\widehat e_{h}^{n},{{\textbf{s}}}_{h}^{n+1},e_{h}^{n+1}) \\ & \leq \|{\widehat e}_{h}^{n}\|_{{{\textbf{L}}}^2}\|\nabla{{\textbf{s}}}_{h}^{n+1}\|_{{{\textbf{L}}}^\infty}\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2} +\|{{\textbf{s}}}_{h}^{n+1}\|_{{{\textbf{L}}}^\infty}\|\nabla\cdot \widehat e_{h}^{n}\|_{L^2}\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2} \\ & \leq \|{\widehat e}_{h}^{n}\|_{{{\textbf{L}}}^2}\|\nabla{{\textbf{s}}}_{h}^{n+1}\|_{{{\textbf{L}}}^\infty}\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2} +\|{{\textbf{s}}}_{h}^{n+1}\|_{{{\textbf{L}}}^\infty}\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2} (2\|\nabla\cdot e_{h}^{n}\|_{L^2}+\|\nabla\cdot e_{h}^{n-1}\|_{L^2}) \end{align*}

    and

    \begin{align} \begin{split} 4\tau \sum\limits_{n = 1}^{m} (I_{4}^{n+1},e_{h}^{n+1}) \leq & \epsilon_2\beta \tau \sum_{n = 1}^{m} (\|\nabla\cdot e_h^{n}\|_{L^2}^{2} + \|\nabla\cdot e_h^{n-1}\|_{L^2}^{2} ) \\ & + C\tau \sum_{n = 1}^{m}(\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2}+\|\widehat e_{h}^{n}\|_{{{\textbf{L}}}^2}^{2}), \end{split} \end{align} (4.18)

    where C > 0 is independent of h, \tau , \nu and \kappa .

    For I_5^{n+1} , using (4.16), the Hölder inequality and the regularity assumption (3.10), we have

    \begin{align*} (I_{5}^{n+1},e_{h}^{n+1}) &\leq C \|e_h^{n+1}\|_{{{\textbf{L}}}^2} (\tau^{3}\int_{t_{n-1}}^{t^{n+1}}\|\partial_{tt}\theta(t)\|_{L^2}^{2}dt)^{\frac{1}{2}}, \end{align*}

    {then}

    \begin{align} 4\tau \sum\limits_{n = 1}^{m} (I_{5}^{n+1},e_{h}^{n+1}) \leq C \tau \sum_{n = 1}^{m} \| e_h^{n+1}\|_{{{\textbf{L}}}^2}^{2} + C\tau^4, \end{align} (4.19)

    where C > 0 is independent of h, \tau , \nu and \kappa .

    We estimate the last term I_6^{n+1} by

    \begin{align*} (I_{6}^{n+1},e_{h}^{n+1}) &\leq \|\widehat \eta_{\theta}^{n}+\widehat e_{\theta}^{n}\|_{L^2}\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}+ \|(\widehat\eta_{\theta}^{n}+\widehat e_{\theta}^{n})\|_{L^{2}} \|\widehat\theta^{n}+\widehat\theta_{h}^{n}\|_{L^{\infty}}\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}\\ &\leq C (1 +\|\widehat \theta_{h}^{n}\|_{L^\infty}^2) ( h^{4}+\|\widehat e_{\theta}^{n}\|_{L^2}^2) +C ( \|\widehat e_{\theta}^{n}\|_{L^2}^{2}+\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2}) , \end{align*}

    where C > 0 is independent of h, \tau , \nu and \kappa . Taking the sum of the above inequality gives

    \begin{align} \begin{split} 4\tau \sum\limits_{n = 0}^{m} (I_{6}^{n+1},e_{h}^{n+1}) &\leq C h^{4} \tau \sum_{n = 1}^{m} (1+\|\widehat \theta_{h}^{n}\|_{L^\infty}^2) + C \tau \sum_{n = 1}^{m} ( \|\widehat e_{\theta}^{n}\|_{L^2}^{2}+\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2}) \\ &+C\tau \sum_{n = 1}^{m} (1+\|\widehat \theta_{h}^{n}\|_{L^\infty}^2) ( \| e_{\theta}^{n}\|_{L^2}^2 +\| e_{\theta}^{n-1}\|_{L^2}^2 ), \end{split} \end{align} (4.20)

    where C > 0 is independent of h, \tau , \nu and \kappa .

    Substituting the estimates (4.14)–(4.20) into (4.13), we obtain

    \begin{align} \begin{split} & \|e_{h}^{m+1}\|_{{{\textbf{L}}}^2}^{2}+\|\widehat e_{h}^{m+1}\|_{{{\textbf{L}}}^2}^{2}+4\tau\sum\limits_{n = 1}^{m} \left ( \nu\|\nabla e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \beta \|\nabla\cdot e_{h}^{n+1}\|_{L^2}^{2} \right ) \\ \leq & C(h^{4}+\tau^4) + C h^{4} \tau \sum_{n = 1}^{m} \|\widehat \theta_{h}^{n}\|_{L^\infty}^2 + C \tau \sum_{n = 1}^{m} ( \|\widehat e_{\theta}^{n}\|_{L^2}^{2}+\|\widehat e_{h}^{n}\|_{{{\textbf{L}}}^2}^{2}+\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2}) \\ & +3\epsilon_2\beta \tau \sum_{n = 1}^{m} \|\nabla\cdot e_h^{n+1}\|_{L^2}^2 + C\tau \sum_{n = 1}^{m} (1+\|\widehat\theta_{h}^{n}\|_{L^\infty}^2) ( \|e_{\theta}^{n}\|_{L^2}^2 +\| e_{\theta}^{n-1}\|_{L^2}^2 ), \end{split} \end{align} (4.21)

    where C > 0 is independent of h, \tau , \nu, \kappa and the estimate (4.3) is used.

    Next, we estimate e_\theta^{n+1} . Taking \psi_h = 4\tau e_\theta^{n+1} in (4.10), summing up the resulting equation from n = 1 to n = m and noticing the skew-symmetric property (2.1), we have

    \begin{align} \|e_{\theta}^{m+1}\|_{L^2}^{2}+\|\widehat e_{\theta}^{m+1}\|_{L^2}^{2}+4\kappa \tau\sum\limits_{n = 1}^{m} \|\nabla e_\theta^{n+1}\|_{{{\textbf{L}}}^2}^{2} \leq 4\tau\sum\limits_{n = 1}^{m} \sum_{j = 1}^3 (X_{j}^{n+1}, e_\theta^{n+1})+C\|e_{\theta}^1\|_{L^2}^{2} , \end{align} (4.22)

    where we noted e_{\theta}^0 = 0 .

    By the regularity assumptions (3.10) and (4.11), it is easy to prove that

    \begin{align*} (X_{1}^{n+1}, e_\theta^{n+1}) & \leq \frac{1}{2\tau}\int_{t_{n-1}}^{t^{n+1}} \left( 2(t-t_{n})^{2}-\frac{1}{2}(t-t_{n-1})^{2}\right) \|\partial_{ttt}\theta(t)\|_{L^2} dt\|e_{\theta}^{n+1}\|_{L^2},\\ \tau \sum_{n = 1}^{m} \|D_{2,\tau}\theta^{n+1}\|_{H^1}^2 & \leq \frac C \tau \sum_{n = 1}^{m} \left( { \int}_{t_{n-1}}^{t_{n+1}} \|\theta_t(t)\|_{H^1}dt \right)^2 \leq C{,} \end{align*}

    then we have

    \begin{align} 4\tau\sum\limits_{n = 1}^{m} (X_{1}^{n+1}, e_\theta^{n+1})&\leq C \tau \sum_{n = 1}^{m} \| e_\theta^{n+1}\|_{L^2} + C\tau^4 , \end{align} (4.23)
    \begin{align} \begin{split} 4\tau\sum\limits_{n = 1}^{m} (X_{2}^{n+1}, e_\theta^{n+1}) & \leq C \tau \sum_{n = 1}^{m} \| e_{\theta}^{n+1}\|_{L^2}^2 + Ch^{4} \tau \sum_{n = 1}^{m} \|D_{2,\tau}\theta^{n+1}\|_{H^1}^2 \\ & \leq C \tau \sum_{n = 1}^{m} \| e_{\theta}^{n+1}\|_{L^2}^2 + Ch^{4}, \end{split} \end{align} (4.24)

    where C > 0 is independent of h, \tau , \nu and \kappa .

    For last term X_3^{n+1} , we estimate it by

    \begin{align*} (X_{3}^{n+1}, e_{\theta}^{n+1}) & = (\widehat{{{\textbf{u}}}}_h^n\cdot\nabla R_h\theta^{n+1},e_{\theta}^{n+1}) +\frac{1}{2}((\nabla\cdot\widehat{{{\textbf{u}}}}_h^n)R_h\theta^{n+1},e_{\theta}^{n+1}) -({{\textbf{u}}}^{n+1}\cdot\nabla\theta^{n+1},e_{\theta}^{n+1})\\ & = -(\widehat{e}_h^n\cdot\nabla R_h\theta^{n+1},e_{\theta}^{n+1}) -(\widehat{{{\textbf{s}}}}_h^n\cdot\nabla\eta_{\theta}^{n+1},e_{\theta}^{n+1}) +((\widehat{{{\textbf{s}}}}_h^n-\widehat{{{\textbf{u}}}}^{n})\cdot\nabla\theta^{n+1},e_{\theta}^{n+1})\\ &+((\widehat{{{\textbf{u}}}}^n-{{\textbf{u}}}^{n+1})\cdot\nabla\theta^{n+1},e_{\theta}^{n+1})\\ &-\frac{1}{2}((\nabla\cdot \widehat{e}_h^n)R_h\theta^{n+1},e_{\theta}^{n+1}) +\frac{1}{2}(\nabla\cdot(\widehat{{{\textbf{s}}}}_{h}^n-\widehat{{{\textbf{u}}}}^n)R_h\theta^{n+1},e_{\theta}^{n+1}) , \end{align*}

    where the skew-symmetric property (2.1) is used. We have

    \begin{align} \begin{split} 4\tau \sum\limits_{n = 1}^{m} (X_{3}^{n+1}, e_\theta^{n+1}) &\leq (\|\widehat{e}_h^n\|_{{{\textbf{L}}}^2}\|\nabla R_h\theta^{n+1}\|_{{{\textbf{L}}}^\infty} +\|\widehat{{{\textbf{s}}}}_h^n\|_{{{\textbf{L}}}^{\infty}}\|\nabla\eta_{\theta}^{n+1}\|_{{{\textbf{L}}}^2})\|e_{\theta}^{n+1}\|_{L^2}\\ &+(\|\widehat{{{\textbf{s}}}}_h^n-\widehat{{{\textbf{u}}}}^{n}\|_{{{\textbf{L}}}^2}+\|\widehat{{{\textbf{u}}}}^n-{{\textbf{u}}}^{n+1}\|_{{{\textbf{L}}}^2})\|\nabla\theta^{n+1}\|_{{{\textbf{L}}}^\infty}\|e_{\theta}^{n+1}\|_{L^2}\\ &+(\|\nabla\cdot \widehat{e}_h^n\|_{L^2}+\|\widehat{{{\textbf{s}}}}_{h}^n-\widehat{{{\textbf{u}}}}^n\|_{{{\textbf{H}}}^1})\|R_h\theta^{n+1}\|_{L^\infty}\|e_{\theta}^{n+1}\|_{L^2}\\ &\leq C(h^{4}+\tau^4) + \epsilon_2\beta \tau \sum_{n = 1}^{m} \|\nabla \cdot e_h^{n}\|_{L^2}^2\\ & + C\tau \sum\limits_{n = 1}^{m} (\| e_{\theta}^{n+1}\|_{L^2}^2 + \|\widehat e_h^n\|_{{{\textbf{L}}}^2}^2), \end{split} \end{align} (4.25)

    where the C > 0 is independent of h, \tau , \nu and \kappa .

    Substituting the estimates (4.23)–(4.25) into (4.22) and using (4.3), we get

    \begin{align} \begin{split} & \|e_{\theta}^{m+1}\|_{L^2}^{2}+4\kappa \tau\sum\limits_{n = 1}^{m} \|\nabla e_\theta^{n+1}\|_{{{\textbf{L}}}^2}^{2} \\ \leq & C(h^{4}+\tau^4) + C\tau \sum\limits_{n = 1}^{m} (\| e_{\theta}^{n+1}\|_{L^2}^2 + \|\widehat e_h^n\|_{{{\textbf{L}}}^2}^2) +\epsilon_2\beta \tau \sum_{n = 1}^{m} \|\nabla \cdot e_h^{n}\|_{L^2}^2 , \end{split} \end{align} (4.26)

    where C > 0 is independent of h, \tau , \nu and \kappa .

    Taking the sum of (4.21) and (4.26), we have

    \begin{align} \begin{split} & \|e_{h}^{m+1}\|_{{{\textbf{L}}}^2}^{2}+\|e_{\theta}^{m+1}\|_{L^2}^{2}+4\tau\sum\limits_{n = 1}^{m} \left ( \nu\|\nabla e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \kappa \|\nabla e_\theta^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \beta \|\nabla\cdot e_{h}^{n+1}\|_{L^2}^{2} \right ) \\ & \leq C(h^{4}+\tau^4) + C \tau \sum_{n = 1}^{m} (\|e_{\theta}^{n+1}\|_{L^2}^{2}+\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \|\widehat e_h^n\|_{{{\textbf{L}}}^2}^2+\|\widehat e_\theta^n\|_{L^2}^2) + C h^{4} \tau \sum_{n = 1}^{m} \|\widehat \theta_{h}^{n}\|_{L^\infty}^2\\ & +4\epsilon_2\beta \tau \sum_{n = 1}^{m} \|\nabla\cdot e_h^{n+1}\|_{L^2}^2 + C\tau \sum_{n = 1}^{m} (1+\|\widehat \theta_{h}^{n}\|_{L^\infty}^2) ( \| e_{\theta}^{n}\|_{L^2}^2+ \| e_{\theta}^{n-1}\|_{L^2}^2), \end{split} \end{align} (4.27)

    where C > 0 is independent of h, \tau , \nu and \kappa .

    Next, we bound uniformly the norm \|\theta_h^n\|_{L^\infty} by the method of mathematical induction as in the proof of Theorem 3.1.

    According to (4.3), we can see that (4.12) holds for m = 0 if we take \widehat C_0\geq C_3 . Now, we assume that (4.12) also holds for m\leq k-1 with 1\leq k\leq N-1 . Under the condition \tau\leq Ch and the inverse inequality, one has

    \begin{align*} \|e_\theta^n\|_{L^\infty}\leq C h^{-3/2}\|e_\theta^n\|_{L^2} \leq C\widehat C_0 h^{1/2} \leq C \quad \forall \ 1\leq n\leq k \end{align*}

    for sufficiently small h with \widehat C_0 h^{1/2}\leq 1 , which further gives

    \begin{align} \|\widehat \theta_h^n\|_{L^\infty} \leq \|\widehat e_\theta^n\|_{L^\infty} + \| \theta^n\|_{L^\infty} \leq C \quad {\mbox{and}} \quad \tau \sum_{n = 1}^{k} \|\widehat \theta_{h}^{n}\|_{L^\infty}^2 \leq C. \end{align} (4.28)

    To finish the mathematical induction, we need to prove that (4.12) also holds for m\leq k . As a result of (4.28), we have

    \begin{align*} & \|e_{h}^{k+1}\|_{{{\textbf{L}}}^2}^{2}+\|e_{\theta}^{k+1}\|_{L^2}^{2}+4\tau\sum\limits_{n = 1}^{k} \left ( \nu\|\nabla e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \kappa \|\nabla e_\theta^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \beta \|\nabla\cdot e_{h}^{n+1}\|_{L^2}^{2} \right ) \\ & \leq C(h^4+\tau^4) + C \tau \sum_{n = 1}^{k} (\|e_{\theta}^{n+1}\|_{L^2}^{2}+\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \|\widehat e_h^n\|_{{{\textbf{L}}}^2}^2+\|\widehat e_\theta^n\|_{L^2}^2) \\ & +4\epsilon_3\beta \tau \sum_{n = 1}^{k} \|\nabla\cdot e_h^{n+1}\|_{L^2}^2 \end{align*}

    by setting m = k in (4.27). Select small \epsilon_2\leq 1 , then we have

    \begin{align*} & \|e_{h}^{k+1}\|_{{{\textbf{L}}}^2}^{2}+\|e_{\theta}^{k+1}\|_{L^2}^{2}+\tau\sum\limits_{n = 1}^{k} \left ( \nu\|\nabla e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \kappa \|\nabla e_\theta^{n+1}\|_{{{\textbf{L}}}^2}^{2} \right ) \\ \leq & C(h^4+\tau^4) + C \tau \sum_{n = 1}^{k} (\|e_{\theta}^{n+1}\|_{L^2}^{2}+\|e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2}+ \|\widehat e_h^n\|_{{{\textbf{L}}}^2}^2), \end{align*}

    where C > 0 is independent of h, \tau , \nu and \kappa .

    Applying the discrete Grönwall inequality to the above inequality, we conclude that there exists some C_4 > 0 , which is independent of h, \tau , \nu and \kappa , such that

    \begin{align*} \|e_{h}^{k+1}\|_{{{\textbf{L}}}^2}^{2}+\|e_{\theta}^{k+1}\|_{L^2}^{2}+\tau\sum\limits_{n = 1}^{k} \left ( \nu\|\nabla e_{h}^{n+1}\|_{{{\textbf{L}}}^2}^{2} + \kappa \|\nabla e_{\theta}^{n+1}\|_{{{\textbf{L}}}^2}^{2} \right ) \leq C_4^2(h^4+\tau^4) \end{align*}

    for sufficiently small \tau . Thus, we prove that the error estimate (4.12) also holds for m\leq k by taking \widehat C_0\geq C_4 , and we finish the mathematical induction.

    Based on (4.12) in Theorem 4.1, we get the following error estimate for the second-order BDF2 grad-div finite element scheme (4.1)–(4.2).

    Theorem 4.2 Under the assumptions in Theorem 4.1, we have

    \begin{align} \|{{\textbf{u}}}^{m+1}-{{\textbf{u}}}_h^{m+1}\|_{{{\textbf{L}}}^2}^{2}+\|\theta^{m+1}-\theta_h^{m+1}\|_{L^2}^{2} \leq C(h^{4}+\tau^4) \end{align} (4.29)

    with 0\leq m\leq N-1 , where C > 0 is independent of h, \tau , \nu and \kappa .

    In this section, we present numerical results of the second-order BDF2 grad-div stabilization method for the penetrative convection problems (1.1)–(1.5) with small viscosity coefficient \nu . For simplicity, we only consider the 2D problem with the domain \Omega = [0, 1]^2 . To check the convergence rates, we take the appropriate {{\textbf{f}}} and g such that the exact solution ({{\textbf{u}}}, p, \theta) is given by

    {{\textbf{u}}}_1 = 10 x^2(x-1)^2 y(y-1)(2y-1)\exp(-t),\quad {{\textbf{u}}}_2 = -10 x(x-1)(2x-1) y^2 (y-1)^2\exp(-t) ,
    p = 10(2x-1)(2y-1)\exp(-t), \quad \theta = \sin(\pi x)\sin(\pi y)\exp(-t).

    In numerical computation, we take the physical parameters \kappa = 10^{-1}, \gamma_{1} = 10^{-1}, \gamma_{2} = 10^{-1} , the stabilized parameter \beta = 0.1 and the final time T = 1 . In addition, we denote

    \begin{eqnarray*} \|{{\textbf{u}}}-{{\textbf{u}}}_h\|_{{{\textbf{L}}}^2} & = & \|{{\textbf{u}}}^N-{{\textbf{u}}}_h^N\|_{{{\textbf{L}}}^2}, \quad \|\theta-\theta_h\|_{L^2} = \|\theta^N-\theta_h^N\|_{L^2} , \\ \|{{\textbf{u}}}-{{\textbf{u}}}_h\|_{l^2(V)} & = & \left( \tau \sum_{n = 1}^N \|\nabla({{\textbf{u}}}^n-{{\textbf{u}}}^n_h)\|_{{{\textbf{L}}}^2}^2 \right)^{1/2}, \\ \|\theta-\theta_h\|_{l^2(Y)} & = & \left( \tau \sum_{n = 1}^N \|\nabla(\theta^n-\theta^n_h)\|_{{{\textbf{L}}}^2}^2 \right)^{1/2}. \end{eqnarray*}

    In terms of error estimates in Theorems 4.1 and 4.2, we have the second-order convergence rates of the velocity and temperature if we select \tau = \mathcal O(h) . We use a uniform mesh on \Omega with the spatial mesh size h in each direction. By taking gradually decreasing mesh sizes h = 1/4, 1/8, \cdots, 1/128 , we choose different iteration numbers N = 1/h such that the time step size \tau = h . We present numerical results with small viscosity coefficients \nu = 10^{-3} and \nu = 10^{-4} in Tables 1 and 2, respectively, from which we can see that the second-order convergence rates are reached. Thus, these numerical results are in good agreement with the theoretical analysis.

    Table 1.  Numerical errors and convergence rates with \nu = 10^{-3} and \tau = h .
    h \|{{\textbf{u}}}-{{\textbf{u}}}_{h}\|_{{{\textbf{L}}}^2} rate \|{{\textbf{u}}}-{{\textbf{u}}}_h\|_{l^2(V)} rate \|\theta-\theta_{h}\|_{L^2} rate \|\theta-\theta_{h}\|_{l^2(Y)} rate
    1/4 4.28913E-03 \quad 6.57526E-02 \quad 2.09701E-03 \quad 4.88982E-02 \quad
    1/8 9.74180E-04 2.14 2.16564E-02 1.60 3.44116E-04 2.61 1.23962E-02 1.98
    1/16 2.34556E-04 2.05 4.99515E-03 2.12 6.70734E-05 2.36 3.11457E-03 1.99
    1/32 5.84811E-05 2.00 1.03426E-03 2.27 1.51182E-05 2.15 7.79712E-04 2.00
    1/64 1.46777E-05 1.99 2.10634E-04 2.30 3.61134E-06 2.07 1.94978E-04 2.00
    1/128 3.67926E-06 2.00 4.68535E-05 2.17 8.84215E-07 2.03 4.87451E-05 2.00

     | Show Table
    DownLoad: CSV
    Table 2.  Numerical errors and convergence rates with \nu = 10^{-4} and \tau = h .
    h \|{{\textbf{u}}}-{{\textbf{u}}}_{h}\|_{{{\textbf{L}}}^2} rate \|{{\textbf{u}}}-{{\textbf{u}}}_h\|_{l^2(V)} rate \|\theta-\theta_{h}\|_{L^2} rate \|\theta-\theta_{h}\|_{l^2(Y)} rate
    1/4 4.75610E-03 \quad 7.67595E-02 \quad 2.10166E-03 \quad 4.89096E-02 \quad
    1/8 1.20543E-03 1.98 3.84952E-02 1.00 3.45921E-04 2.60 1.23987E-02 1.98
    1/16 2.87010E-04 2.07 1.36084E-02 1.50 6.79439E-05 2.35 3.11550E-03 1.99
    1/32 6.89986E-05 2.06 3.67556E-03 1.89 1.54451E-05 2.14 7.80036E-04 2.00
    1/64 1.73110E-05 1.99 8.06317E-04 2.19 3.70433E-06 2.06 1.95075E-04 2.00
    1/128 4.24985E-06 2.03 1.45108E-04 2.47 9.07167E-07 2.03 4.87662E-05 2.00

     | Show Table
    DownLoad: CSV

    On the other hand, the second-order convergence rates in the L^2 norm is sub-optimal since we use the P_2 element to approximate {{\textbf{u}}} and \theta . Theoretically, the optimal error estimate should be

    \begin{align} \|{{\textbf{u}}}-{{\textbf{u}}}_{h}\|_{{{\textbf{L}}}^2}+\|\theta-\theta_{h}\|_{L^2} \leq C (h^3+\tau^2) \end{align} (5.1)

    for 1\leq n\leq N . To check (5.1), we take \tau = h^{3/2} such that

    \begin{align} \|{{\textbf{u}}}-{{\textbf{u}}}_{h}\|_{{{\textbf{L}}}^2}+\|\theta-\theta_{h}\|_{L^2} \leq C h^3. \end{align} (5.2)

    By taking different mesh sizes h = 1/2^2, 1/3^2, \cdots, 1/7^2 , we give numerical results in Tables 3 and 4 with \nu = 10^{-3} and \nu = 10^{-4} , respectively. We can see that the almost third-order convergence rates in L^2 norm are reached. Thus, the L^2 error estimates in (4.29) is not optimal. How to prove the optimal error estimates in L^2 norm will be reported in future work.

    Table 3.  Numerical errors and convergence rates with \nu = 10^{-3} and \tau = h^{3/2} .
    h \|{{\textbf{u}}}-{{\textbf{u}}}_{h}\|_{{{\textbf{L}}}^2} rate \|\theta-\theta_{h}\|_{L^2} rate
    1/4 3.18961E-03 \quad 1.39164E-03 \quad
    1/9 3.03406E-04 2.90 1.22372E-04 3.00
    1/16 5.02327E-05 3.13 2.18932E-05 2.99
    1/25 1.15939E-05 3.29 5.75192E-06 3.00
    1/36 3.31503E-06 3.43 1.93196E-06 2.99
    1/49 1.12367E-06 3.51 7.67713E-07 2.99

     | Show Table
    DownLoad: CSV
    Table 4.  Numerical errors and convergence rates with \nu = 10^{-4} and \tau = h^{3/2} .
    h \|{{\textbf{u}}}-{{\textbf{u}}}_{h}\|_{{{\textbf{L}}}^2} rate \|\theta-\theta_{h}\|_{L^2} rate
    1/4 3.65795E-03 \quad 1.39237E-03 \quad
    1/9 5.23458E-04 2.40 1.22415E-04 3.00
    1/16 1.16969E-04 2.60 2.18917E-05 2.99
    1/25 3.32275E-05 2.82 6.22161E-06 2.82
    1/36 1.12122E-05 2.98 2.25287E-06 2.79
    1/49 4.52035E-06 2.95 9.47731E-07 2.81

     | Show Table
    DownLoad: CSV

    In this paper, we studied the first-order Euler and second-order BDF2 finite element schemes for the approximation of the the time-dependent penetrative convection equations. In designing of numerical scheme, we used the grad-div stabilization method to overcome instabilities from the high Reynolds number. Main advantages of the proposed schemes are two folds. One is that they both are unconditionally stable without any condition of the time step and mesh size. Another is that one only needs to solve linearized systems at each time step. In terms of the analysis technique in [18,19], uniform error estimates in L^2 norm were derived in which the constants are independent of inverse powers of the viscosity coefficient and thermal conductivity coefficient.

    In some related models, the nonlinear term 2\nu\mathcal D({{\textbf{u}}}):\mathcal D({{\textbf{u}}}) , which is used to describe the viscous dissipation, appears in the temperature equation (1.3) [2,9,47]. Here, \mathcal D({{\textbf{u}}}) is the strain tensor having components \mathcal D_{ij}({{\textbf{u}}}) = \frac 12 (\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}) . However, this strongly nonlinear term will result in much more difficulties in well-posedness analysis and numerical analysis. Thus, we consider the Eq (1.3) without the viscous dissipation in this paper. On the other hand, different modified Boussinesq approximations for some practical problems have been considered, such as the heat and mass transfer models in [10,48]. In future works, we can use the grad-div stabilization method for these modified models.

    On the other hand, for the reason of simplicity, we use the zero Dirichlet boundary condition for the temperature. We remark that uniform error estimates derived in this paper can be extended to the problem with the mixed boundary condition for the temperature in [19], where the mixed boundary condition is composed by zero Dirichlet and Neumann boundary conditions.

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

    This work was supported by National Natural Science Foundation of China (No. 11771337).

    On behalf of all authors, the corresponding author states that there is no conflict of interest.



    [1] R. A. Adams, J. F. Fournier, Sobolev spaces, New York: Academic Press, 2009.
    [2] A. Barletta, On the thermal instability induced by viscous dissipation, Int. J. Therm. Sci., 88 (2015), 238–247. https://doi.org/10.1016/j.ijthermalsci.2014.02.009 doi: 10.1016/j.ijthermalsci.2014.02.009
    [3] K. R. Blake, D. Poulikakos, A. Bejan, Natural convection near 4℃ in a horizontal water layer heated from below, Phys. Fluids, 27 (1984), 2608–2616. https://doi.org/10.1063/1.864561 doi: 10.1063/1.864561
    [4] S. C. Brenner, L. R. Scott, The mathematical theory of finite element methods, New York: Springer, 2008. https://doi.org/10.1007/978-0-387-75934-0
    [5] A. N. Brooks, T. J. R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Method. Appl. M., 32 (1982), 199–259. https://doi.org/10.1016/0045-7825(82)90071-8 doi: 10.1016/0045-7825(82)90071-8
    [6] M. Cao, Y. Li, Optimal error analysis of linearized Crank-Nicolson finite element scheme for the time-Dependent penetrative convection problem, Commun. Appl. Math. Comput., 2023 (2023), 7. https://doi.org/10.1007/s42967-023-00269-7 doi: 10.1007/s42967-023-00269-7
    [7] F. Capone, M. Gentile, A. A. Hill, Penetrative convection in a fluid layer with throughflow, Ricerche Mat., 57 (2008), 251–260. https://doi.org/10.1007/s11587-008-0035-8 doi: 10.1007/s11587-008-0035-8
    [8] M. Carr, S. de Putter, Penetrative convection in a horizontallt isotropic porous layer, Continum Mech. Theromodyn., 15 (2003), 33–43. https://doi.org/10.1007/s00161-002-0102-4 doi: 10.1007/s00161-002-0102-4
    [9] S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, Oxford: Oxford University Press, 1961.
    [10] A. A. Domnich, E. S. Baranovskii, M. A. Artemov, A nonlinear model of the non-isothermal slip flow between two parallel plates, J. Phys. Conf. Ser., 1479 (2020), 012005. https://doi.org/10.1088/1742-6596/1479/1/012005 doi: 10.1088/1742-6596/1479/1/012005
    [11] D. Erkmen, A. E. Labovsky, Note on the usage of grad-div stabilization for the penalty-projection algorithm in magnetohydrodynamics, Appl. Math. Comput., 349 (2019), 48–52. http://doi.org/10.1016/j.amc.2018.12.036 doi: 10.1016/j.amc.2018.12.036
    [12] L. P. Franca, T. J. R. Hughes, Two classes of mixed finite element methods, Comput. Method. Appl. M., 69 (1988), 89–129. https://doi.org/10.1016/0045-7825(88)90168-5 doi: 10.1016/0045-7825(88)90168-5
    [13] L. P. Franca, T. J. R. Hughes, Convergence analyses of Galerkin least-squares methods for symmetric advective-diffusive forms of the Stokes and incompressible Navier-Stokes equations, Comput. Method. Appl. M., 105 (1993), 285–298. https://doi.org/10.1016/0045-7825(93)90126-I doi: 10.1016/0045-7825(93)90126-I
    [14] L. P. Franca, A. Russo, Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles, Appl. Math. Lett., 9 (1996), 83–88. https://doi.org/10.1016/0893-9659(96)00078-X doi: 10.1016/0893-9659(96)00078-X
    [15] L. P. Franca, A. Nesliturk, On a two-level finite element method for the incompressible Navier-Stokes equations, Int. J. Numer. Meth. Eng., 52 (2001), 433–453. https://doi.org/10.1002/nme.220 doi: 10.1002/nme.220
    [16] F. Franchi, Stabilization estimates for penetrative motions in porous media, Math. Method. Appl. Sci., 17 (1994), 11–20. https://doi.org/10.1002/mma.1670170103 doi: 10.1002/mma.1670170103
    [17] J. de Frutos, B. García-Archilla, V. John, J. Novo, Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements, J. Sci. Comput., 66 (2016), 991–1024. https://doi.org/10.1007/s10915-015-0052-1 doi: 10.1007/s10915-015-0052-1
    [18] J. de Frutos, B. García-Archilla, V. John, J. Novo, Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements, Adv. Comput. Math., 44 (2018), 195–225. https://doi.org/10.1007/s10444-017-9540-1 doi: 10.1007/s10444-017-9540-1
    [19] J. de Frutos, B. García-Archilla, J. Novo, Grad-div stabilization for the time-dependent Boussinesq equations with inf-sup stable finite elements, Appl. Math. Comput., 349 (2019), 281–291. https://doi.org/10.1016/j.amc.2018.12.062 doi: 10.1016/j.amc.2018.12.062
    [20] B. García-Archilla, V. John, J. Novo, On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows, Comput. Method. Appl. M., 385 (2021), 114032. https://doi.org/10.1016/j.cma.2021.114032 doi: 10.1016/j.cma.2021.114032
    [21] B. García-Archilla, J. Novo, Robust error bounds for the Navier-Stokes equations using implicit-explicit second-order BDF method with variable steps, IMA J. Numer. Anal., 43 (2023), 2892–2933. https://doi.org/10.1093/imanum/drac058 doi: 10.1093/imanum/drac058
    [22] V. Girault, P. Raviart, Finite element methods for Navier-Stokes equations, Berlin Heidelberg: Springer-Verlag, 1986. https://doi.org/10.1007/978-3-642-61623-5
    [23] J. L. Guermond, Stabilization of Galerkin approximations of transport equations by subgrid modeling, ESAIM-Math. Model. Num., 33 (1999), 1293–1316. https://doi.org/10.1051/m2an:1999145 doi: 10.1051/m2an:1999145
    [24] J. G. Heywood, R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem Part Ⅳ: error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353–384. https://doi.org/10.1137/0727022 doi: 10.1137/0727022
    [25] T. J. R. Hughes, L. P. Franca, G. M. Hulbert, A new finite element formulation for computational fluid dynamics: Ⅷ. The Galerkin/least-squares method for advective-diffusive equations, Comput. Method. Appl. M., 73 (1989), 173–189. https://doi.org/10.1016/0045-7825(89)90111-4 doi: 10.1016/0045-7825(89)90111-4
    [26] T. J. R. Hughes, L. Mazzei, K. E. Jansen, Large Eddy simulation and the variational multiscale method, Comput. Visual. Sci., 3 (2000), 47–59. https://doi.org/10.1007/s007910050051 doi: 10.1007/s007910050051
    [27] T. J. R. Hughes, L. Mazzei, A. A. Oberai, A. A. Wray, The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence, Phys. Fluids, 13 (2001), 505–512. https://doi.org/10.1063/1.1332391 doi: 10.1063/1.1332391
    [28] E. W. Jenkins, V. John, A. Linke, L. G. Rebholz, On the parameter choice in grad-div stabilization for the Stokes equations, Adv. Comput. Math, 40 (2014), 491–516. https://doi.org/10.1007/s10444-013-9316-1 doi: 10.1007/s10444-013-9316-1
    [29] L. M. Jiji, Heat convection, Heidelberg: Springer Berlin, 2006. https://doi.org/10.1007/978-3-642-02971-4
    [30] V. John, Finite element methods for incompressible flow problems, Cham: Springer, 2016. https://doi.org/10.1007/978-3-319-45750-5
    [31] W. Layton, A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput., 133 (2002), 147–157. https://doi.org/10.1016/S0096-3003(01)00228-4 doi: 10.1016/S0096-3003(01)00228-4
    [32] W. Layton, C. C. Manica, M. Neda, M. Olshanskii, L. G. Rebholz, On the accuracy of the rotation form in simulations of the Navier-Stokes equations, J. Comput. Phys., 228 (2009), 3433–3447. https://doi.org/10.1016/j.jcp.2009.01.027 doi: 10.1016/j.jcp.2009.01.027
    [33] Y. Li, D. D. Xue, Y. Rong, Y. Qin, A second order partitioned method with grad-div stabilization for the non-stationary dual-porosity-Stokes model, Comput. Math. Appl., 124 (2022), 111–128. https://doi.org/10.1016/j.camwa.2022.08.025 doi: 10.1016/j.camwa.2022.08.025
    [34] Y. Li, R. An, Two-level variational multiscale finite element methods for Navier-Stokes type variational inequality problem, J. Comput. Appl. Math., 290 (2015), 656–669. https://doi.org/10.1016/j.cam.2015.06.018 doi: 10.1016/j.cam.2015.06.018
    [35] J. Liu, Simple and efficient ALE methods with provable temporal accuracy up to fifth order for the Stokes equations on time varying domains, SIAM J. Numer. Anal., 51 (2013), 743–772. https://doi.org/10.1137/110825996 doi: 10.1137/110825996
    [36] A. J. Majda, A. L. Bertozzi, Vorticity and incompressible flow, Cambridge: Cambridge University Press, 2002. https://doi.org/10.1017/CBO9780511613203
    [37] G. Matthies, N. I. Ionkin, G. Lube, L. Röhe, Some remarks on residual-based stabilisation of inf-sup stable discretisations of the generalized Oseen problem, Comput. Meth. Appl. Math., 9 (2009), 368–390. https://doi.org/10.2478/cmam-2009-0024 doi: 10.2478/cmam-2009-0024
    [38] G. L. Mellor, T. Yamada, Development of a turbulence closure model for geophysical fluid problems, Rev. Geophys., 20 (1982), 851–875. https://doi.org/10.1029/RG020i004p00851 doi: 10.1029/RG020i004p00851
    [39] S. Mussman, Penetrative convection, J. Fluid Mech., 31 (1968), 343–360. https://doi.org/10.1017/S0022112068000194 doi: 10.1017/S0022112068000194
    [40] M. A. Olshanskii, A. Reusken, Grad-div stablilization for Stokes equations, Math. Comput., 73 (2004), 1699–1718. https://doi.org/10.1090/S0025-5718-03-01629-6 doi: 10.1090/S0025-5718-03-01629-6
    [41] L. E. Payne, J. C. Song, B. Straughan, Double diffusive porous penetrative convection-thawing subsea permafrost, Int. J. Eng. Sci., 26 (1998), 797–809. https://doi.org/10.1016/0020-7225(88)90031-6 doi: 10.1016/0020-7225(88)90031-6
    [42] J. Pedlosky, Geophysical fluid dynamics, New York: Springer, 1987. https://doi.org/10.1007/978-1-4612-4650-3
    [43] R. Rannacher, R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comput., 38 (1982), 437–445. https://doi.org/10.1090/S0025-5718-1982-0645661-4 doi: 10.1090/S0025-5718-1982-0645661-4
    [44] S. S. Ravindran, Convergence of extrapolated BDF2 finite element schemes for unsteady penetrative convection model, Numer. Func. Anal. Opt., 33 (2012), 48–79. https://doi.org/10.1080/01630563.2011.618899 doi: 10.1080/01630563.2011.618899
    [45] P. Sagaut, Large Eddy simulation for incompressible flows, Heidelberg: Springer Berlin, 2003. https://doi.org/10.1007/b137536
    [46] B. Straughan, Continuous dependence on the heat source and non-linear stability in penetrative convection, Int. J. Nonlin. Mech., 26 (1991), 221–231. https://doi.org/10.1016/0020-7462(91)90053-V doi: 10.1016/0020-7462(91)90053-V
    [47] F. M. White, Fluid mechanics, New York: McGraw-Hill, 2002.
    [48] S. Woo, P. L. F. Liu, Finite-element model for modified boussinesq equations. Ⅰ: Model development, J. Waterway Port Coast., 130 (2004), 1. https://doi.org/10.1061/(ASCE)0733-950X(2004)130:1(1) doi: 10.1061/(ASCE)0733-950X(2004)130:1(1)
    [49] H. Zheng, Y. Hou, F. Shi, L. Song, A finite element variational multiscale method for incompressible flows based on two local Gauss integrations, J. Comput. Phys., 228 (2009), 5961–5977. https://doi.org/10.1016/j.jcp.2009.05.006 doi: 10.1016/j.jcp.2009.05.006
  • This article has been cited by:

    1. 耕耘 田, Temporal Error Estimate of First-Order Fractional Step Algorithm for Unsteady Penetrative Convection Model, 2024, 13, 2324-7991, 3039, 10.12677/aam.2024.137289
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1546) PDF downloads(65) Cited by(1)

Figures and Tables

Tables(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog