Open Access

The Euler implicit/explicit scheme for the Boussinesq equations

Boundary Value Problems20162016:181

https://doi.org/10.1186/s13661-016-0693-5

Received: 2 August 2016

Accepted: 10 October 2016

Published: 18 October 2016

Abstract

In this article, we consider the stability and convergence of the first-order implicit/explicit scheme for the Boussinesq equations. The finite element spatial discretization is based on a MINI element for the velocity and pressure, which satisfies the discrete inf-sup condition, and a linear polynomial for the temperature. The temporal terms are treated by the Euler implicit/explicit scheme, which is implicit for the linear terms and explicit for the nonlinear terms. The advantage of using the implicit/explicit scheme is that a linear system with constant coefficient matrix is obtained, which can save a lot of computational cost. The main novelties of this work are the stability of numerical solutions under the conditions \(k_{1}\Delta t\leq1\) and \(k_{2}\Delta t\leq1\) with two positive constants \(k_{1},k_{2}\) and the optimal error estimates of numerical solutions in different norms. Finally, some numerical results are provided to verify the performances of the Euler implicit/explicit scheme.

Keywords

Boussinesq equationsEuler implicit/explicit schemestabilityerror estimates

MSC

65N1565N3076D07

1 Introduction

In this paper, we consider the following Boussinesq equations in \(\mathbb{R}^{2}\) with coupled equations governing the viscous incompressible flow and heat transfer:
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l@{\quad}l} u_{t}-\nu\Delta u+(u\cdot\nabla)u+\nabla p=-j\theta+f & \mbox{in } \Omega \times(0,T],\\ \operatorname{div} u=0 & \mbox{in} \Omega\times(0,T],\\ \theta_{t}-\lambda\nu\Delta\theta+u\cdot\nabla\theta=g & \mbox{in } \Omega \times(0,T],\\ u=0, \qquad\theta=0 & \mbox{on } \partial\Omega\times(0,T],\\ u(x,0)=u_{0},\qquad\theta(x,0)=\theta_{0}, & \mbox{on } \Omega\times\{0\}, \end{array}\displaystyle \right . \end{aligned}$$
(1.1)
where Ω is a bounded convex polygonal domain, \(u=(u_{1},u_{2})^{T}\) is the fluid velocity, p is the pressure, θ is the temperature, \(\nu>0\) is the viscosity, \(\lambda=Pr^{-1}, Pr\) is the Prandtl number, \(j=(0,1)^{T}\) is the vector of gravitational acceleration, \(T>0\) is the final time, and f and g are the forcing functions.

The Boussinesq equations (1.1) are an important dissipative nonlinear model in the atmospheric dynamics (see [1]). This system not only contains the velocity and pressure but also includes the temperature filed, and therefore finding a numerical solution of problem (1.1) becomes a difficult task. There are numerous works devoted to the development of efficient schemes for this model, for example, the standard Galerkin finite element method (FEM) [2], the projection-based stabilized mixed FEM [3], the precondition techniques [4, 5], the two-level algorithms [68], and the references therein. In the literature mentioned, the suitable stability condition is a key issue for these developed schemes. Generally speaking, we can adopt the fully implicit, semi-implicit, and implicit/explicit schemes to treat the nonlinear equations. The fully implicit schemes are unconditionally stable. However, we have to solve a system of nonlinear equations at each time level. Explicit schemes are much easier in computation, but they suffer from a sever restriction of time step by the stability requirement. A popular approach is based on an implicit scheme for the linear terms and a semi-implicit scheme or an explicit scheme for the nonlinear terms. The semi-implicit scheme for the nonlinear terms results in a linear system with variable coefficient matrix of time, and an explicit treatment for the nonlinear term gives a constant matrix. Many researchers have studied the stability and convergence of these schemes for the Navier-Stokes equations [915]. The main results are summarized by He in his recent works [16, 17].

In this paper, we consider a first-order scheme for the Boussinesq equations. In view of the advantages of the explicit scheme for the nonlinear terms, we adopt the implicit/explicit scheme for the Boussinesq equations (1.1). Under the conditions \(k_{1}\Delta t\leq1\) and \(k_{2}\Delta t\leq1\) with two positive constants \(k_{1},k_{2}\), we present some new stabilities and establish the corresponding convergence for velocity, pressure, and temperature by the Taylor expansion and other skills. This report can be considered as an extension of the existing results [10, 16, 18, 19] from the Navier-Stokes equations to the more complex Boussinesq equations. Our main results can be stated as follows:
$$\begin{aligned}& \bigl\| u-u_{h}^{n}\bigr\| _{0}+\bigl\| \theta- \theta_{h}^{n}\bigr\| _{0}\leq C\bigl(\Delta t+h^{2}\bigr), \end{aligned}$$
(1.2)
$$\begin{aligned}& \bigl\| \nabla\bigl(u-u_{h}^{n}\bigr)\bigr\| _{0}+\bigl\| \nabla \bigl(\theta-\theta_{h}^{n}\bigr)\bigr\| _{0}+ \bigl\| p-p_{h}^{n}\bigr\| _{0}\leq C(\Delta t+h), \end{aligned}$$
(1.3)
where \(C>0\) is a constant depending on the parameters \(f_{\infty },b_{\infty},u_{0},\theta_{0},\Omega,\nu\), and λ, but independent of h and Δt, where \(f_{\infty}=\sup_{t\geq0}\{|f|+|f_{t}|\}\), \(f_{t}=\frac{df}{dt}\), \(g_{\infty}=\sup_{t\geq0}\{|g|+|g_{t}|\}\), \(g_{t}=\frac{dg}{dt}\). Here and thereafter, C denotes a general positive constant, which may take different values at different places. From (1.2)-(1.3) we can see that our results are optimal for both space length h and time step Δt.

The outline of this article is as follows. Some basic notation and results for problem (1.1) are recalled in Section 2. Section 3 is devoted to develop the Euler implicit/explicit scheme. Stabilities and optimal error estimates are established in Sections 4 and 5, respectively. Finally, a series of numerical results are provided to verify the efficiency and effectiveness of the Euler implicit/explicit scheme.

2 Preliminaries

In this section, we construct a variable formulation for problem (1.1) and recall some classical results, which will be frequently used in this paper. To fix the idea, we set
$$\begin{aligned}& X=H_{0}^{1}(\Omega)^{2}, \qquad W=H_{0}^{1}(\Omega),\qquad Y=L^{2}( \Omega)^{2},\qquad Z=L^{2}(\Omega ), \\& M=L_{0}^{2}(\Omega)=\biggl\{ \varphi\in L^{2}( \Omega); \int_{\Omega}\varphi \,dx=0\biggr\} . \end{aligned}$$
Throughout this paper, we adopt \((\cdot,\cdot)\) and \(\|\cdot\|_{0}\) to denote the inner product and norm on \(L^{2}(\Omega)\) or \(L^{2}(\Omega)^{2}\). The spaces \(H^{1}_{0}(\Omega)\) and X are equipped with the usual scalar product and norm \(\|\nabla u\|^{2}_{0}=(\nabla u,\nabla u)\). Define the continuous bilinear forms \(a(\cdot,\cdot),d(\cdot,\cdot)\), and \(\overline{a}(\cdot,\cdot)\) by
$$\begin{aligned} a(u,v)=\nu(\nabla u,\nabla v), \qquad d(v,q)=(q,\operatorname{div}v),\qquad \overline {a}(\theta,\psi)=\lambda\nu(\nabla\theta,\nabla\psi) \end{aligned}$$
for all \(u,v\in X,q\in M\), and \(\theta,\psi\in W\).
Next, we introduce the closed subset V of X given by
$$\begin{aligned} V=\bigl\{ v\in X,d(v,q)=0, \forall q\in M\bigr\} =\{v\in X,\nabla\cdot v=0 \mbox{ in } \Omega\} \end{aligned}$$
and denote by H the closed subset of Y (see [17, 20]) given by
$$\begin{aligned} H=\{v\in Y,\nabla\cdot v=0,v\cdot n|_{\partial\Omega}=0\}. \end{aligned}$$
We denote the Stokes operator by \(A=P\Delta\), where P is the \(L^{2}\)-orthogonal projection of Y onto H or of Z onto W. Assume that Ω is such that the domain of A is given by (see [10, 17, 21, 22])
$$\begin{aligned} D(A)=H^{2}(\Omega)^{2}\cap X \quad \mbox{or} \quad E(A)=H^{2}(\Omega)\cap W. \end{aligned}$$
(2.1)
For instance, (2.1) holds if Γ is of class \(C^{2}\) or if Ω is a convex plane polygonal domain.
Moreover, we can define the trilinear forms for all \(u,v,w\in X\) and \(\theta,\psi\in W\) as follows:
$$\begin{aligned}& b(u,v,w)=\bigl((u\cdot\nabla)v,w\bigr)+\frac{1}{2}\bigl(( \operatorname{div}u)v,w\bigr)=\frac {1}{2}\bigl((u\cdot\nabla)v,w\bigr)- \frac{1}{2}\bigl((u\cdot\nabla)w,v\bigr), \\& \overline{b}(u,\theta,\psi)=\bigl((u\cdot\nabla)\theta,\psi\bigr)+ \frac {1}{2}\bigl((\operatorname{div}u)\theta,\psi\bigr)=\frac{1}{2} \bigl((u\cdot\nabla)\theta,\psi\bigr) -\frac{1}{2}\bigl((u\cdot\nabla) \theta,\psi\bigr). \end{aligned}$$
With these notations, for given \(f\in L^{\infty}(R^{+};Y)\) with \(u_{0}\in D(A)\cap V\) and \(g\in L^{\infty}(R^{+};Z)\) with \(\theta_{0}\in E(A)\), the variational formulation of (1.1) reads as follows: For all \((v,q,\psi)\in X\times M\times W\), find \((u,p,\theta)\in X\times M\times W\) with
$$\begin{aligned}& u\in L^{\infty}\bigl(R^{+};X\bigr)\cap L^{2}(0,T;V), \qquad u_{t}\in L^{2}\bigl(0,T;V'\bigr),\qquad \theta _{t}\in L^{2}\bigl(0,T;W'\bigr), \\& p\in L^{2}(0,T;M),\qquad \theta\in L^{\infty}\bigl(R^{+};W\bigr) \cap L^{2}(0,T;Z), \quad \forall T>0, \end{aligned}$$
such that
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} (u_{t},v)+a(u,v)-d(v,p)+b(u,u,v)=(f,v)-(j\theta,v),\\ d(u,q)=0,\\ (\theta_{t},\psi)+\overline{a}(\theta,\psi)+\overline{b}(u,\theta,\psi )=(g,\psi),\\ u(x,0)=u_{0},\qquad \theta(x,0)=\theta_{0}. \end{array}\displaystyle \right . \end{aligned}$$
(2.2)

Assuming that \(f\in L^{2}(0,T;X'),g\in L^{2}(0,T;W')\), and \(u_{0}\in V,\theta _{0}\in W\), problem (2.2) has at least one solution \((u,p,\theta)\) satisfying \(u\in L^{\infty}(0,T;\Omega)\cap L^{2}(0,T;V)\) and \(\theta\in L^{\infty}(0,T;\Omega)\cap L^{2}(0,T;W)\). The uniqueness and regularity of the solution \((u,p,\theta)\) can also be proved by strengthening the assumptions on the data; see [23] for details.

We recall the following discrete Gronwall lemma, which can be found in [17, 24].

Lemma 2.1

Let \(C_{0}\) and \(a_{k},b_{k},c_{k},d_{k}\) for integer \(k\geq0\) be nonnegative numbers such that
$$\begin{aligned} a_{n}+\triangle t\sum_{k=0}^{n}b_{k} \leq\Delta t\sum_{k=0}^{n-1}d_{k}a_{k}+ \Delta t\sum_{k=0}^{n-1}c_{k}+C_{0} \quad \forall n\geq1. \end{aligned}$$
Then,
$$\begin{aligned} a_{n}+\triangle t\sum_{k=0}^{n}b_{k} \leq\Biggl(\Delta t\sum_{k=0}^{n-1}c_{k}+C_{0} \Biggr)\exp\Biggl(\Delta t\sum_{k=0}^{n-1}d_{k} \Biggr) \quad \forall n\geq1. \end{aligned}$$

Following the proofs provided in [1, 20, 22, 23], we can obtain that problem (2.2) possesses a unique solution \((u,p,\theta)\) with the following regularity properties.

Theorem 2.2

Let \(f\in L^{\infty}(R^{+};Y),f_{t}\in L^{2}(0,T;Y),g\in L^{\infty}(R^{+};Z),g_{t}\in L^{2}(0,T;Z)\) and \(u_{0}\in D(A)\cap V,\theta_{0}\in E(A)\). Then the solution \((u,p,\theta)\) of problem (2.2) satisfies
$$\begin{aligned} \bigl\| Au(t)\bigr\| _{0}+\bigl\| \nabla u_{t}(t)\bigr\| _{0}+ \bigl\| Au_{t}(t)\bigr\| _{0}+\bigl\| A\theta(t)\bigr\| _{0}+\bigl\| \nabla \theta_{t}(t)\bigr\| _{0}+\bigl\| A\theta_{t}(t)\bigr\| _{0} \leq C. \end{aligned}$$
Introduce the following Poincaré inequalities:
$$\begin{aligned} \|v\|_{0}\leq C_{1}\|\nabla v\|_{0} \quad \forall v\in X \textit{ or } W;\qquad \|\nabla v\|_{0}\leq C_{2}\|Av\|_{0} \quad \forall v\in D(A) \textit{ or } H^{2}(\Omega). \end{aligned}$$
(2.3)

We end this section by recalling some properties of the trilinear forms \(b(\cdot,\cdot,\cdot)\) and \(\overline{b}(\cdot,\cdot,\cdot)\), which can be found in [1, 68, 10, 17, 21, 22].

Lemma 2.3

The trilinear forms \(b(\cdot,\cdot,\cdot)\) and \(\overline{b}(\cdot,\cdot,\cdot)\) satisfy:

(1) Under the condition \(\operatorname{div} u=0\), we have that
$$\begin{aligned} b(u,v,v)=0 \quad\forall u,v\in X; \qquad \overline{b}(u,\theta,\theta)=0 \quad \forall u\in X,\theta\in W. \end{aligned}$$
(2) We have the following estimates for trilinear terms \(b(\cdot,\cdot ,\cdot)\) and \(\overline{b}(\cdot,\cdot,\cdot)\):
$$\begin{aligned}& \bigl\vert b(u,v,w)\bigr\vert \leq C_{3}\Vert u\Vert ^{1/2}_{0}\bigl\Vert A^{1/2}u\bigr\Vert ^{1/2}_{0}\bigl\Vert A^{1/2}v\bigr\Vert ^{1/2}_{0}\Vert Av\Vert _{0}^{1/2}\|w \|_{0} \quad \forall u\in V,v\in D(A),w\in X, \\& \bigl\vert b(u,v,w)\bigr\vert \leq C_{4}\|u\|_{0}^{1/2} \|Av\|^{1/2}_{0}\|v\|_{1}\|w\|_{0}\quad \forall u\in V,v\in D(A),w\in X, \\& \bigl\vert \overline{b}(u,\theta,\psi)\bigr\vert \leq C_{5} \Vert u\Vert ^{1/2}_{0}\bigl\Vert A^{1/2}u\bigr\Vert ^{1/2}_{0}\bigl\Vert A^{1/2}\theta\bigr\Vert ^{1/2}_{0}\|A\theta\|_{0}^{1/2}\| \psi\|_{0}\quad \forall u\in V,\theta\in E(A),\psi\in W, \\& \bigl\vert \overline{b}(u,\theta,\psi)\bigr\vert \leq C_{6} \Vert u\Vert _{0}^{1/2} \Vert Au\Vert ^{1/2}_{0}\bigl\Vert A^{1/2}\theta\bigr\Vert _{0}\|\psi\|_{0} \quad\forall u\in D(A),\theta,\psi\in W. \end{aligned}$$

3 The Euler implicit/explicit scheme for the Boussinesq equations

Let \(\mathcal{T}_{h}\) be a family of finite element partitions of Ω into triangles satisfying the usual compatibility conditions [21] with \(h=\max h_{K}\), where \(h_{K}\) is the diameter of an element \(K\in\mathcal{T}_{h}\). We assume that \(\mathcal{T}_{h}\) is shape regular, that is, there exists a constant \(\sigma> 0\) such that \(h_{K}<\sigma\rho _{K}\) for all \(K\in\mathcal{T}_{h}\), where \(h_{K}\) and \(\rho_{h}\) denote the diameter of K and the diameter of the largest ball that can be inscribed into K, respectively.

The finite element subspaces of interest in this paper are defined by the so-called MINI element with the continuous piecewise finite element subspace for the approximation of velocity and pressure and the linear polynomial for temperature, respectively:
$$\begin{aligned}& X_{h}=\bigl\{ v_{h}\in X_{h}:v_{h}|_{K} \in P_{1}(K)^{2}\oplus \operatorname{span}\{ \lambda_{1}\lambda _{2}\lambda_{3}\} \ \forall K \in\mathcal{T}_{h}\bigr\} , \\& M_{h}=\bigl\{ q\in M:q|_{K}\in P_{1}(K) \ \forall K\in\mathcal{T}_{h}\bigr\} , \\& W_{h}=\bigl\{ \psi\in W:\psi|_{K}\in P_{1}(K)\ \forall K\in\mathcal{T}_{h}\bigr\} . \end{aligned}$$
Note that \(\lambda_{1},\lambda_{2}\), and \(\lambda_{3}\) are the barycentric coordinates of the reference element.
We define the subspace \(V_{h}\) of \(X_{h}\) by
$$\begin{aligned} V_{h}=\bigl\{ v_{h}\in X_{h}:d(v_{h},q_{h})=0 \ \forall q_{h}\in M_{h}\bigr\} . \end{aligned}$$
Let \(P_{h}:Y\rightarrow V_{h}\) or \(Z\rightarrow W_{h}\) denote the \(L^{2}\)-orthogonal projection defined by
$$\begin{aligned} (P_{h}\omega,\chi_{h})=(\omega,\chi_{h})\quad \forall \omega\in Y \mbox{ or } Z, \chi_{h}\in V_{h} \mbox{ or } W_{h}. \end{aligned}$$
We introduce a discrete analogue \(A_{h}=-P_{h}\Delta_{h}\) of the Stokes operator A through the condition \((A_{h}\phi_{h},\psi_{h})=(\nabla\phi_{h},\nabla\psi_{h})\) for all \(\phi_{h},\psi_{h}\in X_{h} \mbox{or} W_{h}\). The restriction of \(A_{h}\) to \(V_{h}\) is invertible with the inverse \(A_{h}^{-1}\). Since \(A_{h}^{-1}\) is self-adjoint and positive definite, we may define the “discrete” Sobolev norms on \(V_{h}\) of any order \(r\in R\) by setting
$$\begin{aligned} \|\omega_{h}\|_{r}=\bigl\| A_{h}^{r/2} \omega_{h}\bigr\| _{0}, \quad \omega_{h}\in V_{h}. \end{aligned}$$
These norms will be assumed to have various properties similar to their continuous counterparts, implicitly imposing conditions on the structure of the spaces \(X_{h},M_{h}\), and \(W_{h}\). In particular,
$$\begin{aligned} \|\omega_{h}\|_{1}=\|\nabla\omega_{h} \|_{0}, \qquad \|\omega_{h}\|_{2}= \|A_{h}\omega_{h}\| _{0} \quad\forall \omega_{h}\in V_{h} \mbox{ or } W_{h}. \end{aligned}$$
The discrete Laplace operator \(A_{h}\) is first introduced in [20] to analyze and obtain the optimal estimates for the transient Navier-Stokes equations. Furthermore, from [1, 22] we know that there exists a positive constant \(\beta>0\) independent of h such that, for all \(v_{h}\in X_{h}\) and \(q_{h}\in M_{h}\),
$$\begin{aligned} b(v_{h},q_{h})\geq\beta\|v_{h} \|_{1}\|q_{h}\|_{0}. \end{aligned}$$
(3.1)
The finite element discretization applied to problem (2.2) leads to spatial discrete equations as follows: Find \((u_{h},p_{h},\theta _{h})\in X_{h}\times M_{h}\times W_{h}\) such that, for all \((v_{h},q_{h},\psi_{h})\in X_{h}\times M_{h}\times W_{h}\),
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} (u_{ht},v_{h})+ a(u_{h},v_{h})+b(u_{h},u_{h},v_{h})-d(v_{h},p_{h})=(f,v_{h})-(j\theta _{h},v_{h}),\\ b(u_{h},q_{h})=0,\\ (\theta_{ht},\psi)+\overline{a}(\theta_{h},\psi_{h})+\overline{b}(u_{h},\theta _{h},\psi)=(g,\psi_{h}). \end{array}\displaystyle \right . \end{aligned}$$
(3.2)

For the stability and convergence of problem (3.2), we have the following results.

Theorem 3.1

see [7]

Let \(f\in L^{\infty}(R^{+};Y),f_{t}\in L^{2}(0,T;Y),g\in L^{\infty}(R^{+};Z), g_{t}\in L^{2}(0,T;Z), u_{0}\in D(A)\cap V,\theta_{0}\in E(A)\) and assume that \(u_{h}(0)=u_{0},\theta_{h}(0)=\theta_{0}\). By Theorem  2.2 problem (3.2) admits a unique solution \((u_{h},p_{h},\theta_{h})\in X_{h}\times M_{h}\times W_{h}\) satisfying
$$\begin{aligned}& \|u_{h}\|_{0}^{2}+\nu \int_{0}^{t}\|\nabla u_{h} \|_{0}^{2}\,ds\leq\|u_{0}\|_{0}^{2}+ \frac {2C_{1}^{2}}{\nu} \int_{0}^{t}\|f\|_{0}^{2}\,ds + \frac{2C_{1}^{3}}{\lambda\nu^{2}}\biggl(\theta_{0}+\frac{C_{4}^{2}}{\lambda\nu} \int_{0}^{t}\| g\|_{0}^{2}\,ds \biggr), \\& \|\nabla u_{h}\|_{0}^{2}+\nu \int_{0}^{t}\|A_{h}u_{h} \|_{0}^{2}\,ds\leq\|\nabla u_{0}\| _{0}^{2}+\frac{\nu^{2}C_{4}^{2}}{\lambda}\|\theta_{0} \|_{0}^{2}+\frac{\nu C_{4}^{2}}{\lambda ^{2}} \int_{0}^{t}\|g\|_{0}^{2}\,ds+ \frac{1}{\nu} \int_{0}^{t}\|f\|_{0}^{2}\,ds, \\& \|\theta_{h}\|_{0}^{2}+\lambda\nu \int_{0}^{t}\|\nabla\theta_{h} \|_{0}^{2}\,ds\leq\| \theta_{0}\|_{0}^{2} +\frac{C_{1}^{2}}{\lambda\nu} \int_{0}^{t}\|g\|_{0}^{2}\,ds, \\& \|\nabla\theta_{h}\|_{0}^{2}+\lambda\nu \int_{0}^{t}\|A_{h}\theta_{h} \|_{0}^{2}\,ds\leq\| \nabla\theta_{0} \|_{0}^{2} +\frac{1}{\lambda\nu} \int_{0}^{t}\|g\|_{0}^{2}\,ds, \\& \|u_{ht}\|_{0}^{2}+\nu \int_{0}^{t}\|\nabla u_{h} \|_{0}^{2}\,ds+\|\theta_{ht}\| _{0}^{2}+ \lambda\nu \int_{0}^{t}\|\nabla\theta_{h} \|_{0}^{2}\,ds\leq C, \\& \frac{\nu}{2}\|A_{h}u_{h}\|_{0}^{2}+ \int_{0}^{t}\|\nabla u_{ht} \|_{0}^{2}\,ds +\frac{\lambda\nu}{2}\|A_{h} \theta_{h}\|_{0}^{2}+ \int_{0}^{t}\|\nabla\theta_{ht}\| _{0}^{2}\,ds\leq C, \\& \|u_{htt}\|_{0}^{2}+\nu \int_{0}^{t}\|\nabla u_{ht} \|_{0}^{2}\,ds+\|\theta_{htt}\| _{0}^{2}+ \lambda\nu \int_{0}^{t}\|\nabla\theta_{ht} \|_{0}^{2}\,ds\leq C, \\& \|\nabla u_{ht}\|_{0}^{2}+\frac{\nu}{2} \int_{0}^{t}\|A_{h}u_{ht} \|_{0}^{2}\,ds+\| \nabla\theta_{ht} \|_{0}^{2} +\frac{\lambda\nu}{2} \int_{0}^{t}\|A_{h}\theta_{ht} \|_{0}^{2}\,ds\leq C, \\& \frac{\nu}{2}\|A_{h}u_{ht}\|_{0}^{2}+ \int_{0}^{t}\|\nabla u_{htt} \|_{0}^{2}\,ds +\frac{\lambda\nu}{2}\|A_{h} \theta_{ht}\|_{0}^{2}+ \int_{0}^{t}\|\nabla\theta _{htt} \|_{0}^{2}\,ds\leq C, \\& \bigl\Vert \nabla(u-u_{h})\bigr\Vert _{0}+\Vert p-p_{h}\Vert _{0}+\bigl\Vert \nabla(\theta- \theta_{h})\bigr\Vert _{0}\leq Ch,\qquad \|u-u_{h}\|_{0}+\|\theta-\theta_{h} \|_{0}\leq Ch^{2}. \end{aligned}$$

Let \(\Delta t>0\) be the time-step, and let \(t_{n}=n\Delta t \) (\(0\leq n\leq N=[\frac{T}{\Delta t}]\)), \(u_{h}^{n},p_{h}^{n}\), and \(\theta_{h}^{n}\) denote the numerical solutions of \(u_{h},p_{h}\), and \(\theta_{h}\) at \(t_{n}\), respectively. We consider the Euler implicit/explicit scheme for the Boussinesq equations (1.1). As we have pointed out in Section 1, the advantage of adopting the Euler implicit/explicit scheme is that a linear system with constant coefficient matrix is obtained, and then a lot of computational cost can be saved.

The Euler implicit/explicit scheme for problem (3.2) reads as follows:
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} (\frac{u_{h}^{n}-u_{h}^{n-1}}{\Delta t},v)+\nu(\nabla u_{h}^{n},\nabla v) +b(u_{h}^{n-1},u_{h}^{n-1},v)-(v,\nabla p_{h}^{n})=(f(t_{n}),v)-(j\theta_{h}^{n},v),\\ (\nabla\cdot u_{h}^{n},q)=0, \\ (\frac{\theta_{h}^{n}-\theta_{h}^{n-1}}{\Delta t},\psi)+\lambda\nu(\nabla \theta_{h}^{n},\nabla\psi)+\overline{b}(u_{h}^{n-1},\theta_{h}^{n-1},\psi )=(g(t_{n}),\psi), \end{array}\displaystyle \right . \end{aligned}$$
(3.3)
with \(1\leq n\leq N\). From (3.3) we can see that the discrete system (3.3) is a linear system; for the existence and uniqueness of \(u_{h}^{n},p_{h}^{n}\), and \(\theta_{h}^{n}\), we refer to [25].
For \(n=0\), equations (3.3) can be rewritten as
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} (u_{h}^{0},v)+\Delta t\nu(\nabla u_{h}^{0},\nabla v) -\Delta t(v,\nabla p_{h}^{0})=\Delta t(f(t_{0}),v)-\Delta t(j\theta_{h}^{0},v), \\ (\nabla\cdot u_{h}^{0},q)=0, \\ (\theta_{h}^{0},\psi)+\Delta t\lambda\nu(\nabla\theta_{h}^{0},\nabla\psi )=\Delta t(g(t_{0}),\psi). \end{array}\displaystyle \right . \end{aligned}$$
(3.4)
In order to simplify the expressions, we set \(d_{t}\omega_{h}^{n}=\frac{\omega _{h}^{n}-\omega_{h}^{n-1}}{\Delta t}\), where ω is u or θ. Choosing \(\psi=\theta_{h}^{0},v_{h}=u_{h}^{0}\), and \(q_{h}=p_{h}^{0}\) in (3.4), we have the following estimates:
$$\begin{aligned}& \bigl\Vert u_{h}^{0}\bigr\Vert _{0}^{2} \leq\Delta t\bigl(\Vert f\Vert _{0}+\Delta t\Vert g\Vert _{0}\bigr), \qquad \bigl\Vert \theta_{h}^{0} \bigr\Vert _{0}\leq\Delta t\|g\|_{0}, \end{aligned}$$
(3.5)
$$\begin{aligned}& \bigl\Vert \nabla u_{h}^{0}\bigr\Vert _{0} \leq\frac{C_{1}}{\nu} \Vert f\Vert _{0}+\frac{C_{1}^{3}}{\lambda\nu ^{2}}\Vert g \Vert _{0}, \qquad \bigl\Vert \nabla\theta_{h}^{0} \bigr\Vert _{0}\leq\frac{C_{1}}{\lambda\nu}\|g\|_{0}. \end{aligned}$$
(3.6)
From (3.5)-(3.5) we can see that scheme (3.4) is stable.

4 Stability of the numerical solutions

In this section, we establish the stability of the numerical solutions \(u_{h}^{n},p_{h}^{n}\), and \(\theta_{h}^{n}\) in the Euler implicit/explicit scheme (3.3) for the Boussinesq equations. The mathematical induction has been used to obtain the desired results; this technique has also been used to other problems, for example, for the Dirichlet problems with \((p,q)\)-Laplacian [26] and the mixed initial-boundary value problems [27].

Theorem 4.1

Under the conditions of Theorem  3.1 and the stability conditions \(k_{1}\Delta t\leq1\) and \(k_{2}\Delta t\leq1\), the solutions \(u_{h}^{n},p_{h}^{n}\), and \(\theta_{h}^{n}\) are bounded for any integer \(0\leq n\leq[\frac{T}{\Delta t}]\):
$$\begin{aligned}& \bigl\Vert u_{h}^{n}\bigr\Vert _{0}^{2}+ \nu\Delta t\sum_{i=1}^{n}\bigl\Vert \nabla u_{h}^{i}\bigr\Vert _{0}^{2}\leq\gamma _{0}^{2},\qquad \bigl\Vert \theta_{h}^{n} \bigr\Vert _{0}^{2}+\lambda\nu\Delta t\sum _{i=1}^{n}\bigl\Vert \nabla\theta_{h}^{i} \bigr\Vert _{0}^{2}\leq\gamma_{1}^{2}, \end{aligned}$$
(4.1)
$$\begin{aligned}& \bigl\Vert \nabla u_{h}^{n}\bigr\Vert _{0}^{2}+\nu\Delta t\sum_{i=1}^{n} \bigl\Vert A_{h}u_{h}^{i}\bigr\Vert _{0}^{2}\leq k_{01}, \qquad \bigl\Vert \nabla \theta_{h}^{n}\bigr\Vert _{0}^{2}+ \lambda\nu\Delta t\sum_{i=1}^{n}\bigl\Vert A_{h}\theta _{h}^{i}\bigr\Vert _{0}^{2}\leq k_{02}, \end{aligned}$$
(4.2)
$$\begin{aligned}& \bigl\Vert d_{t}u_{h}^{n}\bigr\Vert _{0}^{2}+\nu\Delta t\sum_{i=1}^{n} \bigl\Vert \nabla d_{t}u_{h}^{i}\bigr\Vert _{0}^{2}\leq k_{03}, \qquad \bigl\Vert d_{t}\theta_{h}^{n}\bigr\Vert _{0}^{2}+\nu\Delta t\sum_{i=1}^{n} \bigl\Vert \nabla d_{t}\theta_{h}^{i}\bigr\Vert _{0}^{2}\leq k_{04}, \end{aligned}$$
(4.3)
$$\begin{aligned}& \bigl\Vert A_{h}u_{h}^{n}\bigr\Vert _{0}^{2}\leq k_{05}=12\nu^{-2} \bigl(k_{03}+f_{\infty}^{2}\bigr)+\gamma _{1}+48\nu^{-4}C_{2}^{4}k_{01}^{4} \gamma_{0}^{2}, \end{aligned}$$
(4.4)
$$\begin{aligned}& \bigl\Vert A_{h}\theta_{h}^{n}\bigr\Vert _{0}^{2}\leq k_{06}=12(\lambda\nu)^{-2} \bigl(k_{04}+b_{\infty }^{2}\bigr)+48(\lambda \nu)^{-4}C_{3}^{4}k_{01}^{4} \gamma_{1}^{2}, \end{aligned}$$
(4.5)
where
$$\begin{aligned}& \gamma_{0}^{2}=2\bigl\Vert u_{h}^{0} \bigr\Vert _{0}^{2}+2(1+4\Delta t)\bigl\Vert \theta_{h}^{0}\bigr\Vert _{0}^{2}+20T(T+ \Delta t)g_{\infty}^{2}+4(1+\Delta t)f_{\infty}^{2}, \\& \gamma_{1}^{2}=2\bigl\Vert \theta_{h}^{0} \bigr\Vert _{0}^{2}+4T(T+\Delta t)g_{\infty}^{2}, \\& \begin{aligned}[b] k_{01}={}&\exp\biggl(\frac{8^{3}}{\nu^{2}}C_{0}^{4} \gamma_{0}^{4}\biggr) \Biggl(2\bigl\Vert \nabla u_{h}^{0}\bigr\Vert _{0}^{2}+ \frac{\nu}{4}\bigl\Vert Au_{h}^{0}\bigr\Vert _{0}^{2} \\ &{}+40\nu^{-1}\Biggl(T\sup _{0\leq t\leq T}\bigl\Vert f(t)\bigr\Vert _{0}^{2}+ \Delta t\sum_{i=0}^{n}\bigl\Vert \theta_{h}^{i}\bigr\Vert _{0}^{2} \Biggr)\Biggr), \end{aligned} \\& k_{02}=\exp\biggl(\frac{8^{3}}{\lambda^{2}\nu^{2}}C_{3}^{4} \gamma_{0}^{2}\gamma_{1}^{2}\biggr) \biggl( 2\bigl\Vert \nabla\theta_{h}^{0}\bigr\Vert _{0}^{2}+\frac{\lambda\nu}{4}\bigl\Vert A \theta_{h}^{n}\bigr\Vert _{0}^{2}\Delta t+24T(\lambda\nu)^{-1}\sup_{0\leq t\leq T}\bigl\Vert g(t) \bigr\Vert _{0}^{2}\biggr), \\& \begin{aligned}[b] k_{03}={}&\exp\biggl(\exp\bigl(16 \nu^{-2}C_{2}^{2}k_{01}\bigr)\exp\biggl( \frac {8C_{3}^{2}k_{02}}{\lambda\nu}\biggr)\frac{16}{\lambda^{2}\nu ^{3}}C^{2}_{0}C_{3}^{2}k_{02}T \biggr) \cdot\exp\bigl(16\nu^{-2}C_{2}^{2}k_{01} \bigr) \\ &{}\times\biggl(\bigl\Vert d_{t}u_{h}^{0}\bigr\Vert _{0}^{2}+4\nu^{-1}C_{0}^{2}T \sup_{0\leq t\leq T}\Vert f_{t}\Vert _{0}^{2}+ \exp\biggl(\frac{8C_{3}^{2}k_{02}}{\lambda\nu}\biggr)\\ &{}\times \biggl(\bigl\Vert d_{t} \theta_{h}^{0}\bigr\Vert _{0}^{2} +4 \frac{C_{0}^{2}T}{\lambda\nu}\sup_{0\leq t\leq T}\bigl\Vert g(t)\bigr\Vert _{0}^{2}\biggr)\biggr), \end{aligned} \\& \begin{aligned}[b] k_{04}={}&\exp\biggl(\frac{8C_{3}^{2}k_{02}}{\lambda\nu}\biggr) \biggl(\bigl\Vert d_{t}\theta_{h}^{0}\bigr\Vert _{0}^{2}+\frac{4C_{0}^{2}T}{\lambda\nu}\sup_{0\leq t\leq T}\bigl\Vert g(t)\bigr\Vert _{0}^{2}\biggr) \\ &{}\times\biggl(1+ \frac{4C_{1}^{2}C_{3}^{2}k_{02}}{\lambda\nu^{2}}\exp\biggl(\frac {8C_{3}^{2}k_{02}}{\lambda\nu}\biggr)\biggr), \end{aligned}\\& k_{1}=2\nu^{-1}C_{4}^{2}k_{05}, \qquad k_{2}=2(\lambda\nu)^{-1}C_{6}^{2}k_{05}. \end{aligned}$$

Proof

We prove this theorem by induction. From (3.5)-(3.6) we know that (4.1)-(4.5) hold for \(n=0\). Assume that (4.1)-(4.5) hold for \(n=0,\ldots,J\) with \(0\leq J< N=[\frac{T}{\Delta t}]\). We need to prove (4.1)-(4.5) for \(n=J+1\).

First, taking \(v_{h}=2\Delta tu_{h}^{n},q_{h}=2\Delta t p_{h}^{n}\), and \(\psi _{h}=2\Delta t\theta_{h}^{n}\) in (3.3), we obtain
$$\begin{aligned} &\bigl(u_{h}^{n}-u_{h}^{n-1},2u_{h}^{n} \bigr)+2\nu\Delta t\bigl\Vert \nabla u_{h}^{n}\bigr\Vert _{0}^{2} +2\Delta tb\bigl(u_{h}^{n-1},u_{h}^{n-1},u_{h}^{n} \bigr) \\ &\quad=2\Delta t\bigl(f(t_{n}),u_{h}^{n}\bigr)-2 \Delta t\bigl(j\theta_{h}^{n},u_{h}^{n} \bigr) \end{aligned}$$
(4.6)
and
$$\begin{aligned} \bigl(\theta_{h}^{n}-\theta_{h}^{n-1},2 \theta_{h}^{n}\bigr)+2\lambda\nu\Delta t\bigl\| \nabla \theta_{h}^{n}\bigr\| _{0}^{2} +2\Delta t \overline{b}\bigl(u_{h}^{n-1},\theta_{h}^{n-1}, \theta_{h}^{n}\bigr) =2\Delta t\bigl(g(t_{n}), \theta_{h}^{n}\bigr). \end{aligned}$$
(4.7)
By using of the identities
$$\begin{aligned} (a-b,2a)=|a|^{2}-|b|^{2}+|a-b|^{2} \quad\mbox{and} \quad 2(a,b)=|a|^{2}+|b|^{2}-|a-b|^{2}, \end{aligned}$$
(4.8)
equations (4.6)-(4.7) can be transformed into
$$\begin{aligned} &\bigl\Vert u_{h}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert u_{h}^{n-1}\bigr\Vert _{0}^{2}+\bigl\Vert u_{h}^{n}-u_{h}^{n-1} \bigr\Vert _{0}^{2}+2\Delta t\nu\bigl\Vert \nabla u_{h}^{n}\bigr\Vert _{0}^{2} +2\Delta tb\bigl(u_{h}^{n-1},u_{h}^{n-1},u_{h}^{n} \bigr) \\ &\quad=2\Delta t\bigl(f(t_{n}),u_{h}^{n}\bigr)-2 \Delta t\bigl(j\theta_{h}^{n},u_{h}^{n} \bigr) \end{aligned}$$
(4.9)
and
$$\begin{aligned} &\bigl\Vert \theta_{h}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert \theta_{h}^{n-1} \bigr\Vert _{0}^{2} +\bigl\Vert \theta_{h}^{n}- \theta_{h}^{n-1}\bigr\Vert _{0}^{2}+2 \Delta t\lambda\nu\bigl\Vert \nabla\theta_{h}^{n}\bigr\Vert _{0}^{2} +2\Delta tb\bigl(u_{h}^{n-1}, \theta_{h}^{n-1},\theta_{h}^{n}\bigr) \\ &\quad=2 \Delta t\bigl(g(t_{n}),\theta_{h}^{n}\bigr). \end{aligned}$$
(4.10)
For the right-hand side terms of (4.9)-(4.10), we have
$$\begin{aligned}& \begin{aligned}[b] \bigl\vert 2\Delta t\bigl(f,u_{h}^{n} \bigr)\bigr\vert &=\bigl\vert 2\Delta t\bigl(f,u_{h}^{n-1} \bigr)+2\Delta t\bigl(f,u_{h}^{n}-u_{h}^{n-1} \bigr)\bigr\vert \\ &\leq2\Delta t\Vert f\Vert _{0}\bigl\Vert u_{h}^{n-1} \bigr\Vert _{0}+\frac{1}{4}\bigl\Vert u_{h}^{n}-u_{h}^{n-1} \bigr\Vert _{0}^{2}+4\Delta t^{2}\Vert f\Vert _{0}^{2}, \end{aligned} \\& \begin{aligned}[b] \bigl\vert 2\Delta t\bigl(j\theta_{h}^{n},u_{h}^{n} \bigr)\bigr\vert &=\bigl\vert 2\Delta t\bigl(j\theta _{h}^{n},u_{h}^{n-1} \bigr)+2\Delta t\bigl(j\theta_{h}^{n},u_{h}^{n}-u_{h}^{n-1} \bigr)\bigr\vert \\ &\leq2\Delta t\bigl\Vert \theta_{h}^{n}\bigr\Vert _{0}\bigl\Vert u_{h}^{n-1}\bigr\Vert _{0}+\frac{1}{4}\bigl\Vert u_{h}^{n}-u_{h}^{n-1} \bigr\Vert _{0}^{2}+4\Delta t^{2}\bigl\Vert \theta_{h}^{n}\bigr\Vert _{0}^{2}, \end{aligned} \\& \begin{aligned}[b] \bigl\vert 2\Delta t\bigl(g,\theta_{h}^{n} \bigr)\bigr\vert &=\bigl\vert 2\Delta t\bigl(g,\theta_{h}^{n-1} \bigr)+2\Delta t\bigl(g,\theta_{h}^{n}-\theta_{h}^{n-1} \bigr)\bigr\vert \\ &\leq2\Delta t\Vert g\Vert _{0}\bigl\Vert \theta_{h}^{n-1} \bigr\Vert _{0}+\frac{1}{2}\bigl\Vert \theta _{h}^{n}-\theta_{h}^{n-1}\bigr\Vert _{0}^{2}+2\Delta t^{2}\Vert g\Vert _{0}^{2}. \end{aligned} \end{aligned}$$
For the trilinear terms, thanks to Lemma 2.3, we deduce that
$$\begin{aligned}& \begin{aligned}[b] 2\bigl\vert b\bigl(u_{h}^{n-1},u_{h}^{n-1},u_{h}^{n} \bigr)\bigr\vert & =2\bigl\vert b\bigl(u_{h}^{n-1},u_{h}^{n},u_{h}^{n}-u_{h}^{n-1} \bigr)\bigr\vert \\ &\leq 2C_{4}\bigl\Vert A_{h}u_{h}^{n-1} \bigr\Vert _{0}\bigl\Vert \nabla u_{h}^{n}\bigr\Vert _{0}\bigl\Vert u_{h}^{n}-u_{h}^{n-1} \bigr\Vert _{0} \\ &\leq\nu\bigl\Vert \nabla u_{h}^{n}\bigr\Vert _{0}^{2}+\nu^{-1}C_{4}^{2}\bigl\Vert A_{h}u_{h}^{n-1}\bigr\Vert _{0}^{2}\bigl\Vert u_{h}^{n+1}-u_{h}^{n} \bigr\Vert _{0}^{2}, \end{aligned} \\& \begin{aligned}[b] 2\bigl\vert \overline{b}\bigl(u_{h}^{n-1}, \theta_{h}^{n-1},\theta _{h}^{n}\bigr) \bigr\vert &=2\bigl\vert b\bigl(u_{h}^{n-1}, \theta_{h}^{n},\theta_{h}^{n}- \theta_{h}^{n-1}\bigr)\bigr\vert \\ &\leq2C_{6}\bigl\Vert A_{h}u_{h}^{n-1} \bigr\Vert _{0}\bigl\Vert \nabla\theta_{h}^{n} \bigr\Vert _{0}\bigl\Vert \theta_{h}^{n}-\theta _{h}^{n-1}\bigr\Vert _{0} \\ &\leq\lambda\nu\bigl\Vert \nabla\theta_{h}^{n}\bigr\Vert _{0}^{2}+(\lambda\nu)^{-1}C_{6}^{2} \bigl\Vert A_{h}u_{h}^{n-1}\bigr\Vert _{0}^{2}\bigl\Vert \theta_{h}^{n}- \theta_{h}^{n-1}\bigr\Vert _{0}^{2}. \end{aligned} \end{aligned}$$
Combining these estimates with (4.9)-(4.10), we arrive at
$$\begin{aligned}& \begin{aligned}[b] &\bigl\Vert u_{h}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert u_{h}^{n-1}\bigr\Vert _{0}^{2}+\nu\Delta t\bigl\Vert \nabla u_{h}^{n}\bigr\Vert _{0}^{2} \\ &\quad\leq \biggl(\nu^{-1}C_{4}^{2}\bigl\Vert A_{h}u_{h}^{n-1}\bigr\Vert _{0}^{2} \Delta t-\frac{1}{2}\biggr) \bigl\Vert u_{h}^{n}-u_{h}^{n-1} \bigr\Vert _{0}^{2} \\ &\qquad{}+2\Delta t\bigl\Vert u_{h}^{n}\bigr\Vert _{0}\bigl(\Vert f\Vert _{0}+\bigl\Vert \theta_{h}^{n}\bigr\Vert _{0}\bigr)+4\Delta t^{2}\bigl(\Vert f\Vert _{0}^{2}+\bigl\Vert \theta_{h}^{n}\bigr\Vert _{0}^{2} \bigr), \end{aligned} \end{aligned}$$
(4.11)
$$\begin{aligned}& \begin{aligned}[b] &\bigl\Vert \theta_{h}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert \theta_{h}^{n-1} \bigr\Vert _{0}^{2}+\lambda\nu\Delta t\bigl\Vert \nabla \theta_{h}^{n}\bigr\Vert _{0}^{2} \\ &\quad\leq \biggl((\lambda\nu)^{-1}C_{6}^{2}\bigl\Vert A_{h}u_{h}^{n-1}\bigr\Vert _{0}^{2}\Delta t-\frac {1}{2}\biggr)\bigl\Vert \theta_{h}^{n}-\theta_{h}^{n-1}\bigr\Vert _{0}^{2} \\ &\qquad{}+2\Delta t\bigl\Vert \theta_{h}^{n}\bigr\Vert _{0}\Vert g\Vert _{0}+2\Delta t^{2}\Vert g \Vert _{0}^{2}, \end{aligned} \end{aligned}$$
(4.12)
for all \(1\leq n\leq N\). Under the stability conditions \(k_{1}\Delta t\leq1,k_{2}\Delta t\leq1\) and the induction assumption on \(n=0,1,\ldots,J\), we have
$$\begin{aligned}& \nu^{-1}C_{4}^{2}\bigl\| A_{h}u_{h}^{n-1} \bigr\| _{0}^{2}\Delta t-\frac{1}{2}\leq\nu ^{-1}C_{4}^{2}k_{05}\Delta t- \frac{1}{2} \leq\frac{1}{2}\Delta tk_{1}-\frac{1}{2} \leq0, \end{aligned}$$
(4.13)
$$\begin{aligned}& (\lambda\nu)^{-1}C_{6}^{2}\bigl\| A_{h}u_{h}^{n-1} \bigr\| _{0}^{2}\Delta t-\frac{1}{2}\leq (\lambda \nu)^{-1}C_{6}^{2}k_{05}\Delta t- \frac{1}{2} \leq\frac{1}{2}\Delta tk_{2}-\frac{1}{2} \leq0. \end{aligned}$$
(4.14)
Summing (4.11)-(4.12) for n from 1 to \(J+1\) and using (4.13)-(4.14), we obtain
$$\begin{aligned} \bigl\Vert \theta_{h}^{J+1}\bigr\Vert _{0}^{2}+\Delta t\lambda\nu\sum_{n=1}^{J+1} \bigl\Vert \nabla\theta _{h}^{n}\bigr\Vert _{0}^{2} \leq\bigl\Vert \theta_{h}^{0} \bigr\Vert _{0}^{2}+2\gamma_{1}Tg_{\infty}+2T \Delta tg_{\infty}\leq \gamma_{1}^{2} \end{aligned}$$
and
$$\begin{aligned} \bigl\Vert u_{h}^{J+1}\bigr\Vert _{0}^{2}+ \Delta t\nu\sum_{n=1}^{J+1}\bigl\Vert \nabla u_{h}^{n}\bigr\Vert _{0}^{2} \leq\bigl\Vert u_{h}^{0}\bigr\Vert _{0}^{2}+4 \gamma_{0}Tf_{\infty}+4T\Delta tf_{\infty}+2T\gamma _{0}\gamma_{1}+4T\Delta t\gamma_{1}^{2} \leq\gamma_{0}^{2}, \end{aligned}$$
which is (4.1) with \(n=J+1\).
Next, taking \(v_{h}=(\frac{1}{\nu}d_{t}u_{h}^{n}+A_{h}u_{h}^{n})\Delta t\in V_{h}\) and \(\psi_{h}=(\frac{1}{\lambda\nu}d_{t}\theta_{h}^{n}+A_{h}\theta_{h}^{n})\Delta t\) in (3.3), we get
$$\begin{aligned} &\nu^{-1}\Delta t\bigl\Vert d_{t}u_{h}^{n} \bigr\Vert _{0}^{2}+\nu\Delta t\bigl\Vert A_{h}u_{h}^{n}\bigr\Vert _{0}^{2} +\bigl\Vert \nabla u_{h}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert \nabla u_{h}^{n-1} \bigr\Vert _{0}^{2}+\bigl\Vert \nabla \bigl(u_{h}^{n}-u_{h}^{n-1}\bigr)\bigr\Vert _{0}^{2} \\ &\qquad{}+b\biggl(u_{h}^{n-1},u_{h}^{n-1}, \frac{1}{\nu}d_{t}u_{h}^{n}+Au_{h}^{n} \biggr)\Delta t \\ &\quad=\bigl(f(t_{n}),\nu ^{-1}d_{t}u_{h}^{n}+Au_{h}^{n} \bigr)\Delta t -\bigl(j\theta_{h}^{n},\nu^{-1}d_{t}u_{h}^{n}+Au_{h}^{n} \bigr) \end{aligned}$$
(4.15)
and
$$\begin{aligned} &(\lambda\nu)^{-1}\Delta t\bigl\Vert d_{t} \theta_{h}^{n}\bigr\Vert _{0}^{2}+ \lambda\nu\Delta t\bigl\Vert A_{h}\theta_{h}^{n} \bigr\Vert _{0}^{2} +\bigl\Vert \nabla \theta_{h}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert \nabla\theta_{h}^{n-1}\bigr\Vert _{0}^{2}+\bigl\Vert \nabla\bigl(\theta _{h}^{n}- \theta_{h}^{n-1}\bigr)\bigr\Vert _{0}^{2} \\ &\qquad{}+\overline{b}\bigl(u_{h}^{n-1}, \theta_{h}^{n-1},(\lambda\nu)^{-1}d_{t} \theta _{h}^{n}+A\theta_{h}^{n}\bigr) \Delta t \\ &\quad=\bigl(g(t_{n}),(\lambda\nu)^{-1}d_{t} \theta_{h}^{n}+A\theta _{h}^{n}\bigr) \Delta t. \end{aligned}$$
(4.16)
For the right-hand side terms and the trilinear terms, by using Lemma 2.3 we obtain
$$\begin{aligned} &\bigl\vert b\bigl(u_{h}^{n-1},u_{h}^{n-1}, \nu^{-1}d_{t}u_{h}^{n}+A_{h}u_{h}^{n} \bigr)\bigr\vert \\ &\quad\leq2C_{3}\bigl\Vert A_{h}^{1/2}u_{h}^{n-1} \bigr\Vert _{0}\bigl\Vert u_{h}^{n-1}\bigr\Vert ^{1/2}_{0}\bigl\Vert A_{h}u_{h}^{n-1} \bigr\Vert ^{1/2}_{0}\bigl(\nu^{-1}\bigl\Vert d_{t}u_{h}^{n}\bigr\Vert _{0}+\bigl\Vert A_{h}u_{h}^{n}\bigr\Vert _{0} \bigr) \\ &\quad\leq\frac{1}{4\nu}\bigl\Vert d_{t}u_{h}^{n} \bigr\Vert _{0}^{2}+\frac{\nu}{4}\bigl\Vert A_{h}u_{h}^{n}\bigr\Vert _{0}^{2}+ \frac{8}{\nu}C_{3}^{2}\bigl\Vert A_{h}^{1/2}u_{h}^{n-1} \bigr\Vert _{0}^{2}\bigl\Vert u_{h}^{n-1} \bigr\Vert _{0}\bigl\Vert A_{h}u_{h}^{n-1} \bigr\Vert _{0} \\ &\quad\leq\frac{1}{4\nu}\bigl\Vert d_{t}u_{h}^{n} \bigr\Vert _{0}^{2}+\frac{\nu}{4}\bigl\Vert A_{h}u_{h}^{n}\bigr\Vert _{0}^{2}+ \frac {\nu}{8}\bigl\Vert A_{h}u_{h}^{n-1} \bigr\Vert _{0}^{2}+ \frac{1}{2}\biggl( \frac{8}{\nu}\biggr)^{3}C_{3}^{4}\bigl\Vert A_{h}^{1/2}u_{h}^{n-1}\bigr\Vert _{0}^{4}\bigl\Vert u_{h}^{n-1}\bigr\Vert _{0}^{2}, \\ &\bigl\vert \overline{b}\bigl(u_{h}^{n-1}, \theta_{h}^{n-1},(\lambda\nu)^{-1}d_{t} \theta _{h}^{n}+A_{h}\theta_{h}^{n} \bigr)\bigr\vert \\ &\quad\leq2C_{5}\bigl\Vert A_{h}^{1/2}u_{h}^{n-1} \bigr\Vert ^{1/2}_{0}\bigl\Vert u_{h}^{n-1} \bigr\Vert ^{1/2}_{0}\bigl\Vert A_{h}^{1/2} \theta_{h}^{n-1}\bigr\Vert ^{1/2}_{0}\\ &\qquad{}\times\bigl\Vert A_{h}\theta_{h}^{n-1}\bigr\Vert ^{1/2}_{0} \bigl((\lambda\nu)^{-1}\bigl\Vert d_{t}\theta_{h}^{n}\bigr\Vert _{0}+ \bigl\Vert A_{h}\theta_{h}^{n}\bigr\Vert _{0}\bigr) \\ &\quad\leq\frac{1}{4\lambda\nu}\bigl\Vert d_{t}\theta_{h}^{n} \bigr\Vert _{0}^{2}+\frac{\lambda\nu}{4}\bigl\Vert A_{h}\theta_{h}^{n}\bigr\Vert _{0}^{2}+\frac{\lambda\nu}{8}\bigl\Vert A_{h} \theta_{h}^{n-1}\bigr\Vert _{0}^{2}\\ &\qquad{}+ \frac{1}{2}\biggl(\frac{8}{\lambda\nu}\biggr)^{3}C_{5}^{4} \bigl\Vert A_{h}^{1/2}u_{h}^{n-1}\bigr\Vert _{0}^{2}\bigl\Vert \nabla\theta_{h}^{n-1} \bigr\Vert _{0}^{2}\bigl\Vert u_{h}^{n-1} \bigr\Vert _{0}^{2}, \\ &\bigl\vert \bigl(f(t_{n}),\nu^{-1}d_{t}u_{h}^{n}+A_{h}u_{h}^{n} \bigr)\bigr\vert \leq\frac{1}{4\nu}\bigl\Vert d_{t}u_{h}^{n} \bigr\Vert _{0}^{2}+\frac{\nu}{16}\bigl\Vert A_{h}u_{h}^{n}\bigr\Vert _{0}^{2}+ \frac{20}{\nu}\bigl\Vert f(t_{n})\bigr\Vert _{0}^{2}, \\ &\bigl\vert \bigl(j\theta_{h}^{n},\nu^{-1}d_{t}u_{h}^{n}+A_{h}u_{h}^{n} \bigr)\bigr\vert \leq\frac{1}{4\nu}\bigl\Vert d_{t}u_{h}^{n} \bigr\Vert _{0}^{2}+\frac{\nu}{16}\bigl\Vert A_{h}u_{h}^{n}\bigr\Vert _{0}^{2}+ \frac{20}{\nu}\bigl\Vert \theta_{h}^{n}\bigr\Vert _{0}^{2}, \\ &\bigl\vert \bigl(g(t_{n}),(\lambda\nu)^{-1}d_{t} \theta_{h}^{n}+A_{h}\theta_{h}^{n} \bigr)\bigr\vert \leq\frac{1}{4\lambda\nu}\bigl\Vert d_{t} \theta_{h}^{n}\bigr\Vert _{0}^{2}+ \frac{\lambda\nu}{8}\bigl\Vert A_{h}\theta_{h}^{n} \bigr\Vert _{0}^{2}+ \frac{12}{\lambda\nu}\bigl\Vert g(t_{n})\bigr\Vert _{0}^{2}. \end{aligned}$$
Combining these inequalities with (4.15)-(4.16), we find
$$\begin{aligned} &(2\nu)^{-1}\Delta t\bigl\Vert d_{t}u_{h}^{n} \bigr\Vert _{0}^{2} +2\bigl\Vert \nabla u_{h}^{n}\bigr\Vert _{0}^{2}-2\bigl\Vert \nabla u_{h}^{n-1}\bigr\Vert _{0}^{2}+ \nu\Delta t\biggl(\frac {5}{4}\bigl\Vert A_{h}u_{h}^{n} \bigr\Vert _{0}^{2}-\frac{1}{4}\bigl\Vert A_{h}u_{h}^{n-1}\bigr\Vert _{0}^{2} \biggr) \\ &\quad\leq \biggl(\frac{8}{\nu}\biggr)^{3}C_{3}^{4} \bigl\Vert \nabla u_{h}^{n-1}\bigr\Vert _{0}^{2}\bigl\Vert u_{h}^{n-1}\bigr\Vert _{0}^{2}\bigl\Vert \nabla u_{h}^{n-1} \bigr\Vert _{0}^{2}\Delta t+\frac{40}{\nu}\bigl\Vert f(t_{n})\bigr\Vert _{0}^{2}\Delta t + \frac{40}{\nu}\bigl\Vert \theta_{h}^{n}\bigr\Vert _{0}^{2}\Delta t \end{aligned}$$
(4.17)
and
$$ \begin{aligned}[b] &(2\lambda\nu)^{-1}\Delta t\bigl\Vert d_{t} \theta_{h}^{n}\bigr\Vert _{0}^{2} +2 \bigl\Vert \nabla\theta_{h}^{n}\bigr\Vert _{0}^{2}-2\bigl\Vert \nabla\theta_{h}^{n-1} \bigr\Vert _{0}^{2}\\ &\qquad{}+\lambda\nu \Delta t\biggl( \frac{5}{4}\bigl\Vert A_{h}\theta_{h}^{n} \bigr\Vert _{0}^{2}-\frac{1}{4}\bigl\Vert A_{h}\theta _{h}^{n-1}\bigr\Vert _{0}^{2}\biggr) \\ &\quad\leq \biggl(\frac{8}{\lambda\nu}\biggr)^{3}C_{5}^{4} \bigl\Vert \nabla u_{h}^{n-1}\bigr\Vert _{0}^{2}\bigl\Vert \nabla\theta_{h}^{n-1} \bigr\Vert _{0}^{2}\bigl\Vert u_{h}^{n-1} \bigr\Vert _{0}^{2}\Delta t+\frac{24}{\lambda\nu }\bigl\Vert g(t_{n})\bigr\Vert _{0}^{2}\Delta t. \end{aligned} $$
(4.18)
Summing (4.17)-(4.18) for n from 1 to \(J+1\) and using Lemma 2.3, we finish the proof of (4.2).
Moreover, for all \(v\in V_{h}\) and \(\psi\in W_{h}\) with \(2\leq n\leq[\frac {T}{\Delta t}]-1\), we deduce from (3.3) that
$$\begin{aligned}& \begin{aligned}[b] &\bigl(d_{tt}u_{h}^{n},v\bigr)+a \bigl(d_{t}u_{h}^{n},v\bigr)+b\bigl(d_{t}u_{h}^{n-1},u_{h}^{n-1},v \bigr)+b\bigl(u_{h}^{n-2},d_{t}u_{h}^{n-1},v \bigr) \\ &\quad=\frac{1}{\Delta t} \int_{t_{n-1}}^{t_{n}}(f_{t},v)\,dt- \bigl(jd_{t}\theta_{h}^{n},v\bigr), \end{aligned} \end{aligned}$$
(4.19)
$$\begin{aligned}& \bigl(d_{tt}u_{h}^{1},v\bigr)+a \bigl(d_{t}u_{h}^{1},v\bigr)=-\bigl(jd_{t} \theta_{h}^{1},v\bigr), \end{aligned}$$
(4.20)
$$\begin{aligned}& \begin{aligned}[b] &\bigl(d_{tt}\theta_{h}^{n},\psi\bigr)+ \overline{a}\bigl(d_{t}\theta_{h}^{n},\psi\bigr)+ \overline {b}\bigl(d_{t}u_{h}^{n-1}, \theta_{h}^{n-1},\psi\bigr) +\overline{b}\bigl(u_{h}^{n-2},d_{t} \theta_{h}^{n-1},\psi\bigr) \\ &\quad=\frac{1}{\Delta t} \int_{t_{n-1}}^{t_{n}}(g_{t},\psi) \,dt, \end{aligned} \end{aligned}$$
(4.21)
$$\begin{aligned}& \bigl(d_{tt}\theta_{h}^{1},\psi\bigr)+ \overline{a}\bigl(d_{t}\theta_{h}^{1},\psi \bigr)=0. \end{aligned}$$
(4.22)
Choosing \(v=d_{t}u_{h}^{1}\Delta t\) and \(\psi=d_{t}\theta_{h}^{1}\Delta t\) in (4.20) and (4.22), respectively, we obtain
$$\begin{aligned}& \bigl\Vert d_{t}\theta_{h}^{1}\bigr\Vert _{0}^{2}+\bigl\Vert d_{tt}\theta_{h}^{1} \bigr\Vert _{0}^{2}\Delta t^{2}+\lambda\nu\bigl\Vert \nabla d_{t}\theta_{h}^{1}\bigr\Vert _{0}^{2}\Delta t=\bigl\Vert d_{t} \theta_{h}^{0}\bigr\Vert _{0}^{2}, \\& \bigl\Vert d_{t}u_{h}^{1}\bigr\Vert _{0}^{2}+\bigl\Vert d_{tt}u_{h}^{1} \bigr\Vert _{0}^{2}\Delta t^{2}+\nu\bigl\Vert \nabla d_{t}u_{h}^{1}\bigr\Vert _{0}^{2}\Delta t\leq\bigl\Vert d_{t}u_{h}^{0} \bigr\Vert _{0}^{2}+\frac{C_{1}}{2\lambda\nu}\bigl\Vert d_{t}\theta _{h}^{0}\bigr\Vert _{0}^{2}. \end{aligned}$$
Taking \(v=2d_{t}u_{h}^{n}\Delta t\) in (4.19) and \(\psi=2d_{t}\theta _{h}^{n}\Delta t\) in (4.21) with \(2\leq n\leq[\frac{T}{\Delta t}]\), we get
$$\begin{aligned} &\bigl\Vert d_{t}u_{h}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert d_{t}u_{h}^{n-1} \bigr\Vert _{0}^{2}+2\nu\Delta t\bigl\Vert \nabla d_{t}u_{h}^{n}\bigr\Vert _{0}^{2} +2b\bigl(d_{t}u_{h}^{n-1},u_{h}^{n-1},d_{t}u_{h}^{n} \bigr)\Delta t \\ &\qquad{}+2b\bigl(u_{h}^{n-2},d_{t}u_{h}^{n-1},d_{t}u_{h}^{n} \bigr)\Delta t \leq2 \int_{t_{n-1}}^{t_{n}}\bigl(f_{t},d_{t}u_{h}^{n} \bigr)\,dt-\bigl(jd_{t}\theta _{h}^{n},d_{t}u_{h}^{n} \bigr)\Delta t \end{aligned}$$
(4.23)
and
$$\begin{aligned} &\bigl\Vert d_{t}\theta_{h}^{n} \bigr\Vert _{0}^{2}-\bigl\Vert d_{t} \theta_{h}^{n-1}\bigr\Vert _{0}^{2}+2 \lambda\nu\Delta t\bigl\Vert \nabla d_{t}\theta_{h}^{n} \bigr\Vert _{0}^{2} \\ &\qquad{}+2\overline{b}\bigl(d_{t}u_{h}^{n-1}, \theta_{h}^{n-1},d_{t}\theta_{h}^{n} \bigr)\Delta t +2\overline{b}\bigl(u_{h}^{n-2},d_{t} \theta_{h}^{n-1},d_{t}\theta_{h}^{n} \bigr)\Delta t \\ &\quad\leq2 \int_{t_{n-1}}^{t_{n}}\bigl(g_{t},d_{t} \theta_{h}^{n}\bigr)\,dt. \end{aligned}$$
(4.24)
By Lemma 2.3 and the Poincaré inequality we have
$$\begin{aligned} &2\bigl\vert b\bigl(d_{t}u_{h}^{n-1},u_{h}^{n-1},d_{t}u_{h}^{n} \bigr)\bigr\vert +\bigl\vert 2b\bigl(u_{h}^{n-2},d_{t}u_{h}^{n-1},d_{t}u_{h}^{n} \bigr)\bigr\vert \\ &\quad\leq2C_{4}\bigl(\bigl\Vert A_{h}u_{h}^{n-1} \bigr\Vert _{0}+\bigl\Vert A_{h}u_{h}^{n-2} \bigr\Vert _{0}\bigr)\bigl\Vert \nabla d_{t}u_{h}^{n} \bigr\Vert _{0}\bigl\Vert d_{t}u_{h}^{n-1} \bigr\Vert _{0} \\ &\quad\leq\frac{\nu}{4}\bigl\Vert \nabla d_{t}u_{h}^{n} \bigr\Vert _{0}^{2}+4\nu^{-1}C_{4}^{2} \bigl(\bigl\Vert Au_{h}^{n-1}\bigr\Vert ^{2}_{0}+\bigl\Vert Au_{h}^{n-2}\bigr\Vert _{0}^{2}\bigr)\bigl\Vert d_{t}u_{h}^{n-1} \bigr\Vert _{0}^{2}, \\ &2\bigl\vert \overline{b}\bigl(d_{t}u_{h}^{n-1}, \theta_{h}^{n-1},d_{t}\theta_{h}^{n} \bigr)\bigr\vert +\bigl\vert 2\overline {b}\bigl(u_{h}^{n-2},d_{t} \theta_{h}^{n-1},d_{t}\theta_{h}^{n} \bigr)\bigr\vert \\ &\quad\leq2C_{6}\bigl\Vert A_{h}\theta_{h}^{n-1} \bigr\Vert _{0}\bigl\Vert \nabla d_{t} \theta_{h}^{n}\bigr\Vert _{0}\bigl\Vert d_{t}u_{h}^{n-1}\bigr\Vert _{0}+2C_{6} \bigl\Vert A_{h}u_{h}^{n-2}\bigr\Vert _{0}\bigl\Vert \nabla d_{t}\theta_{h}^{n} \bigr\Vert _{0}\bigl\Vert d_{t}\theta_{h}^{n-1} \bigr\Vert _{0} \\ &\quad\leq\frac{\lambda\nu}{4}\bigl\Vert \nabla d_{t} \theta_{h}^{n}\bigr\Vert _{0}^{2}+ \frac {4C_{6}^{2}}{\lambda\nu}\bigl(\bigl\Vert A_{h}\theta_{h}^{n-1} \bigr\Vert ^{2}_{0}\bigl\Vert d_{t}u_{h}^{n-1} \bigr\Vert _{0}^{2} +\bigl\Vert A_{h}u_{h}^{n-2} \bigr\Vert _{0}^{2}\bigl\Vert d_{t} \theta_{h}^{n-1}\bigr\Vert _{0}^{2} \bigr), \\ &2\biggl\vert \int_{t_{n-1}}^{t_{n}}\bigl(f_{t},d_{t}u_{h}^{n} \bigr)\,dt\biggr\vert \leq\frac{\nu\Delta t}{4}\bigl\Vert \nabla d_{t}u_{h}^{n}\bigr\Vert _{0}^{2} +4\nu^{-1}C_{1}^{2} \int_{t_{n-1}}^{t_{n}}\Vert f_{t}\Vert _{0}^{2}\,dt, \\ &2\biggl\vert \int_{t_{n-1}}^{t_{n}}\bigl(g_{t},d_{t} \theta_{h}^{n}\bigr)\,dt\biggr\vert \leq\frac{\lambda\nu \Delta t}{4} \bigl\Vert \nabla d_{t}\theta_{h}^{n}\bigr\Vert _{0}^{2} +4(\lambda\nu)^{-1}C_{1}^{2} \int_{t_{n-1}}^{t_{n}}\Vert g_{t}\Vert _{0}^{2}\,dt, \\ &\bigl\vert \bigl(jd_{t}\theta_{h}^{n},d_{t}u_{h}^{n} \bigr)\bigr\vert \leq\frac{\nu}{4}\bigl\Vert \nabla d_{t}u_{h}^{n} \bigr\Vert _{0}^{2}+\frac{4}{\nu}C_{1}^{2} \bigl\Vert d_{t}\theta_{h}^{n}\bigr\Vert _{0}^{2}. \end{aligned}$$
It follows from these inequalities that (4.23) and (4.24) can be transformed into
$$\begin{aligned} &\bigl\Vert d_{t}u_{h}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert d_{t}u_{h}^{n-1} \bigr\Vert _{0}^{2}+\nu\Delta t\bigl\Vert \nabla d_{t}u_{h}^{n}\bigr\Vert _{0}^{2} \\ &\quad\leq4\nu^{-1}C_{1}^{2}\biggl( \int_{t_{n-1}}^{t_{n}}\Vert f_{t}\Vert _{0}^{2}\,dt+\bigl\Vert d_{t} \theta_{h}^{n}\bigr\Vert _{0}^{2} \biggr) \\ &\qquad{}+4\nu^{-1}C_{4}^{2}\bigl(\bigl\Vert A_{h}u_{h}^{n-1}\bigr\Vert _{0}^{2}+ \bigl\Vert A_{h}u_{h}^{n-2}\bigr\Vert _{0}^{2}\bigr)\bigl\Vert d_{t}u_{h}^{n-1} \bigr\Vert _{0}^{2} \end{aligned}$$
(4.25)
and
$$\begin{aligned} &\bigl\Vert d_{t}\theta_{h}^{n} \bigr\Vert _{0}^{2}-\bigl\Vert d_{t} \theta_{h}^{n-1}\bigr\Vert _{0}^{2}+ \lambda\nu\Delta t\bigl\Vert \nabla d_{t}\theta_{h}^{n} \bigr\Vert _{0}^{2} \\ &\quad\leq4(\lambda\nu)^{-1}C_{1}^{2} \int_{t_{n-1}}^{t_{n}}\Vert g_{t}\Vert _{0}^{2}\,dt \\ &\qquad{}+\frac {4C_{6}^{2}}{\lambda\nu}\bigl(\bigl\Vert A_{h}\theta_{h}^{n-1}\bigr\Vert ^{2}_{0}\bigl\Vert d_{t}u_{h}^{n-1} \bigr\Vert _{0}^{2} +\bigl\Vert A_{h}u_{h}^{n-2} \bigr\Vert _{0}^{2}\bigl\Vert d_{t} \theta_{h}^{n-1}\bigr\Vert _{0}^{2} \bigr). \end{aligned}$$
(4.26)
Summing (4.25)-(4.26) for n from 1 to \(J+1\) and using Lemma 2.3, we finish the proof of (4.3).
Using again Lemma 2.3 and (3.3), we deduce
$$\begin{aligned} \nu\bigl\Vert A_{h}u_{h}^{n}\bigr\Vert _{0}\leq\bigl\Vert d_{t}u_{h}^{n}\bigr\Vert _{0}+\bigl\Vert f(t_{n})\bigr\Vert _{0} +2C_{3}\bigl\Vert \nabla u_{h}^{n-1}\bigr\Vert _{0}\bigl\Vert u_{h}^{n-1}\bigr\Vert ^{1/2}_{0}\bigl\Vert A_{h}u_{h}^{n-1} \bigr\Vert ^{1/2}_{0} \end{aligned}$$
and
$$\begin{aligned} \lambda\nu\bigl\| A_{h}\theta_{h}^{n}\bigr\| _{0} \leq\bigl\Vert d_{t}\theta_{h}^{n}\bigr\Vert _{0}+\bigl\Vert g(t_{n})\bigr\Vert _{0} +2C_{5}\bigl\Vert u_{h}^{n-1}\bigr\Vert ^{1/2}_{0}\bigl\Vert \nabla u_{h}^{n-1} \bigr\Vert ^{1/2}_{0}\bigl\Vert \nabla\theta _{h}^{n-1}\bigr\Vert ^{1/2}_{0}\bigl\Vert A_{h}\theta_{h}^{n-1}\bigr\Vert ^{1/2}_{0}. \end{aligned}$$
If \(\|A_{h}u_{h}^{n}\|_{0}\leq\|A_{h}u_{h}^{n-1}\|_{0}\) and \(\|A_{h}\theta_{h}^{n}\|_{0}\leq\| A_{h}\theta_{h}^{n-1}\|_{0}\), then by (3.5)-(3.6) we have that (4.4)-(4.5) hold. Otherwise, setting \(k_{*}=\sup_{0\leq n\leq J+1}\|A_{h}u_{h}^{n}\|_{0}\) and \(k_{**}=\sup_{0\leq n\leq J+1}\|A_{h}\theta_{h}^{n}\|_{0}\) and using (4.1)-(4.3), the obtained inequalities give
$$\begin{aligned}& \bigl\Vert A_{h}u_{h}^{J+1}\bigr\Vert _{0}^{2}\leq k_{*}^{2}\leq\frac{12}{\nu^{2}} \Bigl(\sup_{1\leq n\leq J+1}\bigl\Vert d_{t}u_{h}^{n} \bigr\Vert ^{2}_{0}+f^{2}_{\infty}\Bigr) +48 \nu^{-4}C_{3}^{4}\sup_{1\leq n\leq J}\bigl\Vert \nabla u^{n}\bigr\Vert ^{4}_{0}\bigl\Vert u^{n}\bigr\Vert ^{2}_{0}+ \gamma_{1}, \\& \bigl\Vert A_{h}\theta_{h}^{J+1}\bigr\Vert _{0}^{2}\leq k_{*}^{2}\leq\frac{12}{(\lambda\nu)^{2}} \Bigl(\sup_{1\leq n\leq J+1}\bigl\Vert d_{t} \theta_{h}^{n}\bigr\Vert ^{2}_{0}+g^{2}_{\infty} \Bigr) +48(\lambda\nu)^{-4}C_{5}^{4}\sup _{1\leq n\leq J+1}\bigl\Vert \nabla u^{n}\bigr\Vert ^{4}_{0}\bigl\Vert \theta^{n}\bigr\Vert ^{2}_{0}. \end{aligned}$$
Combining these estimates with (4.1)-(4.3), we finish the proof of (4.4)-(4.5) with \(n=J+1\). □

5 Error estimates

This section is devoted to present the optimal error estimates of velocity, pressure, and temperature in the Euler implicit/explicit scheme (3.3). In order to simplify the descriptions, we denote
$$\begin{aligned} E_{u}^{n}=u_{h}(t_{n})-u_{h}^{n}, \qquad E_{p}^{n}=p_{h}(t_{n})-p_{h}^{n}, \qquad E_{\theta}^{n}=\theta _{h}(t_{n})- \theta_{h}^{n}, \end{aligned}$$
where \((u_{h}(t_{n}),p_{h}(t_{n}),\theta_{h}(t_{n}))\) and \((u^{n}_{h},p^{n}_{h},\theta ^{n}_{h})\) be the solutions of problems (3.2) and (3.3), respectively. Furthermore, we set \(E_{u}^{0}=E_{\theta}^{0}=0\).
Let us define the truncation errors \(R_{u}^{n}\) and \(R_{\theta}^{n}\) by
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} \frac{u_{h}(t_{n})-u_{h}(t_{n-1})}{\Delta t}-\nu\Delta u_{h}(t_{n}) +(u_{h}(t_{n})\cdot\nabla)u_{h}(t_{n})+\nabla p_{h}(t_{n}) =f(t_{n})-j\theta_{h}(t_{n})+R_{u}^{n},\\ \nabla\cdot u_{h}(t_{n})=0,\\ \frac{\theta_{h}(t_{n})-\theta_{h}(t_{n-1})}{\Delta t}-\lambda\nu\Delta\theta_{h}(t_{n}) +(u_{h}(t_{n})\cdot\nabla)\theta_{h}(t_{n})=g(t_{n})+R_{\theta}^{n}, \end{array}\displaystyle \right . \end{aligned}$$
(5.1)
where
$$\begin{aligned} R_{u}^{n}=-\frac{1}{\Delta t} \int_{t_{n-1}}^{t_{n}}(t-t_{n})u_{htt}(t) \,dt \quad\mbox{and}\quad R_{\theta}^{n}=-\frac{1}{\Delta t} \int_{t_{n-1}}^{t_{n}}(t-t_{n})\theta _{htt}(t)\,dt. \end{aligned}$$
By subtracting (3.3) from (5.1) we obtain the following error equations:
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} \frac{E_{u}^{n}-E_{u}^{n-1}}{\Delta t}-\nu\Delta E_{u}^{n}+(u_{h}(t_{n})\cdot\nabla )u_{h}(t_{n})-(u_{h}^{n-1}\cdot\nabla)u_{h}^{n-1} +\nabla E_{p}^{n} =R_{u}^{n}-jE_{\theta}^{n},\\ \nabla\cdot E_{u}^{n}=0,\\ \frac{E_{\theta}^{n}-E_{\theta}^{n-1}}{\Delta t}-\lambda\nu\Delta E_{\theta}^{n} +(u_{h}(t_{n})\cdot\nabla)\theta_{h}(t_{n})-(u_{h}^{n-1}\cdot\nabla)\theta _{h}^{n-1}=R_{\theta}^{n}. \end{array}\displaystyle \right . \end{aligned}$$
(5.2)
Now, we present the error estimates for \(E_{u}^{n},E_{p}^{n}\), and \(E_{\theta }^{n}\) in different norms. In order to simplify the expressions, we denote
$$\begin{aligned}& Z_{u}^{n}=u_{h}(t_{n})-u_{h}(t_{n-1})= \int_{t_{n-1}}^{t_{n}}u_{ht}\,dt, \\& Z_{\theta}^{n}=\theta_{h}(t_{n})- \theta_{h}(t_{n-1})= \int_{t_{n-1}}^{t_{n}}\theta_{ht}\,dt. \end{aligned}$$
Then
$$\begin{aligned}& u_{h}(t_{n})-u_{h}^{n-1}=u_{h}(t_{n})-u_{h}(t_{n-1})+u_{h}(t_{n-1})-u_{h}^{n-1}=Z_{u}^{n}+E_{u}^{n-1}, \\& \theta_{h}(t_{n})-\theta_{h}^{n-1}= \theta_{h}(t_{n})-\theta_{h}(t_{n-1})+ \theta _{h}(t_{n-1})-\theta_{h}^{n-1}=Z_{\theta}^{n}+E_{\theta}^{n-1}. \end{aligned}$$
As a consequence, we find
$$\begin{aligned} &\bigl(u_{h}(t_{n})\cdot\nabla\bigr)u_{h}(t_{n})- \bigl(u_{h}^{n-1}\cdot\nabla\bigr)u_{h}^{n-1} \\ &\quad=\bigl(E_{u}^{n-1}\cdot\nabla\bigr)u_{h}^{n-1}+ \bigl(Z_{u}^{n}\cdot\nabla \bigr)u_{h}(t_{n-1})+ \bigl(u_{h}(t_{n-1})\cdot\nabla\bigr)E_{u}^{n-1}+ \bigl(u_{h}(t_{n})\cdot\nabla\bigr)Z_{u}^{n} \end{aligned}$$
and
$$\begin{aligned}[b] &\bigl(u_{h}(t_{n})\cdot\nabla\bigr)\theta_{h}(t_{n})- \bigl(u_{h}^{n-1}\cdot\nabla\bigr)\theta_{h}^{n-1} \\ &\quad=\bigl(E_{u}^{n-1}\cdot\nabla\bigr) \theta_{h}^{n-1}+\bigl(Z_{u}^{n}\cdot \nabla\bigr)\theta _{h}(t_{n-1})+\bigl(u_{h}(t_{n-1}) \cdot\nabla\bigr)E_{\theta}^{n-1}+\bigl(u_{h}(t_{n}) \cdot\nabla \bigr)Z_{\theta}^{n}. \end{aligned} $$

Lemma 5.1

Under the assumptions of Theorems 3.1 and 4.1, we have
$$\begin{aligned} \bigl\Vert E_{u}^{J+1}\bigr\Vert _{0}^{2}+ \bigl\Vert E_{\theta}^{J+1}\bigr\Vert _{0}^{2}+ \nu\Delta t\sum_{n=1}^{J+1}\bigl\Vert \nabla E_{u}^{n}\bigr\Vert _{0}^{2}+ \lambda \nu\Delta t\sum_{n=1}^{J+1}\bigl\Vert \nabla E_{\theta}^{n}\bigr\Vert _{0}^{2}\leq C \Delta t^{2}. \end{aligned}$$

Proof

Taking the inner product of (5.2) with \(2\Delta tE_{u}^{n}\) and \(2\Delta tE_{\theta}^{n}\) and using the fact that \(\nabla\cdot E_{u}^{n}=0\), we obtain
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} \|E_{u}^{n}\|_{0}^{2}-\|E_{u}^{n-1}\|_{0}^{2} +2\nu\Delta t\|\nabla E_{u}^{n}\|_{0}^{2}+2\Delta tb(E_{u}^{n-1},u_{h}^{n-1},E_{u}^{n}) +2\Delta tb(Z_{u}^{n},u_{h}(t_{n-1}),E_{u}^{n}) \\ \quad{}+2\Delta tb(u_{h}(t_{n-1}),E_{u}^{n-1},E_{u}^{n})+2\Delta tb(u_{h}(t_{n}),Z_{u}^{n},E_{u}^{n})\\ \qquad{}\leq2\Delta t(R_{u}^{n},E_{u}^{n})-2\Delta t(jE_{\theta}^{n},E_{u}^{n}),\\ \|E_{\theta}^{n}\|_{0}^{2}-\|E_{\theta}^{n-1}\|_{0}^{2}+2\lambda\nu\Delta t\|\nabla E_{\theta}^{n}\|_{0}^{2} +2\Delta t\overline{b}(E_{u}^{n-1},\theta_{h}^{n-1},E_{\theta}^{n})\\ \quad{}+2\Delta t\overline{b}(Z_{u}^{n},\theta_{h}(t_{n-1}),E_{\theta}^{n}) +2\Delta t\overline{b}(u_{h}(t_{n-1}),E_{\theta}^{n-1},E_{\theta}^{n})+2\Delta t\overline{b}(u_{h}(t_{n}),Z_{\theta}^{n},E_{\theta}^{n})\\ \qquad\leq2\Delta t(R_{\theta}^{n},E_{\theta}^{n}). \end{array}\displaystyle \right . \end{aligned}$$
(5.3)
The right-hand side terms of (5.3) can be treated as follows:
$$\begin{aligned}& \begin{aligned}[b] \bigl\vert 2\Delta t\bigl(R_{u}^{n},E_{u}^{n} \bigr)\bigr\vert &\leq\frac{6\Delta tC_{1}^{2}}{\nu}\biggl\| \int _{t_{n-1}}^{t_{n}}(t-t_{n-1})u_{htt} \,dt\biggr\| _{0}^{2}+\frac{\nu\Delta t}{6}\bigl\| \nabla E_{u}^{n}\bigr\| _{0}^{2}\\ &\leq C\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\|u_{ntt}\|_{0}^{2} \,dt+\frac{\nu\Delta t}{6}\bigl\| \nabla E_{u}^{n} \bigr\| _{0}^{2}, \end{aligned} \\& \begin{aligned}[b] \bigl|2\Delta t\bigl(R_{\theta}^{n},E_{\theta}^{n} \bigr)\bigr| &\leq\frac{5\Delta tC_{1}^{2}}{\lambda\nu}\biggl\| \int _{t_{n-1}}^{t_{n}}(t-t_{n-1}) \theta_{htt}\,dt\biggr\| _{0}^{2}+\frac{\lambda\nu\Delta t}{5}\bigl\| \nabla E_{\theta}^{n}\bigr\| _{0}^{2} \\ &\leq C\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\|\theta_{htt} \|_{0}^{2}\,dt+\frac {\lambda\nu\Delta t}{5}\bigl\| \nabla E_{\theta}^{n} \bigr\| _{0}^{2}, \end{aligned} \\& \bigl|-2\Delta t\bigl(jE_{\theta}^{n},E_{u}^{n} \bigr)\bigr| \leq2\Delta t\bigl\| E_{\theta}^{n}\bigr\| _{0} \bigl\| E_{u}^{n}\bigr\| _{0} \leq\frac{6\Delta t}{\nu} \bigl\| E_{\theta}^{n}\bigr\| _{0}^{2}+\frac{\nu\Delta t}{6} \bigl\| E_{u}^{n}\bigr\| _{0}^{2}. \end{aligned}$$
For the nonlinear terms, by Lemma 2.3 we have
$$\begin{aligned}& \begin{aligned}[b] \bigl\vert 2\Delta tb\bigl(E_{u}^{n-1},u_{h}^{n-1},E_{u}^{n} \bigr)\bigr\vert &\leq2C_{4}\Delta t\bigl\Vert E_{u}^{n-1} \bigr\Vert _{0}\bigl\Vert A_{h}u_{h}^{n-1} \bigr\Vert _{0}\bigl\Vert \nabla E_{u}^{n}\bigr\Vert _{0} \\ &\leq\frac{6C_{4}^{2}}{\nu}\Delta t\bigl\Vert E_{u}^{n-1} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}^{n-1} \bigr\Vert _{0}^{2}+\frac{\nu\Delta t}{6}\bigl\Vert \nabla E_{u}^{n}\bigr\Vert _{0}^{2}, \end{aligned}\\& \begin{aligned}[b] \bigl\vert 2\Delta tb\bigl(Z_{u}^{n},u_{h}(t_{n-1}),E_{u}^{n} \bigr)\bigr\vert &\leq2C_{4}\Delta t\bigl\Vert Z_{u}^{n} \bigr\Vert _{0}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}\bigl\Vert \nabla E_{u}^{n}\bigr\Vert _{0} \\ &\leq\frac{6C_{4}^{2}}{\nu}\Delta t\bigl\Vert Z_{u}^{n} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}^{2}+\frac{\nu\Delta t}{6}\bigl\Vert \nabla E_{u}^{n}\bigr\Vert _{0}^{2}\\ & \leq C \Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert u_{ht}\Vert _{0}^{2}\,dt+\frac{\nu\Delta t}{6}\bigl\Vert \nabla E_{u}^{n}\bigr\Vert _{0}^{2}, \end{aligned}\\& \begin{aligned}[b] \bigl\vert 2\Delta tb\bigl(u_{h}(t_{n-1}),E_{u}^{n-1},E_{u}^{n} \bigr)\bigr\vert &\leq2C_{4}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}\bigl\Vert E_{u}^{n-1}\bigr\Vert _{0}\bigl\Vert \nabla E_{u}^{n}\bigr\Vert _{0} \\ &\leq\frac{6C_{4}^{2}}{\nu}\Delta t\bigl\Vert E_{u}^{n-1} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}^{2}+\frac{\nu\Delta t}{6}\bigl\Vert \nabla E_{u}^{n}\bigr\Vert _{0}^{2}, \end{aligned}\\& \begin{aligned}[b] \bigl\vert 2\Delta tb\bigl(u_{h}(t_{n}),Z_{u}^{n},E_{u}^{n} \bigr)\bigr\vert &\leq2C_{4}\Delta t\bigl\Vert Z_{u}^{n} \bigr\Vert _{0}\bigl\Vert A_{h}u_{h}(t_{n}) \bigr\Vert _{0}\bigl\Vert \nabla E_{u}^{n}\bigr\Vert _{0} \\ &\leq\frac{6C_{4}^{2}}{\nu}\Delta t\bigl\Vert Z_{u}^{n} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}(t_{n}) \bigr\Vert _{0}^{2}+\frac {\nu\Delta t}{6}\bigl\Vert \nabla E_{u}^{n}\bigr\Vert _{0}^{2} \\ &\leq C \Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert u_{ht}\Vert _{0}^{2}\,dt+\frac{\nu\Delta t}{6}\bigl\Vert \nabla E_{u}^{n}\bigr\Vert _{0}^{2}, \end{aligned}\\& \begin{aligned}[b] \bigl\vert 2\Delta t\overline{b}\bigl(E_{u}^{n-1}, \theta_{h}^{n-1},E_{\theta}^{n}\bigr)\bigr\vert &\leq2C_{6}\Delta t\bigl\Vert E_{u}^{n-1} \bigr\Vert _{0}\bigl\Vert A_{h}\theta_{h}^{n-1} \bigr\Vert _{0}\bigl\Vert \nabla E_{\theta}^{n} \bigr\Vert _{0} \\ &\leq\frac{6C_{6}^{2}}{\lambda\nu}\Delta t\bigl\Vert E_{u}^{n-1} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}\theta _{h}^{n-1}\bigr\Vert _{0}^{2}+ \frac{\lambda\nu\Delta t}{6}\bigl\Vert \nabla E_{\theta}^{n}\bigr\Vert _{0}^{2}, \end{aligned}\\& \begin{aligned}[b] \bigl\vert 2\Delta t\overline{b}\bigl(Z_{u}^{n}, \theta_{h}(t_{n-1}),E_{\theta}^{n}\bigr)\bigr\vert &\leq2C_{6}\Delta t\bigl\Vert Z_{u}^{n} \bigr\Vert _{0}\bigl\Vert A_{h}\theta_{h}(t_{n-1}) \bigr\Vert _{0}\bigl\Vert \nabla E_{\theta}^{n} \bigr\Vert _{0} \\ &\leq\frac{6C_{6}^{2}}{\lambda\nu}\Delta t\bigl\Vert Z_{u}^{n} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}\theta _{h}(t_{n-1})\bigr\Vert _{0}^{2}+ \frac{\lambda\nu\Delta t}{6}\bigl\Vert \nabla E_{\theta}^{n}\bigr\Vert _{0}^{2} \\ &\leq C\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert u_{ht}\Vert _{0}^{2}\,dt+\frac{\lambda\nu \Delta t}{6}\bigl\Vert \nabla E_{\theta}^{n}\bigr\Vert _{0}^{2}, \end{aligned}\\& \begin{aligned}[b] \bigl\vert 2\Delta t\overline{b}\bigl(u_{h}(t_{n-1}),E_{\theta}^{n-1},E_{\theta}^{n} \bigr)\bigr\vert &\leq 2C_{6}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}\bigl\Vert E_{\theta}^{n-1}\bigr\Vert _{0}\bigl\Vert \nabla E_{\theta}^{n}\bigr\Vert _{0}\\ &\leq\frac{6C_{6}^{2}}{\lambda\nu}\Delta t\bigl\Vert E_{\theta}^{n-1} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}^{2}+\frac{\lambda\nu\Delta t}{6}\bigl\Vert \nabla E_{\theta}^{n}\bigr\Vert _{0}^{2}, \end{aligned}\\& \begin{aligned}[b] \bigl\vert 2\Delta t\overline{b}\bigl(u_{h}(t_{n}),Z_{\theta}^{n},E_{\theta}^{n} \bigr)\bigr\vert &\leq2C_{6}\Delta t\bigl\Vert Z_{\theta}^{n} \bigr\Vert _{0}\bigl\Vert A_{h}u_{h}(t_{n}) \bigr\Vert _{0}\bigl\Vert \nabla E_{\theta}^{n}\bigr\Vert _{0}\\ &\leq\frac{6C_{6}^{2}}{\lambda\nu}\Delta t\bigl\Vert Z_{\theta}^{n} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}(t_{n}) \bigr\Vert _{0}^{2}+\frac{\lambda\nu\Delta t}{6}\bigl\Vert \nabla E_{\theta}^{n}\bigr\Vert _{0}^{2} \\ &\leq C \Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert \theta_{ht}\Vert _{0}^{2}\,dt+\frac{\lambda \nu\Delta t}{6}\bigl\Vert \nabla E_{\theta}^{n}\bigr\Vert _{0}^{2}. \end{aligned} \end{aligned}$$
From all these inequalities and Theorems 3.1 and 4.1 we obtain
$$\begin{aligned} &\bigl\Vert E_{u}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert E_{u}^{n-1}\bigr\Vert _{0}^{2}+\bigl\Vert E_{u}^{n}-E_{u}^{n-1} \bigr\Vert _{0}^{2} +\nu\Delta t\bigl\Vert \nabla E_{u}^{n}\bigr\Vert _{0}^{2} \\ &\quad\leq C\Delta t\bigl\Vert E_{u}^{n-1}\bigr\Vert _{0}^{2} +C\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\bigl(\Vert u_{htt}\Vert _{0}^{2}+\Vert u_{ht}\Vert _{0}^{2} \bigr)\,dt+\frac {6}{\nu}\Delta t\bigl\Vert E_{\theta}^{n} \bigr\Vert _{0}^{2} \end{aligned}$$
(5.4)
and
$$\begin{aligned} &\bigl\Vert E_{\theta}^{n+1}\bigr\Vert _{0}^{2}-\bigl\Vert E_{\theta}^{n}\bigr\Vert _{0}^{2}+\bigl\Vert E_{\theta}^{n+1}-E_{\theta}^{n}\bigr\Vert _{0}^{2}+\lambda\nu\Delta t\bigl\Vert \nabla E_{\theta}^{n+1}\bigr\Vert _{0}^{2} \\ &\quad\leq C\Delta t\bigl\Vert E_{u}^{n-1}\bigr\Vert _{0}^{2} +C\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\bigl(\Vert \theta_{htt} \Vert _{0}^{2}+\Vert \theta_{ht}\Vert _{0}^{2}+\Vert u_{ht}\Vert _{0}^{2} \bigr)\,dt. \end{aligned}$$
(5.5)
Summing (5.4) and (5.5) from \(n=1\) to \(J+1\) and using Lemma 2.1, we finish the proof. □

Lemma 5.2

Under the assumptions of Theorems 3.1 and 4.1, we have
$$\begin{aligned} \bigl\Vert \nabla E_{u}^{J+1}\bigr\Vert _{0}^{2}+\bigl\Vert \nabla E_{\theta}^{J+1} \bigr\Vert _{0}^{2}+\nu\Delta t\sum _{n=1}^{J+1}\bigl\Vert A_{h}E_{u}^{n} \bigr\Vert _{0}^{2}+ \lambda\nu\Delta t\sum _{n=1}^{J+1}\bigl\Vert A_{h}E_{\theta}^{n} \bigr\Vert _{0}^{2}\leq C\Delta t^{2}. \end{aligned}$$

Proof

Taking the inner product of (5.2) with \(-2\Delta tA_{h}E_{u}^{n}\in V_{h}\) and \(-2\Delta tA_{h}E_{\theta}^{n}\) and using the fact that \(\nabla\cdot E_{u}^{n+1}=0\), we obtain
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} \|\nabla E_{u}^{n}\|_{0}^{2}-\|\nabla E_{u}^{n-1}\|_{0}^{2} +2\nu\Delta t\|A_{h}E_{u}^{n}\|_{0}^{2}\\ \quad\leq2\Delta t(b(E_{u}^{n-1},u_{n}^{n-1},A_{h}E_{u}^{n}) +b(Z_{u}^{n},u_{h}(t_{n-1}),A_{h}E_{u}^{n})+b(u_{h}(t_{n-1}),E_{u}^{n-1},A_{h}E_{u}^{n})\\ \qquad{}+b(u_{h}(t_{n}),Z_{u}^{n},A_{h}E_{u}^{n}) +(jE_{\theta}^{n},A_{h}E_{u}^{n})-(R_{u}^{n},A_{h}E_{u}^{n})),\\ \|\nabla E_{\theta}^{n}\|_{0}^{2}-\|\nabla E_{\theta}^{n-1}\|_{0}^{2}+2\lambda\nu \Delta t\|A_{h}E_{\theta}^{n}\|_{0}^{2}\\ \quad\leq2\Delta t(\overline{b}(E_{u}^{n-1},\theta_{h}^{n-1},A_{h}E_{\theta}^{n}) +\overline{b}(Z_{u}^{n},\theta_{h}(t_{n-1}),A_{h}E_{\theta}^{n})+\overline {b}(u_{h}(t_{n-1}),E_{\theta}^{n-1},A_{h}E_{\theta}^{n})\\ \qquad{}+\overline{b}(u_{h}(t_{n}),Z_{\theta}^{n},A_{h}E_{\theta}^{n})-(R_{\theta}^{n},A_{h}E_{\theta}^{n})). \end{array}\displaystyle \right . \end{aligned}$$
(5.6)
Now, we treat the linear terms in the right-hand side of (5.6) as follows:
$$\begin{aligned}& \bigl\vert 2\Delta t\bigl(R_{u}^{n},A_{h}E_{u}^{n} \bigr)\bigr\vert \leq2\Delta t\bigl\Vert R_{u}^{n}\bigr\Vert _{0}\bigl\Vert A_{h}E_{u}^{n} \bigr\Vert _{0} \leq C\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert u_{htt}\Vert _{0}^{2}\,dt+\frac{\nu\Delta t}{2}\bigl\Vert AE_{u}^{n}\bigr\Vert _{0}^{2}, \\& \bigl\vert 2\Delta t\bigl(R_{\theta}^{n},A_{h}E_{\theta}^{n} \bigr)\bigr\vert \leq2\Delta t\bigl\Vert R_{\theta}^{n}\bigr\Vert _{0}\bigl\Vert AE_{\theta}^{n+1}\bigr\Vert _{0} \leq C\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert \theta_{htt}\Vert _{0}^{2}\,dt+\frac{\lambda \nu\Delta t}{2}\bigl\Vert A_{h}E_{\theta}^{n}\bigr\Vert _{0}^{2}, \\& \bigl\vert -2\Delta t\bigl(jE_{\theta}^{n},A_{h}E_{u}^{n} \bigr)\bigr\vert \leq2\Delta t\bigl\Vert E_{\theta}^{n}\bigr\Vert _{0}\bigl\Vert AE_{u}^{n}\bigr\Vert _{0} \leq4\nu^{-1}\Delta t\bigl\Vert E_{\theta}^{n} \bigr\Vert _{0}^{2}+\frac{\nu\Delta t}{4}\bigl\Vert AE_{u}^{n}\bigr\Vert _{0}^{2}. \end{aligned}$$
For the trilinear terms of (5.6), by applying Lemma 2.3 we have
$$\begin{aligned}& \begin{aligned}[b] \bigl\vert 2\Delta tb\bigl(E_{u}^{n-1},u_{h}^{n-1},A_{h}E_{u}^{n} \bigr)\bigr\vert &\leq2C_{4}\Delta t\bigl\Vert \nabla E_{u}^{n-1} \bigr\Vert _{0}\bigl\Vert A_{h}u_{h}^{n-1} \bigr\Vert _{0}\bigl\Vert A_{h}E_{u}^{n} \bigr\Vert _{0} \\ &\leq\frac{6C_{4}^{2}}{\nu}\Delta t\bigl\Vert \nabla E_{u}^{n-1} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}^{n-1} \bigr\Vert _{0}^{2}+\frac{\nu\Delta t}{6}\bigl\Vert A_{h}E_{u}^{n}\bigr\Vert _{0}^{2}, \end{aligned}\\& \begin{aligned}[b] \bigl\vert 2\Delta tb\bigl(Z_{u}^{n},u_{h}(t_{n-1}),A_{h}E_{u}^{n} \bigr)\bigr\vert &\leq2C_{4}\Delta t\bigl\Vert \nabla Z_{u}^{n} \bigr\Vert _{0}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}\bigl\Vert A_{h}E_{u}^{n} \bigr\Vert _{0} \\ &\leq\frac{6C_{4}^{2}}{\nu}\Delta t\bigl\Vert \nabla Z_{u}^{n} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}^{2}+\frac{\nu\Delta t}{6}\bigl\Vert A_{h}E_{u}^{n}\bigr\Vert _{0}^{2}\\ &\leq C\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert \nabla u_{ht}\Vert _{0}^{2}\,dt+\frac{\nu \Delta t}{6}\bigl\Vert A_{h}E_{u}^{n}\bigr\Vert _{0}^{2}, \end{aligned}\\& \begin{aligned}[b] \bigl\vert 2\Delta tb\bigl(u_{h}(t_{n-1}),E_{u}^{n-1},A_{h}E_{u}^{n} \bigr)\bigr\vert &\leq2C_{4}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}\bigl\Vert \nabla E_{u}^{n-1} \bigr\Vert _{0}\bigl\Vert A_{h}E_{u}^{n} \bigr\Vert _{0}\\ &\leq\frac{6C_{4}^{2}}{\nu}\Delta t\bigl\Vert \nabla E_{u}^{n-1} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}^{2}+\frac{\nu\Delta t}{6}\bigl\Vert A_{h}E_{u}^{n}\bigr\Vert _{0}^{2}, \end{aligned}\\& \begin{aligned}[b] \bigl\vert 2\Delta tb\bigl(u_{h}(t_{n}),Z_{u}^{n},A_{h}E_{u}^{n} \bigr)\bigr\vert &\leq2C_{4}\Delta t\bigl\Vert \nabla Z_{u}^{n} \bigr\Vert _{0}\bigl\Vert A_{h}u_{h}(t_{n}) \bigr\Vert _{0}\bigl\Vert A_{h}E_{u}^{n} \bigr\Vert _{0} \\ &\leq\frac{6C_{4}^{2}}{\nu}\Delta t\bigl\Vert \nabla Z_{u}^{n} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}(t_{n}) \bigr\Vert _{0}^{2}+\frac{\nu\Delta t}{6}\bigl\Vert A_{h}E_{u}^{n}\bigr\Vert _{0}^{2}\\ &\leq C\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert \nabla u_{ht}\Vert _{0}^{2}\,dt+\frac{\nu \Delta t}{6}\bigl\Vert A_{h}E_{u}^{n}\bigr\Vert _{0}^{2}, \end{aligned}\\& \begin{aligned}[b] \bigl\vert 2\Delta t\overline{b}\bigl(E_{u}^{n-1}, \theta_{h}^{n-1},A_{h}E_{\theta}^{n} \bigr)\bigr\vert &\leq2C_{6}\Delta t\bigl\Vert \nabla E_{u}^{n-1} \bigr\Vert _{0}\bigl\Vert A_{h}\theta_{h}^{n-1} \bigr\Vert _{0}\bigl\Vert A_{h}E_{\theta}^{n} \bigr\Vert _{0} \\ &\leq\frac{6C_{6}^{2}}{\lambda\nu}\Delta t\bigl\Vert \nabla E_{u}^{n-1} \bigr\Vert _{0}^{2}\bigl\Vert A_{h} \theta_{h}^{n-1}\bigr\Vert _{0}^{2}+ \frac{\lambda\nu\Delta t}{6}\bigl\Vert A_{h}E_{\theta}^{n}\bigr\Vert _{0}^{2}, \end{aligned}\\& \bigl\vert 2\Delta t\overline{b}\bigl(Z_{u}^{n}, \theta_{h}(t_{n-1}),A_{h}E_{\theta}^{n} \bigr)\bigr\vert \leq2C_{6}\Delta t\bigl\Vert \nabla Z_{u}^{n} \bigr\Vert _{0}\bigl\Vert A_{h}\theta_{h}(t_{n-1}) \bigr\Vert _{0}\bigl\Vert A_{h}E_{\theta}^{n} \bigr\Vert _{0}\\& \hphantom{\bigl\vert 2\Delta t\overline{b}\bigl(Z_{u}^{n}, \theta_{h}(t_{n-1}),A_{h}E_{\theta}^{n} \bigr)\bigr\vert }\leq\frac{6C_{6}^{2}}{\lambda\nu}\Delta t\bigl\Vert \nabla Z_{u}^{n} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}\theta _{h}(t_{n-1})\bigr\Vert _{0}^{2}+ \frac{\lambda\nu\Delta t}{6}\bigl\Vert A_{h}E_{\theta}^{n}\bigr\Vert _{0}^{2} \\& \hphantom{\bigl\vert 2\Delta t\overline{b}\bigl(Z_{u}^{n}, \theta_{h}(t_{n-1}),A_{h}E_{\theta}^{n} \bigr)\bigr\vert }\leq C\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert \nabla u_{ht}\Vert _{0}^{2}\,dt+\frac {\lambda\nu\Delta t}{6}\bigl\Vert A_{h}E_{\theta}^{n}\bigr\Vert _{0}^{2}, \\& \begin{aligned}[b] \bigl\vert 2\Delta t\overline{b}\bigl(u_{h}(t_{n-1}), E_{\theta}^{n-1},A_{h}E_{\theta}^{n} \bigr)\bigr\vert &\leq 2C_{6}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}\bigl\Vert \nabla E_{\theta}^{n-1} \bigr\Vert _{0}\bigl\Vert A_{h}E_{\theta}^{n} \bigr\Vert _{0}\\ &\leq\frac{6C_{6}^{2}}{\lambda\nu}\Delta t\bigl\Vert \nabla E_{\theta}^{n-1} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}^{2}+\frac{\lambda\nu\Delta t}{6}\bigl\Vert A_{h}E_{\theta}^{n}\bigr\Vert _{0}^{2}, \end{aligned}\\& \begin{aligned}[b] \bigl\vert 2\Delta t\overline{b}\bigl(u_{h}(t_{n}),Z_{\theta}^{n},A_{h}E_{\theta}^{n} \bigr)\bigr\vert &\leq2C_{6}\Delta t\bigl\Vert \nabla Z_{\theta}^{n} \bigr\Vert _{0}\bigl\Vert A_{h}u_{h}(t_{n}) \bigr\Vert _{0}\bigl\Vert A_{h}E_{\theta}^{n}\bigr\Vert _{0}\\ &\leq\frac{6C_{6}^{2}}{\lambda\nu}\Delta t\bigl\Vert \nabla Z_{\theta}^{n} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}(t_{n}) \bigr\Vert _{0}^{2}+\frac{\lambda\nu\Delta t}{6}\bigl\Vert A_{h}E_{\theta}^{n}\bigr\Vert _{0}^{2}\\ &\leq C\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert \nabla\theta_{ht} \Vert _{0}^{2}\,dt+\frac {\lambda\nu\Delta t}{6}\bigl\Vert A_{h}E_{\theta}^{n}\bigr\Vert _{0}^{2}. \end{aligned} \end{aligned}$$
Combining these inequalities with (5.6) and summing n from 1 to \(J+1\), we obtain
$$\begin{aligned} \bigl\Vert \nabla E_{\theta}^{J+1}\bigr\Vert _{0}^{2}+\frac{\lambda\nu}{2}\Delta t\sum _{n=1}^{J+1}\bigl\Vert A_{h}E_{\theta}^{n} \bigr\Vert _{0}^{2} \leq& C\Delta t^{2} \int_{0}^{T}\bigl(\Vert \theta_{htt} \Vert ^{2}_{0}+\Vert \nabla\theta_{ht}\Vert ^{2}_{0}+\Vert \nabla u_{ht}\Vert ^{2}_{0}\bigr)\,dt \\ &{}+C\Delta t\sum_{n=1}^{J+1}\bigl\Vert \nabla E_{u}^{n-1}\bigr\Vert _{0}^{2} +C \Delta t\sum_{n=1}^{J+1}\bigl\Vert \nabla E_{\theta}^{n}\bigr\Vert _{0}^{2}, \\ \bigl\Vert \nabla E_{u}^{J+1}\bigr\Vert _{0}^{2}+\frac{\nu}{2}\Delta t\sum _{n=1}^{J+1}\bigl\Vert AE_{u}^{n+1} \bigr\Vert _{0}^{2} \leq& C\Delta t^{2} \int_{0}^{T}\bigl(\Vert u_{htt}\Vert ^{2}_{0}+\Vert \nabla u_{ht}\Vert ^{2}_{0}\bigr)\,dt \\ &{}+C\Delta t\sum_{n=1}^{J+1}\bigl\Vert \nabla E_{u}^{n-1}\bigr\Vert _{0}^{2} + \frac{4}{\nu}\Delta t\sum_{n=1}^{J+1} \bigl\Vert \nabla E_{\theta}^{n-1}\bigr\Vert _{0}^{2}. \end{aligned}$$
By Lemma 2.1 we complete the proof. □

Now, we present the error estimates for \(E_{p}^{n}\), which show that \(p_{h}^{n}\) is first-order approximations to p in the \(L^{\infty}(L^{2})\) norm. In order to achieve this aim, we provide some estimates for \(d_{t}E_{u}^{n}=\frac {E_{u}^{n}-E_{u}^{n-1}}{\Delta t}\) and \(d_{t}E_{\theta}^{n}=\frac{E_{\theta}^{n}-E_{\theta}^{n-1}}{\Delta t}\).

Lemma 5.3

Under the assumptions of Theorems 3.1 and 4.1, we have
$$\begin{aligned} \bigl\Vert d_{t}E_{u}^{J+1}\bigr\Vert _{0}^{2}+\bigl\Vert d_{t}E_{\theta}^{J+1} \bigr\Vert _{0}^{2}+\nu\Delta t\sum _{n=1}^{J+1}\bigl\Vert \nabla d_{t}E_{u}^{n+1} \bigr\Vert _{0}^{2} + \lambda\nu\Delta t\sum _{n=1}^{J+1}\bigl\Vert \nabla d_{t}E_{\theta}^{n+1} \bigr\Vert _{0}^{2} \leq C\Delta t^{2}. \end{aligned}$$

Proof

From problem (5.2) we obtain that, for all \(v\in V\) and \(\psi\in W\),
$$\begin{aligned} &\bigl(d_{tt}E_{u}^{n},v\bigr)-\nu \bigl(\Delta d_{t}E_{u}^{n},v\bigr) \\ &\quad=\bigl(d_{t}R_{u}^{n},v\bigr)-\bigl(jd_{t}E_{\theta}^{n},v\bigr)-b\bigl(d_{t}Z_{u}^{n},u_{h}(t_{n-1}),v \bigr)-b\bigl(Z_{h}^{n-1},d_{t}u_{h}(t_{n-1}),v \bigr) \\ &\qquad{}-b\bigl(d_{t}E_{u}^{n-1},u_{h}^{n-1},v \bigr) -b\bigl(E_{u}^{n-2},d_{t}u_{h}^{n-1},v \bigr)-b\bigl(d_{t}u_{h}(t_{n-1}),E_{u}^{n-1},v \bigr) \\ &\qquad{}-b\bigl(u_{h}(t_{n-2}),d_{t}E_{u}^{n-1},v \bigr)-b\bigl(d_{t}u(t_{n}),Z_{u}^{n},v \bigr) -b\bigl(u(t_{n-1}),d_{t}Z_{u}^{n},v \bigr) \end{aligned}$$
(5.7)
and
$$ \begin{aligned}[b] &\bigl(d_{tt}E_{\theta}^{n},v\bigr)- \lambda\nu\bigl(\Delta d_{t}E_{\theta}^{n},v \bigr)\\ &\quad=-b\bigl(d_{t}Z_{u}^{n}, \theta_{h}(t_{n-1}),v\bigr)-b\bigl(Z_{h}^{n-1},d_{t} \theta_{h}(t_{n-1}),v\bigr)\\ &\qquad{}+\bigl(d_{t}R_{\theta}^{n},v\bigr)-b \bigl(d_{t}E_{u}^{n-1},\theta _{h}^{n-1},v \bigr)-b\bigl(E_{u}^{n-2},d_{t}\theta_{h}^{n-1},v \bigr)-b\bigl(d_{t}u_{h}(t_{n-1}),E_{\theta}^{n-1},v \bigr)\\ &\qquad{}-b\bigl(u_{h}(t_{n-2}),d_{t}E_{\theta}^{n-1},v \bigr)-b\bigl(d_{t}u(t_{n}),Z_{\theta}^{n},v \bigr)-b\bigl(u(t_{n-1}),d_{t}Z_{\theta}^{n},v \bigr). \end{aligned} $$
(5.8)
Choosing \(v=2\Delta td_{t}E_{u}^{n}\) and \(\psi=2\Delta td_{t}E_{\theta}^{n}\) in (5.7)-(5.8), respectively, we deduce that
$$\begin{aligned} &\bigl\Vert d_{t}E_{u}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert d_{t}E_{u}^{n-1} \bigr\Vert _{0}^{2}+\bigl\Vert d_{t}E_{u}^{n}-d_{t}E_{u}^{n-1} \bigr\Vert _{0}^{2}+\nu\Delta t\bigl\Vert \nabla d_{t}E_{u}^{n}\bigr\Vert _{0}^{2} \\ &\quad= 2\Delta t\bigl\{ \bigl(d_{t}R_{u}^{n},d_{t}E_{u}^{n} \bigr)-\bigl(jd_{t}E_{\theta}^{n},d_{t}E_{u}^{n} \bigr)-b\bigl(d_{t}Z_{u}^{n},u_{h}(t_{n-1}),d_{t}E_{u}^{n} \bigr) \\ &\qquad{}-b\bigl(Z_{h}^{n-1},d_{t}u_{h}(t_{n-1}),d_{t}E_{u}^{n} \bigr)-b\bigl(d_{t}E_{u}^{n-1},u_{h}^{n-1},d_{t}E_{u}^{n} \bigr)-b\bigl(E_{u}^{n-2},d_{t}u_{h}^{n-1},v \bigr) \\ &\qquad{}-b\bigl(d_{t}u_{h}(t_{n-1}),E_{u}^{n-1},d_{t}E_{u}^{n} \bigr)-b\bigl(u_{h}(t_{n-2}),d_{t}E_{u}^{n-1},d_{t}E_{u}^{n} \bigr) \\ &\qquad{}-b\bigl(d_{t}u(t_{n}), Z_{u}^{n},d_{t}E_{u}^{n} \bigr)-b\bigl(u(t_{n-1}),d_{t}Z_{u}^{n},d_{t}E_{u}^{n} \bigr)\bigr\} \end{aligned}$$
(5.9)
and
$$\begin{aligned} &\bigl\Vert d_{t}E_{\theta}^{n}\bigr\Vert _{0}^{2}-\bigl\Vert d_{t}E_{\theta}^{n-1} \bigr\Vert _{0}^{2}+\bigl\Vert d_{t}E_{\theta}^{n}-d_{t}E_{\theta}^{n-1}\bigr\Vert _{0}^{2}+2\lambda\nu\Delta t\bigl\Vert \nabla d_{t}E_{\theta}^{n}\bigr\Vert _{0}^{2} \\ &\quad= 2\Delta t\bigl\{ \bigl(d_{t}R_{\theta}^{n},d_{t}E_{\theta}^{n} \bigr)-b\bigl(d_{t}Z_{u}^{n},\theta _{h}(t_{n-1}),d_{t}E_{\theta}^{n} \bigr)-b\bigl(Z_{h}^{n-1},d_{t}\theta_{h}(t_{n-1}),d_{t}E_{\theta}^{n} \bigr) \\ &\qquad{}-b\bigl(d_{t}E_{u}^{n-1}, \theta_{h}^{n-1},d_{t}E_{\theta}^{n} \bigr)-b\bigl(E_{u}^{n-2},d_{t}\theta _{h}^{n-1},d_{t}E_{\theta}^{n} \bigr)-b\bigl(d_{t}u_{h}(t_{n-1}),E_{\theta}^{n-1},d_{t}E_{\theta}^{n} \bigr) \\ &\qquad{}-b\bigl(u_{h}(t_{n-2}),d_{t}E_{\theta}^{n-1},d_{t}E_{\theta}^{n} \bigr)-b\bigl(d_{t}u(t_{n}),Z_{\theta}^{n},d_{t}E_{\theta}^{n}\bigr) -b \bigl(u(t_{n-1}),d_{t}Z_{\theta}^{n},d_{t}E_{\theta}^{n} \bigr)\bigr\} . \end{aligned}$$
(5.10)
Now, we estimate the right-hand side terms of (5.9)-(5.10) separately. For \((d_{t}R_{u}^{n},d_{t}E_{u}^{n})\) and \((d_{t}R_{\theta}^{n},d_{t}E_{\theta}^{n})\), using the techniques adopted by He [16], we arrive at
$$\begin{aligned} \bigl(d_{t}R_{u}^{n},d_{t}E_{u}^{n} \bigr)=-\frac{1}{\Delta t^{2}} \int _{t_{n-1}}^{t_{n}}(t-t_{n-1}) \int_{t-\Delta t}^{t}\bigl(u_{httt}(s),d_{t}E_{u}^{n} \bigr)\,ds\,dt \end{aligned}$$
and
$$\begin{aligned} \bigl(d_{t}R_{\theta}^{n},d_{t}E_{\theta}^{n} \bigr)=-\frac{1}{\Delta t^{2}} \int _{t_{n-1}}^{t_{n}}(t-t_{n-1}) \int_{t-\Delta t}^{t}\bigl(\theta _{httt}(s),d_{t}E_{\theta}^{n} \bigr)\,ds\,dt \end{aligned}$$
for all \(2\leq n\leq J\). We deduce from these equalities that
$$\begin{aligned} &\bigl\vert 2\Delta t\bigl(d_{t}R_{u}^{n},d_{t}E_{u}^{n} \bigr)\bigr\vert \\ &\quad\leq2\Delta t\bigl\Vert d_{t}R_{u}^{n} \bigr\Vert _{0}\bigl\Vert d_{t}E_{u}^{n} \bigr\Vert _{0} \leq C(\nu)\Delta t\bigl\Vert d_{t}R_{u}^{n} \bigr\Vert _{0}^{2}+\frac{\nu}{4}\Delta t\bigl\Vert \nabla d_{t}E_{u}^{n}\bigr\Vert ^{2}_{0} \\ &\quad\leq c(\nu)\Delta t {\biggl[}\Delta t^{-3/2}\biggl( \int _{t_{n-1}}^{t_{n}}(t-t_{n-1})^{2} \biggl\Vert \int_{t-\Delta t}^{t}u_{httt}(s)\,ds\biggr\Vert _{0}^{2}\,dt\biggr)^{1/2} {\biggr]}^{2}+ \frac{\nu}{4}\Delta t\bigl\Vert \nabla d_{t}E_{u}^{n} \bigr\Vert ^{2}_{0} \\ &\quad\leq C\Delta t^{2} \int_{t_{n-2}}^{t_{n}}\Vert u_{httt}\Vert _{0}^{2}\,dt+\frac{\nu }{4}\Delta t\bigl\Vert \nabla d_{t}E_{u}^{n}\bigr\Vert ^{2}_{0}. \end{aligned}$$
In the same way, we have
$$\begin{aligned}& \bigl\vert 2\Delta t\bigl(d_{t}R_{\theta}^{n},d_{t}E_{\theta}^{n+1} \bigr)\bigr\vert \leq C(\lambda\nu)\Delta t^{2} \int_{t_{n-2}}^{t_{n}}\Vert \theta_{httt}\Vert _{0}^{2}\,dt+\frac{\lambda\nu}{4}\Delta t\bigl\Vert \nabla d_{t}E_{\theta}^{n}\bigr\Vert ^{2}_{0}, \\& \bigl\vert \bigl(jd_{t}E_{\theta}^{n},d_{t}E_{u}^{n} \bigr)\bigr\vert \leq\bigl\Vert d_{t}E_{\theta}^{n} \bigr\Vert _{0}\bigl\Vert d_{t}E_{u}^{n} \bigr\Vert _{0}\leq C(\nu)\bigl\Vert d_{t}E_{\theta}^{n} \bigr\Vert ^{2}_{0}+\frac{\nu}{4}\bigl\Vert \nabla d_{t}E_{u}^{n}\bigr\Vert ^{2}_{0}. \end{aligned}$$
For the nonlinear terms, with the help of Lemma 2.3, we find that
$$\begin{aligned}& \begin{aligned}[b] &2\Delta t\bigl\vert b\bigl(d_{t}Z_{u}^{n},u_{h}(t_{n-1}),d_{t}E_{u}^{n} \bigr)\bigr\vert \\ &\quad\leq2C_{4}\Delta t\bigl\Vert d_{t}Z_{u}^{n} \bigr\Vert _{0}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}\bigl\Vert \nabla d_{t}E_{u}^{n} \bigr\Vert _{0} \\ &\quad\leq2C_{4}\Delta t^{2}\bigl\Vert u_{htt}(t_{n-1})+ \mathcal{O}\bigl(\Delta t^{2}\bigr)\bigr\Vert _{0}\bigl\Vert A_{h}u_{h}(t_{n-1})\bigr\Vert _{0}\bigl\Vert \nabla d_{t}E_{u}^{n} \bigr\Vert _{0} \\ &\quad\leq\frac{20C_{4}^{2}}{\nu}\Delta t^{3}\bigl\Vert u_{htt}(t_{n-1})+ \mathcal {O}\bigl(\Delta t^{2}\bigr)\bigr\Vert ^{2}_{0} \bigl\Vert A_{h}u_{h}(t_{n-1})\bigr\Vert ^{2}_{0} +\frac{\nu}{20}\bigl\Vert \nabla d_{t}E_{u}^{n}\bigr\Vert ^{2}_{0} \Delta t, \end{aligned} \\& \begin{aligned}[b] &2\Delta t\bigl\vert b\bigl(Z_{h}^{n-1},d_{t}u_{h}(t_{n-1}),d_{t}E_{u}^{n} \bigr)\bigr\vert \\ &\quad\leq2C_{4}\Delta t\bigl\Vert Z_{h}^{n-1} \bigr\Vert _{0}\bigl\Vert A_{h}d_{t}u_{h}(t_{n-1}) \bigr\Vert _{0}\bigl\Vert \nabla d_{t}E_{u}^{n} \bigr\Vert _{0} \\ &\quad\leq2C_{4}\Delta t\bigl\Vert Z_{h}^{n-1}\bigr\Vert _{0}\bigl\Vert A_{h}u_{ht}(t_{n-2})+ \mathcal {O}(\Delta t)\bigr\Vert _{0}\bigl\Vert \nabla d_{t}E_{u}^{n}\bigr\Vert _{0} \\ &\quad\leq\frac{20C_{4}^{2}}{\nu}\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert u_{ht}\Vert ^{2}_{0}\,dt\cdot\bigl\Vert A_{h}u_{ht}(t_{n-2})+ \mathcal{O}(\Delta t)\bigr\Vert ^{2}_{0} +\frac{\nu}{20} \bigl\Vert \nabla d_{t}E_{u}^{n}\bigr\Vert ^{2}_{0}\Delta t, \end{aligned} \\& \begin{aligned}[b] 2\Delta t\bigl\vert b\bigl(d_{t}E_{u}^{n-1},u_{h}^{n-1},d_{t}E_{u}^{n} \bigr)\bigr\vert &\leq2C_{4}\Delta t\bigl\Vert d_{t}E_{u}^{n-1} \bigr\Vert _{0}\bigl\Vert A_{h} u_{h}^{n-1} \bigr\Vert _{0}\bigl\Vert \nabla d_{t}E_{u}^{n} \bigr\Vert _{0} \\ &\leq\frac{20C_{4}^{2}}{\nu}\Delta t\bigl\Vert d_{t}E_{u}^{n-1} \bigr\Vert ^{2}_{0}\bigl\Vert A_{h} u_{h}^{n-1}\bigr\Vert ^{2}_{0} + \frac{\nu}{20}\bigl\Vert \nabla d_{t}E_{u}^{n} \bigr\Vert ^{2}_{0}\Delta t, \end{aligned} \\& \begin{aligned}[b] 2\Delta t\bigl\vert b\bigl(E_{u}^{n-2},d_{t}u_{h}^{n-1},d_{t}E_{u}^{n} \bigr)\bigr\vert &\leq2C_{4}\Delta t\bigl\Vert E_{u}^{n-2} \bigr\Vert _{0}\bigl\Vert A_{h}d_{t}u_{h}^{n-1} \bigr\Vert _{0}\bigl\Vert \nabla d_{t}E_{u}^{n} \bigr\Vert _{0} \\ &\leq2C_{4}\Delta t\bigl\Vert E_{u}^{n-2}\bigr\Vert _{0}\bigl\Vert A_{h}u_{ht}^{n-2}+ \mathcal{O}(\Delta t)\bigr\Vert _{0}\bigl\Vert \nabla d_{t}E_{u}^{n}\bigr\Vert _{0} \\ &\leq\frac{20C_{4}^{2}}{\nu}\Delta t\bigl\Vert E_{u}^{n-2}\bigr\Vert ^{2}_{0}\bigl\Vert A_{h}u_{ht}^{n-2}+ \mathcal{O}(\Delta t)\bigr\Vert ^{2}_{0} +\frac{\nu}{20} \bigl\Vert \nabla d_{t}E_{u}^{n}\bigr\Vert ^{2}_{0}\Delta t, \end{aligned} \\& \begin{aligned}[b] &2\Delta t\bigl\vert b\bigl(d_{t}u_{h}(t_{n-1}),E_{u}^{n-1},d_{t}E_{u}^{n} \bigr)\bigr\vert \\ &\quad\leq\frac{20C_{4}^{2}}{\nu }\Delta t\bigl\Vert E_{u}^{n-1} \bigr\Vert ^{2}_{0}\bigl\Vert A_{h}u_{ht}(t_{n-2})+ \mathcal{O}(\Delta t)\bigr\Vert ^{2}_{0} +\frac{\nu}{20} \bigl\Vert \nabla d_{t}E_{u}^{n}\bigr\Vert ^{2}_{0}\Delta t, \end{aligned} \\& 2\Delta t\bigl\vert b\bigl(u_{h}(t_{n-2}),d_{t}E_{u}^{n-1},d_{t}E_{u}^{n} \bigr)\bigr\vert \leq\frac{20C_{4}^{2}}{\nu }\Delta t\bigl\Vert d_{t}E_{u}^{n-1} \bigr\Vert ^{2}_{0}\bigl\Vert A_{h}u_{h}(t_{n-2}) \bigr\Vert ^{2}_{0} +\frac{\nu}{20}\bigl\Vert \nabla d_{t}E_{u}^{n}\bigr\Vert ^{2}_{0} \Delta t, \\& \begin{aligned}[b] &2\Delta t\bigl\vert b\bigl(d_{t}u_{h}(t_{n}),Z_{u}^{n},d_{t}E_{u}^{n} \bigr)\bigr\vert \\ &\quad\leq2C_{4}\bigl\Vert A_{h}d_{t}u_{h}(t_{n}) \bigr\Vert _{0}\bigl\Vert Z_{u}^{n}\bigr\Vert _{0}\bigl\Vert \nabla d_{t}E_{u}^{n} \bigr\Vert _{0} \\ &\quad\leq 2C_{4}\bigl\Vert A_{h}u_{ht}(t_{n-1})+ \mathcal{O}(\Delta t)\bigr\Vert _{0}\bigl\Vert Z_{u}^{n} \bigr\Vert _{0}\bigl\Vert \nabla d_{t}E_{u}^{n} \bigr\Vert _{0} \\ &\quad\leq \frac{20C_{4}^{2}}{\nu}\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert u_{ht}\Vert _{0}^{2}\,dt\bigl\Vert A_{h}u_{ht}(t_{n-1})+ \mathcal{O}(\Delta t)\bigr\Vert ^{2}_{0} +\frac{\nu}{20} \bigl\Vert \nabla d_{t}E_{u}^{n}\bigr\Vert ^{2}_{0}\Delta t, \end{aligned} \\& \begin{aligned}[b] &2\Delta t\bigl\vert b\bigl(u(t_{n-1}),d_{t}Z_{u}^{n},d_{t}E_{u}^{n} \bigr)\bigr\vert \\ &\quad\leq \frac{20C_{4}^{2}}{\nu}\Delta t^{3}\bigl\Vert u_{htt}(t_{n-1})+\mathcal{O}(\Delta t)^{2}\bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert ^{2}_{0} +\frac{\nu}{20}\bigl\Vert \nabla d_{t}E_{u}^{n}\bigr\Vert ^{2}_{0} \Delta t. \end{aligned} \end{aligned}$$
In the same way, we have
$$\begin{aligned}& 2\Delta t\bigl\vert \overline{b}\bigl(d_{t}Z_{u}^{n}, \theta_{h}(t_{n-1}),d_{t}E_{\theta}^{n} \bigr)\bigr\vert \\& \quad\leq \frac{20C_{6}^{2}}{\lambda\nu}\Delta t^{3}\bigl\Vert u_{htt}(t_{n-1})+\mathcal {O}\bigl(\Delta t^{2}\bigr) \bigr\Vert ^{2}_{0}\bigl\Vert A_{h} \theta_{h}(t_{n-1})\bigr\Vert ^{2}_{0} + \frac{\lambda\nu}{20}\bigl\Vert \nabla d_{t}E_{\theta}^{n} \bigr\Vert ^{2}_{0}\Delta t, \\& 2\Delta t\bigl\vert \overline{b}\bigl(Z_{h}^{n-1},d_{t} \theta_{h}(t_{n-1}),d_{t}E_{\theta}^{n} \bigr)\bigr\vert \\& \quad\leq \frac{20C_{6}^{2}}{\lambda\nu}\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert u_{ht}\Vert ^{2}_{0}\,dt\cdot\bigl\Vert A_{h} \theta_{ht}(t_{n-2})+\mathcal{O}(\Delta t)\bigr\Vert ^{2}_{0} +\frac{\lambda\nu}{20}\bigl\Vert \nabla d_{t}E_{\theta}^{n}\bigr\Vert ^{2}_{0} \Delta t, \\& 2\Delta t\bigl\vert \overline{b}\bigl(d_{t}E_{u}^{n-1}, \theta_{h}^{n-1},d_{t}E_{\theta}^{n} \bigr)\bigr\vert \leq \frac{20C_{6}^{2}}{\lambda\nu}\Delta t\bigl\Vert d_{t}E_{u}^{n-1}\bigr\Vert ^{2}_{0} \bigl\Vert A_{h}\theta _{h}^{n-1}\bigr\Vert ^{2}_{0} +\frac{\lambda\nu}{20}\bigl\Vert \nabla d_{t}E_{\theta}^{n}\bigr\Vert ^{2}_{0} \Delta t, \\& 2\Delta t\bigl\vert \overline{b}\bigl(E_{u}^{n-2},d_{t} \theta_{h}^{n-1},d_{t}E_{\theta}^{n} \bigr)\bigr\vert \leq \frac{20C_{6}^{2}}{\lambda\nu}\Delta t\bigl\Vert E_{u}^{n-2}\bigr\Vert ^{2}_{0}\bigl\Vert A_{h}\theta _{ht}^{n-2}+\mathcal{O}(\Delta t)\bigr\Vert ^{2}_{0} +\frac{\lambda\nu}{20}\bigl\Vert \nabla d_{t}E_{\theta}^{n}\bigr\Vert ^{2}_{0}\Delta t, \\& 2\Delta t\bigl\vert \overline{b}\bigl(d_{t}u_{h}(t_{n-1}),E_{\theta}^{n-1},d_{t}E_{\theta}^{n} \bigr)\bigr\vert \\& \quad\leq \frac{20C_{6}^{2}}{\lambda\nu}\Delta t\bigl\Vert E_{\theta}^{n-1}\bigr\Vert ^{2}_{0}\bigl\Vert A_{h}u_{ht}(t_{n-2})+\mathcal{O}(\Delta t) \bigr\Vert ^{2}_{0} +\frac{\lambda\nu}{20}\bigl\Vert \nabla d_{t}E_{\theta}^{n}\bigr\Vert ^{2}_{0} \Delta t, \\& 2\Delta t\bigl\vert \overline{b}\bigl(u_{h}(t_{n-2}),d_{t}E_{\theta}^{n-1},d_{t}E_{\theta}^{n} \bigr)\bigr\vert \leq \frac{20C_{6}^{2}}{\lambda\nu}\Delta t\bigl\Vert d_{t}E_{\theta}^{n-1}\bigr\Vert ^{2}_{0} \bigl\Vert A_{h}u_{h}(t_{n-2})\bigr\Vert ^{2}_{0} +\frac{\lambda\nu}{20}\bigl\Vert \nabla d_{t}E_{\theta}^{n}\bigr\Vert ^{2}_{0} \Delta t, \\& 2\Delta t\bigl\vert \overline{b}\bigl(d_{t}u_{h}(t_{n}),Z_{\theta}^{n},d_{t}E_{\theta}^{n} \bigr)\bigr\vert \\& \quad\leq \frac{20C_{6}^{2}}{\lambda\nu}\Delta t^{2} \int_{t_{n-1}}^{t_{n}}\Vert \theta_{ht}\Vert _{0}^{2}\,dt\bigl\Vert A_{h}u_{ht}(t_{n-1})+ \mathcal{O}(\Delta t)\bigr\Vert ^{2}_{0} +\frac{\lambda\nu}{20} \bigl\Vert \nabla d_{t}E_{\theta}^{n}\bigr\Vert ^{2}_{0}\Delta t, \\& 2\Delta t\bigl\vert \overline{b}\bigl(u_{h}(t_{n-1}),d_{t}Z_{\theta}^{n},d_{t}E_{\theta}^{n} \bigr)\bigr\vert \\& \quad\leq \frac{20C_{6}^{2}}{\lambda\nu}\Delta t^{3}\bigl\Vert \theta_{htt}(t_{n-1})+\mathcal {O}(\Delta t)^{2}\bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert ^{2}_{0} +\frac{\lambda\nu}{20}\bigl\Vert \nabla d_{t}E_{\theta}^{n}\bigr\Vert ^{2}_{0} \Delta t. \end{aligned}$$
Combining these inequalities with (5.9)-(5.10) and summing from \(n=1\) to \(J+1\), we obtain
$$\begin{aligned} &\bigl\Vert d_{t}E_{u}^{J+1}\bigr\Vert _{0}^{2}+\sum_{n=1}^{J+1} \bigl\Vert d_{t}E_{u}^{n}-d_{t}E_{u}^{n-1} \bigr\Vert _{0}^{2}+\nu \Delta t\sum _{n=1}^{J+1}\bigl\Vert \nabla d_{t}E_{u}^{n} \bigr\Vert _{0}^{2} \\ &\quad\leq\frac{40TC_{4}^{2}}{\nu}\Delta t^{2}\bigl\Vert u_{htt}(t_{n-1})\bigr\Vert _{0}^{2}\bigl\Vert A_{h}u_{h}(t_{n-1})\bigr\Vert _{0}^{2} +\frac{20C_{4}^{2}}{\nu}\Delta t\sum _{n=1}^{J+1}\bigl\Vert E_{u}^{n-1} \bigr\Vert _{0}^{2}\bigl\Vert A_{h}u^{n-2}_{ht} \bigr\Vert _{0}^{2} \\ &\qquad{}+C\Delta t^{2} \int_{0}^{T}\Vert u_{httt}\Vert _{0}^{2}\,dt +\frac{20C_{4}^{2}}{\nu}\Delta t\sum _{n=1}^{J+1}\bigl\Vert d_{t}E_{u}^{n-1} \bigr\Vert _{0}^{2}\bigl(\bigl\Vert A_{h}u_{h}^{n-1} \bigr\Vert _{0}^{2}+\Vert A_{h}u_{ht} \Vert _{0}^{2}\bigr) \\ &\qquad{}+\frac{20}{\nu}\Delta t^{2}\Vert A_{h}u_{ht} \Vert _{0}^{2} \int_{0}^{T}\Vert u_{ht}\Vert _{0}^{2}\,dt+C\Delta t\sum_{n=1}^{J+1} \bigl\Vert d_{t}E_{\theta}^{n}\bigr\Vert _{0}^{2} \end{aligned}$$
(5.11)
and
$$\begin{aligned} &\bigl\Vert d_{t}E_{\theta}^{J+1}\bigr\Vert _{0}^{2}+\sum_{n=1}^{J+1} \bigl\Vert d_{t}E_{\theta}^{n}-d_{t}E_{\theta}^{n-1}\bigr\Vert _{0}^{2}+\nu\Delta t\sum _{n=1}^{J+1}\bigl\Vert \nabla d_{t}E_{\theta}^{n} \bigr\Vert _{0}^{2} \\ &\quad\leq\frac{20C_{6}^{2}T}{\lambda\nu}\Delta t^{2}\bigl(\Vert \theta_{htt}\Vert _{0}^{2}\Vert A_{h}u_{h}\Vert _{0}^{2}+\Vert u_{htt}\Vert _{0}^{2}\Vert A_{h} \theta_{h}\Vert _{0}\bigr) \\ &\qquad{}+\frac{20C_{6}^{2}}{\lambda\nu}\Delta t^{2}\Vert A_{h}\theta_{ht}\Vert _{0}^{2} \int_{0}^{T}\Vert u_{ht}\Vert _{0}^{2}\,dt \\ &\qquad{}+\frac{20C_{6}^{2}}{\lambda\nu}\Delta t\sum_{n=1}^{J+1} \bigl(\bigl\Vert d_{t}E_{u}^{n-1}\bigr\Vert _{0}^{2}\bigl\Vert A_{h}\theta_{h}^{n-1} \bigr\Vert _{0}^{2} +\bigl\Vert d_{t}E_{\theta}^{n-1} \bigr\Vert _{0}^{2}\Vert A_{h}u_{h} \Vert _{0}^{2} +\bigl\Vert E_{\theta}^{n-1} \bigr\Vert _{0}^{2}\Vert A_{h}u_{ht} \Vert _{0}^{2} \\ &\qquad{}+\bigl\Vert E_{u}^{n-2}\bigr\Vert _{0}^{2}\bigl\Vert A_{h}\theta_{ht}^{n-2} \bigr\Vert _{0}^{2} \bigr) +\frac{20C_{6}^{2}}{\lambda\nu}\Delta t^{2}\Vert A_{h}u_{ht}\Vert _{0}^{2} \int_{0}^{T}\Vert \theta _{ht}\Vert _{0}^{2}\,dt \\ &\qquad{}+C\Delta t^{2} \int_{0}^{T}\Vert u_{httt}\Vert _{0}^{2}\,dt. \end{aligned}$$
(5.12)
Substituting (5.12) into (5.11) and using Lemma 2.1, we obtain the desired results. □

Remark 5.1

In the estimates of trilinear terms, the bounds of \(\int_{0}^{T}\|u_{httt}\|_{0}^{2}\,dt\) and \(\int_{0}^{T}\|\theta_{httt}\| _{0}^{2}\,dt\) are used. We can prove them by differentiating (3.2) twice with respect to time and following the proofs provided in [7]. As for the bounds of \(\|A_{h}u_{ht}^{n-2}\|_{0}\) and \(\|A_{h}\theta _{ht}^{n-2}\|_{0}\), we can obtain them as we have done in Section 4. Here, we omit these proofs for simplification.

Now, we are in the position of establishing the optimal error estimate for pressure in \(L^{\infty}(L^{2})\) norm based on the results presented in Theorems 3.1 and 4.1 and Lemmas 5.1-5.3.

Theorem 5.4

Under the assumptions of Theorems 3.1 and 4.1, we have
$$\begin{aligned} \bigl\Vert p_{h}(t_{n})-p_{h}^{n} \bigr\Vert _{0}\leq C\Delta t. \end{aligned}$$

Proof

We rewrite the first equation of (5.2) as follows:
$$\begin{aligned} -\nabla E_{p}^{n}=d_{t}E_{u}^{n}- \nu\Delta E_{u}^{n}-R_{u}^{n}+jE_{\theta}^{n} +\bigl(u_{h}(t_{n})\cdot\nabla\bigr)u_{h}(t_{n})- \bigl(u_{h}^{n-1}\cdot\nabla\bigr)u_{h}^{n-1}. \end{aligned}$$
(5.13)
Taking the inner product of (5.13) with arbitrary \(v\in X\) and using the Poincaré inequality, we obtain
$$\begin{aligned}& \bigl\vert \bigl(d_{t}E_{u}^{n},v\bigr)\bigr\vert \leq\bigl\Vert d_{t}E_{u}^{n}\bigr\Vert _{0}\Vert v\Vert _{0}\leq C_{1}\bigl\Vert d_{t}E_{u}^{n}\bigr\Vert _{0}\Vert \nabla v\Vert _{0}, \\& \bigl\vert \nu\bigl(\Delta E_{u}^{n},v\bigr)\bigr\vert \leq\nu\bigl\Vert \nabla E_{u}^{n}\bigr\Vert _{0}\Vert \nabla v\Vert _{0}, \\& \bigl\vert \bigl(R_{u}^{n},v\bigr)\bigr\vert \leq\bigl\Vert R_{u}^{n}\bigr\Vert _{0}\Vert v\Vert _{0}\leq C\Delta t\biggl( \int_{t_{n}}^{t_{n+1}}\Vert u_{htt}\Vert _{0}^{2}\biggr)^{1/2}\Vert \nabla v\Vert _{0}, \\& \bigl\vert \bigl(jE_{\theta}^{n},v\bigr)\bigr\vert \leq\bigl\Vert E_{\theta}^{n}\bigr\Vert _{0}\Vert v\Vert _{0}\leq C_{1}\bigl\Vert E_{\theta}^{n} \bigr\Vert _{0}\Vert \nabla v\Vert _{0}. \end{aligned}$$
For the nonlinear terms, applying Lemma 2.3 and Theorems 3.1 and 4.1, we arrive at
$$\begin{aligned}& \bigl\vert b\bigl(Z_{u}^{n},u_{h}(t_{n-1}),v \bigr)\bigr\vert \leq C_{4}\bigl\Vert Z_{u}^{n} \bigr\Vert _{0}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}\Vert \nabla v\Vert _{0} \leq C\biggl( \Delta t \int_{t_{n-1}}^{t_{n}}\Vert u_{ht}\Vert _{0}^{2}\,dt\biggr)^{1/2}\Vert \nabla v\Vert _{0}, \\& \bigl\vert b\bigl(u_{h}(t_{n-1}),E_{u}^{n-1},v \bigr)\bigr\vert \leq C_{4}\bigl\Vert E_{u}^{n-1} \bigr\Vert _{0}\bigl\Vert A_{h}u_{h}(t_{n-1}) \bigr\Vert _{0}\Vert \nabla v\Vert _{0} \leq C\bigl\Vert E_{u}^{n-1}\bigr\Vert _{0}\Vert \nabla v \Vert _{0}, \\& \bigl\vert b\bigl(u_{h}(t_{n}),Z_{u}^{n},v \bigr)\bigr\vert \leq C\bigl\Vert Z_{u}^{n}\bigr\Vert _{0}\bigl\Vert A_{h}u_{h}(t_{n})\bigr\Vert _{0}\Vert \nabla v\Vert _{0} \leq C\biggl(\Delta t \int_{t_{n-1}}^{t_{n}}\Vert u_{ht}\Vert _{0}^{2}\,dt\biggr)^{1/2}\Vert \nabla v\Vert _{0}, \\& \bigl\vert b\bigl(E_{u}^{n-1},u_{h}^{n-1},v \bigr)\bigr\vert \leq C_{4}\bigl\Vert E_{u}^{n-1} \bigr\Vert _{0}\bigl\Vert A_{h}u_{h}^{n-1} \bigr\Vert _{0}\Vert \nabla v\Vert _{0} \leq C\bigl\Vert E_{u}^{n-1}\bigr\Vert _{0}\Vert \nabla v \Vert _{0}. \end{aligned}$$
By the discrete inf-sup condition (3.1) we obtain
$$\begin{aligned} \bigl\Vert E_{p}^{n+1}\bigr\Vert _{0} \leq& C \bigl(\beta^{-1}\bigr)\biggl\{ \bigl\Vert d_{t}E_{u}^{n+1} \bigr\Vert _{0} +\nu\bigl\Vert \nabla E_{u}^{n} \bigr\Vert _{0}+\bigl\Vert E_{\theta}^{n}\bigr\Vert _{0} +\biggl(\Delta t \int_{t_{n-1}}^{t_{n}}\Vert u_{ht}\Vert _{0}^{2}\,dt\biggr)^{1/2} \\ &{}+\Delta t\biggl( \int_{t_{n}}^{t_{n+1}}\Vert u_{htt}\Vert _{0}^{2}\,dt\biggr)^{1/2}\biggr\} . \end{aligned}$$
With the results of Theorem 3.1 and Lemmas 5.1, 5.2, and 5.3, we complete the proof. □

6 Numerical experiments

In order to gain insights on the established convergence results in Section 5, in this section, we present some numerical tests. Our main interest is to verify and compare the performances of the Euler implicit/explicit scheme (3.3) for the Boussinesq equations. In all experiments, the Boussinesq equations are defined on the convex domain \(\Omega=[0,1]\times[0,1]\). The mesh consists of triangular elements that are obtained by dividing Ω into subsquares of equal size and then drawing the diagonal in each subsquare. The model parameters ν and λ are simply set to 1. We use the MINI element that satisfies the discrete inf-sup condition to approximate the velocity u and pressure p and the linear polynomial to approximate the temperature θ. The boundary and initial conditions and right-hand side functions f and g are selected such that the exact solutions are given by
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} u_{1}=10x^{2}(x-1)^{2}y(y-1)(2y-1)\cos(t),\\ u_{2}=-10x(x-1)(2x-1)y^{2}(y-1)^{2}\cos(t),\\ p=10(2x-1)(2y-1)\cos(t),\\ \theta=10x^{2}(x-1)^{2}y(y-1)(2y-1)\cos(t)-10x(x-1)(2x-1)y^{2}(y-1)^{2}\cos(t), \end{array}\displaystyle \right . \end{aligned}$$
where the components of u are denoted by \((u_{1},u_{2})\) for convenience.
First, we compare the errors and CPU time of the standard Galerkin finite element method using the backward Euler scheme and Newton iteration to treat the temporal term and nonlinear term and the Euler implicit/explicit scheme with varying time step Δt or mesh length h. From Tables 1-4 we can see that two kinds of numerical methods almost get the same accuracy, but the Euler implicit/explicit scheme takes less CPU time than the standard Galerkin FEM. In other words, the Euler implicit/explicit scheme is comparable with the standard Galerkin FEM but cheaper and more efficient.
Table 1

Numerical results of the standard Galerkin FEM at time \(\pmb{T=1.0}\) with varying time step Δ t but fixed mesh size \(\pmb{h=\frac{1}{32}}\)

Δ t

\(\boldsymbol {\frac{\|u-u^{n}\|_{0}}{\|u\|_{0}}}\)

\(\boldsymbol {\frac{\|\nabla(u-u^{n})\|_{0}}{\|\nabla u\|_{0}}}\)

\(\boldsymbol {\frac{\|p-p^{n}\|_{0}}{\|p\|_{0}}}\)

\(\boldsymbol {\frac{\|\theta-\theta^{n}\|_{0}}{\|\theta\|_{0}}}\)

\(\boldsymbol {\frac{\|\nabla(\theta-\theta^{n})\|_{0}}{\|\nabla\theta\|_{0}}}\)

CPU(S)

0.1

0.000171956

0.0127757

0.00415879

0.000115897

0.0104146

189.625

0.05

0.000162403

0.0127748

0.00416427

0.000107522

0.0104136

306.596

0.025

0.00015784

0.0127745

0.00416706

0.000103686

0.0104134

476.16

0.0125

0.000155623

0.0127744

0.0041685

0.000101859

0.0104133

965.143

0.00625

0.000154529

0.0127744

0.00416949

0.000100969

0.0104133

1,807.32

Table 2

Numerical results of the Euler implicit/explicit scheme at time \(\pmb{T=1.0}\) with varying time step Δ t but fixed mesh size \(\pmb{h=\frac{1}{32}}\)

Δ t

\(\boldsymbol {\frac{\|u-u^{n}\|_{0}}{\|u\|_{0}}}\)

\(\boldsymbol {\frac{\|\nabla(u-u^{n})\|_{0}}{\|\nabla u\|_{0}}}\)

\(\boldsymbol {\frac{\|p-p^{n}\|_{0}}{\|p\|_{0}}}\)

\(\boldsymbol {\frac{\|\theta-\theta^{n}\|_{0}}{\|\theta\|_{0}}}\)

\(\boldsymbol {\frac{\|\nabla(\theta-\theta^{n})\|_{0}}{\|\nabla\theta\|_{0}}}\)

CPU(S)

0.1

0.000172356

0.0127759

0.00425416

0.000134799

0.0104262

190.068

0.05

0.000162794

0.0127749

0.00425975

0.000127682

0.0104252

291.062

0.025

0.000158237

0.0127747

0.00426272

0.000124474

0.0104249

457.629

0.0125

0.000156015

0.0127746

0.00426424

0.000122958

0.0104249

875.082

0.00625

0.000154919

0.0127745

0.00426517

0.000122223

0.0104248

1,665.94

Table 3

Numerical results of the standard Galerkin FEM at time \(\pmb{T=1.0}\) with varying mesh size h but fixed time step \(\pmb{\Delta t=0.01}\)

\(\boldsymbol {\frac{1}{h}}\)

\(\boldsymbol {\frac{\|u-u^{n}\|_{0}}{\|u\|_{0}}}\)

\(\boldsymbol {\frac{\|\nabla(u-u^{n})\|_{0}}{\|\nabla u\|_{0}}}\)

\(\boldsymbol {\frac{\|p-p^{n}\|_{0}}{\|p\|_{0}}}\)

\(\boldsymbol {\frac{\|\theta-\theta^{n}\|_{0}}{\|\theta\|_{0}}}\)

\(\boldsymbol {\frac{\|\nabla(\theta-\theta^{n})\|_{0}}{\|\nabla\theta\|_{0}}}\)

CPU(S)

4

0.0225932

0.24839

0.698555

0.0209496

0.15429

14.898

9

0.0085461

0.116535

0.152122

0.00558951

0.078966

23.744

16

0.00247039

0.054131

0.0461534

0.00147279

0.0410255

46.129

25

0.000621059

0.0259783

0.0136212

0.000377335

0.0207576

208.163

36

0.000155184

0.0127744

0.00416879

0.000101501

0.0104133

1,063.48

Table 4

Numerical results of the Euler implicit/explicit scheme at time \(\pmb{T=1.0}\) with varying mesh size h but fixed time step \(\pmb{\Delta t=0.01}\)

\(\boldsymbol {\frac{1}{h}}\)

\(\boldsymbol {\frac{\|u-u^{n}\|_{0}}{\|u\|_{0}}}\)

\(\boldsymbol {\frac{\|\nabla(u-u^{n})\|_{0}}{\|\nabla u\|_{0}}}\)

\(\boldsymbol {\frac{\|p-p^{n}\|_{0}}{\|p\|_{0}}}\)

\(\boldsymbol {\frac{\|\theta-\theta^{n}\|_{0}}{\|\theta\|_{0}}}\)

\(\boldsymbol {\frac{\|\nabla(\theta-\theta^{n})\|_{0}}{\|\nabla\theta\|_{0}}}\)

CPU(S)

4

0.0225932

0.24839

0.698555

0.0209496

0.15429

8.439

9

0.00854615

0.116536

0.152154

0.00558949

0.0789666

10.203

16

0.00247067

0.0541318

0.0461762

0.00147402

0.0410279

27.69

25

0.0006214

0.0259784

0.0136427

0.000383256

0.0207631

129.902

36

0.000155559

0.0127745

0.00420583

0.000122663

0.0104249

703.171

Next, we focus on examining the orders of convergence for both standard Galerkin FEM and the Euler implicit/explicit scheme with respect to the time step Δt or the mesh size h. Following [28], we introduce the following way to examine the orders of convergence with respect to the time step Δt or the mesh size h due to the approximation errors \(\mathcal{O}(\Delta t^{\gamma})+\mathcal{O}(\Delta t^{\mu})\). For example, assuming that
$$\begin{aligned} v_{h}^{\Delta t}\thickapprox v(x,t_{n})+C_{1}(x,t_{n})h^{\mu}+C_{2}(x,t_{n}) \Delta t^{\gamma}, \end{aligned}$$
we have
$$\begin{aligned}& \rho_{v,h,j}=\frac{\|v_{h}^{\Delta t}(x,t_{n})-v^{\Delta t}_{\frac {h}{2}}(x,t_{n})\|_{j}}{\|v^{\Delta t}_{\frac{h}{2}}(x,t_{n})-v^{\Delta t}_{\frac{h}{4}}(x,t_{n})\|_{j}}\thickapprox\frac{4^{\mu}-2^{\mu}}{2^{\mu}-1}, \\& \rho_{v,\Delta t,j}=\frac{\|v_{h}^{\Delta t}(x,t_{n})-v_{h}^{\frac{\Delta t}{2}}(x,t_{n})\|_{j}}{\|v_{h}^{\frac{\Delta t}{2}}(x,t_{n})-v_{h}^{\frac{\Delta t}{4}}(x,t_{n})\|_{j}}\thickapprox\frac{4^{\gamma}-2^{\gamma}}{2^{\gamma}-1}. \end{aligned}$$
Here,v is \(u,p\), or θ, and j is 0 or 1. Since \(\rho _{v,h,j}\) and \(\rho_{v,\Delta t,j}\) approach 4.0 or 2.0, the convergent order will be 2.0 or 1.0, respectively.
In Tables 5-6, we present the convergent orders with fixed spacing \(h=\frac{1}{32}\) and varying time steps \(\Delta t=0.1,0.05,0.025,0.0125\). From these results we can see that the Euler implicit/explicit scheme almost gets the same accuracy with the standard Galerkin finite element method and shows optimal convergent orders on Δt. In Tables 7-8, we study the convergence orders with fixed time step \(\Delta t=0.01\) with varying spacing \(\frac {1}{h}=2,4,8,16\). Observe that \(\rho_{u,h,0}\) and \(\rho_{\theta,h,0}\) are close to 4.0 and \(\rho_{u,h,1}\), \(\rho_{\theta,h,1}\) approach 2.0, which suggests that the orders of convergence are \(\mathcal{O}(h^{2})\) for the \(L^{2}\)-norm of u and θ and \(\mathcal{O}(h)\) for the \(H^{1}\)-norm of u and θ in space. For the convergence order of pressure\(,\rho_{p,h,0}\) is close to 3.2, which shows the superconvergence. From these numerical results we can conclude that the Euler implicit/explicit scheme not only has a good accuracy, but also saves a lot of computational cost.
Table 5

Convergence orders of the standard Galerkin FEM at time \(\pmb{T=1.0}\) with fixed mesh \(\pmb{h=\frac{1}{32}}\)

Δ t

\(\boldsymbol {\|u^{n,\Delta t}-u^{n,\frac{\Delta t}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{u,\Delta t,0}}\)

\(\boldsymbol {\|u^{n,\Delta t}-u^{n,\frac{\Delta t}{2}}\|_{1}}\)

\(\boldsymbol {\rho_{u,\Delta t,1}}\)

\(\boldsymbol {\|p^{n,\Delta t}-p^{n,\frac{\Delta t}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{p,\Delta t,0}}\)

0.1

1.09819e−005

2.06348

7.97307e−005

2.06314

2.09931e−005

2.0025

0.05

5.32202e−006

2.04257

3.86453e−005

2.03837

1.04834e−005

1.64187

0.025

2.60556e−006

2.01799

1.89589e−005

1.91156

6.38507e−006

0.618958

0.0125

1.29116e−006

 

9.91804e−006

 

1.03158e−005

 

Δ t

\(\boldsymbol {\|\theta^{n,\Delta t}-\theta^{n,\frac{\Delta t}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{\theta,\Delta t,0}}\)

\(\boldsymbol {\|\theta^{n,\Delta t}-\theta^{n,\frac{\Delta t}{2}}\|_{1}}\)

\(\boldsymbol {\rho_{\theta,\Delta t,1}}\)

0.1

1.14663e−005

2.06739

8.10132e−005

2.06742

0.05

5.54628e−006

2.03617

3.91857e−005

2.03617

0.025

2.72388e−006

2.01728

1.92448e−005

2.01729

0.0125

1.35028e−006

 

9.53992e−006

 
Table 6

Convergence orders of the Euler implicit/explicit scheme at time \(\pmb{T=1.0}\) with fixed mesh \(\pmb{h=\frac{1}{32}}\)

Δ t

\(\boldsymbol {\|u^{E,n,\Delta t}-u^{E,n,\frac{\Delta t}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{u,\Delta t,0}}\)

\(\boldsymbol {\|u^{E,n,\Delta t}-u^{E,n,\frac{\Delta t}{2}}\|_{1}}\)

\(\boldsymbol {\rho_{u,\Delta t,1}}\)

\(\boldsymbol {\|p^{E,n,\Delta t}-p^{E,n,\frac{\Delta t}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{p,\Delta t,0}}\)

0.1

1.09855e−005

2.06806

7.97559e−005

2.06786

2.10207e−005

2.04139

0.05

5.31197e−006

2.03533

3.85693e−005

2.03005

1.02973e−005

1.57437

0.025

2.60989e−006

2.01716

1.89991e−005

1.91612

6.54057e−006

0.654705

0.0125

1.29384e−006

 

9.91544e−006

 

9.9901e−006

 

Δ t

\(\boldsymbol {\|\theta^{E,n,\Delta t}-\theta^{E,n,\frac{\Delta t}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{\theta,\Delta t,0}}\)

\(\boldsymbol {\|\theta^{E,n,\Delta t}-\theta^{E,n,\frac{\Delta t}{2}}\|_{1}}\)

\(\boldsymbol {\rho_{\theta,\Delta t,1}}\)

0.1

1.14669e−005

2.06806

8.10171e−005

2.06809

0.05

5.54475e−006

2.0351

3.91749e−005

2.0351

0.025

2.72457e−006

2.01847

1.92496e−005

2.01848

0.0125

1.34982e−006

 

9.53665e−006

 
Table 7

Convergence orders of the standard Galerkin FEM at time \(\pmb{T=1.0}\) with time step \(\pmb{\Delta t=0.01}\)

\(\boldsymbol {\frac{1}{h}}\)

\(\boldsymbol {\|u^{n,h}-u^{n,\frac{h}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{u,h,0}}\)

\(\boldsymbol {\|u^{n,h}-u^{n,\frac{h}{2}}\|_{1}}\)

\(\boldsymbol {\rho_{u,h,1}}\)

\(\boldsymbol {\|p^{n,h}-p^{n,\frac{h}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{p,h,0}}\)

2

0.0172184

2.65194

0.230412

2.14169

0.63202

4.86123

4

0.00649278

3.37905

0.107585

2.20926

0.130012

3.19887

8

0.00192148

3.98286

0.0486971

2.12864

0.0406432

3.29174

16

0.000482436

 

0.0228771

 

0.012347

 

\(\boldsymbol {\frac{1}{h}}\)

\(\boldsymbol {\|\theta^{n,h}-\theta^{n,\frac{h}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{\theta,h,0}}\)

\(\boldsymbol {\|\theta^{n,h}-\theta^{n,\frac{h}{2}}\|_{1}}\)

\(\boldsymbol {\rho_{\theta,h,1}}\)

2

0.0166398

3.77383

0.131697

1.96256

4

0.00440926

3.80314

0.0671046

1.91516

8

0.00115937

3.92768

0.0350386

1.99231

16

0.000295181

 

0.0175869

 
Table 8

Convergence orders of the Euler implicit/explicit scheme at time \(\pmb{T=1.0}\) with time step \(\pmb{\Delta t=0.01}\)

\(\boldsymbol {\frac{1}{h}}\)

\(\boldsymbol {\|u^{E,n,h}-u^{E,n,\frac{h}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{u,h,0}}\)

\(\boldsymbol {\|u^{E,n,h}-u^{E,n,\frac{h}{2}}\|_{1}}\)

\(\boldsymbol {\rho_{u,h,1}}\)

\(\boldsymbol {\|p^{E,n,h}-p^{E,n,\frac{h}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{p,h,0}}\)

2

0.0172184

2.65201

0.230411

2.14169

0.631987

4.86024

4

0.00649258

3.37906

0.107584

2.20925

0.130032

3.19873

8

0.00192142

3.98294

0.0486971

2.12866

0.0406512

3.29108

16

0.000482412

 

0.0228769

 

0.0123519

 

\(\boldsymbol {\frac{1}{h}}\)

\(\boldsymbol {\|\theta^{E,n,h}-\theta^{E,n,\frac{h}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{\theta,h,0}}\)

\(\boldsymbol {\|\theta^{E,n,h}-\theta^{E,n,\frac{h}{2}}\|_{1}}\)

\(\boldsymbol {\rho_{\theta,h,1}}\)

2

0.01664

3.77398

0.131698

1.96258

4

0.00440914

3.80302

0.0671049

1.91516

8

0.00115938

3.92765

0.0350387

1.99231

16

0.000295184

 

0.017587

 

Declarations

Acknowledgements

The authors are grateful to the editor and two referees for a number of helpful suggestions, which have greatly improved our original manuscript. This research is supported by CAPES and CNPq of Brazil (No. 88881.068004/2014.01), the NSF of China (No. 11301157), and the Foundation of Distinguished Young Scientists of Henan Polytechnic University (J2015-05).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Information Science, Henan Polytechnic University
(2)
Departamento de Matemática, Centro Politécnico, Universidade Federal do Paraná

References

  1. Brezzi, F, Fortin, M: Mixed and Hybrid Finite Element Methods. Springer, New York (1991) View ArticleMATHGoogle Scholar
  2. Shen, SM: The finite element analysis of conduction-convection problem. J. Comput. Math. 16, 170-182 (1994) (in Chinese) MATHGoogle Scholar
  3. Cibik, A, Kaya, S: A projection-based stabilized finite element method for steady-state natural convection problem. J. Math. Anal. Appl. 381, 469-484 (2011) MathSciNetView ArticleMATHGoogle Scholar
  4. Damanik, H, Hron, J, Ouazzi, A, Turek, S: A monolithic FEM-multigrid solver for non-isothermal incompressible flow on general meshes. J. Comput. Phys. 228, 3869-3881 (2009) MathSciNetView ArticleMATHGoogle Scholar
  5. Elman, HC, Mihajlovic, MD, Silvester, DJ: Fast iterative solvers for buoyancy driven flow problems. J. Comput. Phys. 230, 3900-3914 (2011) MathSciNetView ArticleMATHGoogle Scholar
  6. Zhang, T, Tao, ZZ: Decoupled scheme for time-dependent natural convection problem II time semidiscreteness. Math. Probl. Eng. (2014). doi:10.1155/2014/726249 MathSciNetGoogle Scholar
  7. Zhang, T, Yuan, JY, Si, ZY: Decoupling two grid finite element method for the non-steady natural convection problem I: space discretization. Numer. Methods Partial Differ. Equ. 31, 2135-2168 (2015) MathSciNetView ArticleMATHGoogle Scholar
  8. Zhang, T, Zhao, X, Huang, PZ: Decoupled two level finite element methods for the steady natural convection problem. Numer. Algorithms 68, 837-866 (2015) MathSciNetView ArticleMATHGoogle Scholar
  9. Baker, GA, Dougalis, VA, Karakashian, OA: On a high order accurate fully discrete Galerkin approximation to the Navier-Stokes equations. Math. Comput. 39, 339-375 (1982) MathSciNetView ArticleMATHGoogle Scholar
  10. He, YN: Euler implicit/explicit iterative scheme for the stationary Navier-Stokes equations. Numer. Math. 123, 67-96 (2013) MathSciNetView ArticleMATHGoogle Scholar
  11. He, YN, Li, KT: Nonlinear Galerkin method and two-step method for the Navier- Stokes equations. Numer. Methods Partial Differ. Equ. 12, 283-305 (1996) MathSciNetView ArticleMATHGoogle Scholar
  12. He, YN, Sun, WW: Stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 45, 837-869 (2007) MathSciNetView ArticleMATHGoogle Scholar
  13. Johnston, H, Liu, JG: Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term. J. Comput. Phys. 199, 221-259 (2004) MathSciNetView ArticleMATHGoogle Scholar
  14. Marion, M, Temam, R: Navier-Stokes equations: theory and approximation. In: Handbook of Numerical Analysis, Vol. VI, pp. 503-688. North-Holland, Amsterdam (1998) Google Scholar
  15. Tone, F: Error analysis for a second scheme for the Navier-Stokes equations. Appl. Numer. Math. 50, 93-119 (2004) MathSciNetView ArticleMATHGoogle Scholar
  16. He, YN: The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data. Math. Comput. 77, 2097-2124 (2008) MathSciNetView ArticleMATHGoogle Scholar
  17. He, YN: Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 41, 1263-1285 (2003) MathSciNetView ArticleMATHGoogle Scholar
  18. He, YN, Li, J: A penalty finite element method based on the Euler implicit/explicit scheme for the time-dependent Navier-Stokes equations. J. Comput. Appl. Math. 235, 708-725 (2010) MathSciNetView ArticleMATHGoogle Scholar
  19. Zhang, T, Pedro, D, Yuan, JY: A large time stepping viscosity-splitting finite element method for the viscoelastic flow problem. Adv. Comput. Math. 41, 149-190 (2015) MathSciNetView ArticleMATHGoogle Scholar
  20. Heywood, J, Rannacher, R: Finite element approximation of the nonstationary Navier-Stokes problem I: regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275-311 (1982) MathSciNetView ArticleMATHGoogle Scholar
  21. Ciarlet, PG: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978) MATHGoogle Scholar
  22. Temam, R: Navier-Stokes Equation: Theory and Numerical Analysis, 3rd edn. North-Holland, Amsterdam (1984) MATHGoogle Scholar
  23. Luo, ZD: The Bases and Applications of Mixed Finite Element Methods. Chinese Science Press, Beijing (2006) Google Scholar
  24. Shen, J: On error estimates of projection methods for Navier-Stokes equations: first order schemes. SIAM J. Numer. Anal. 29, 57-77 (1992) MathSciNetView ArticleMATHGoogle Scholar
  25. Thomee, V: Galerkin Finite Element Methods for Parabolic Problem. Springer, Berlin (2010) MATHGoogle Scholar
  26. Cringanu, J, Paşca, D: Existence results for Dirichlet problems with \((p,q)\)-Laplacian. J. Math. Anal. Appl. 387, 828-836 (2012) MathSciNetView ArticleMATHGoogle Scholar
  27. Marin, M, Craciun, EM, Pop, N: Considerations on mixed initial-boundary value problems for micropolar porous bodies. Dyn. Syst. Appl. 25, 175-196 (2016) MATHGoogle Scholar
  28. Mu, M, Zhu, XH: Decoupled schemes for a non-stationary mixed Stokes-Darcy model. Math. Comput. 79, 707-731 (2010) MathSciNetView ArticleMATHGoogle Scholar

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