A Priori Error Analysis and Finite Element Approximations for a Coupled Model Under Nonlinear Slip Boundary Conditions

Abstract

We consider the time-dependent Navier–Stokes system coupled with the heat equation governed by the nonlinear Tresca boundary conditions. We propose a discretization of these equations that combines Euler implicit scheme in time and finite element approximations in space. We present optimal error estimates for velocity, pressure and temperature. Numerical examples are displayed to illustrate the theoretical results.

1 Introduction

Let $\Omega$ be a connected bounded open set in ${\mathbb {R}}^d, d=2$ or 3, with a Lipschitz-continuous boundary $\partial \Omega$ divided in two parts S and $\Gamma =\partial \Omega \backslash {\bar{S}}$ with $\bar{\Gamma } \cap {\bar{S}} = \emptyset$. And let [0, T] denote an interval in ${\mathbb {R}}$ where T is a positive real number.

We consider the time-dependent Navier–Stokes problem coupled with the heat equation,

$$\begin{aligned} \left\{ \begin{array}{lrcl} \displaystyle \frac{\partial \varvec{u}}{\partial t}-\text {div}\ (2\nu (\theta )\varvec{D}\varvec{u})+(\varvec{u}\cdot \nabla )\varvec{u} + \nabla p = \varvec{f} \ {} &{}\text {in} \quad \Omega \times [0,T],&{} \\ \text {div} \, \varvec{u} = 0 \ {} &{}\text {in} \quad \Omega \times [0,T],&{} \\ \displaystyle \frac{\partial \theta }{\partial t}-\text {div}\ (\mu (\theta )\nabla \theta ) \,+(\varvec{u} . \ \nabla )\theta = {k}\ {} &{}\text {in} \quad \Omega \times [0,T].&{} \end{array} \right. \end{aligned}$$

(1.1)

The unknowns in (1.1) are the velocity $\varvec{u}$, the pressure p and the temperature $\theta$. The function $\varvec{f}$ represents the external volumic forces and the function k represents the external heat source. They depends only on the position vector $(\varvec{x},t)\in {\mathbb {R}}^d\times [0,T]$. The functions $\nu$ and $\mu$ represents respectively the viscosity and the diffusion coefficient, they are both positive, bounded and depends on the temperature.

The first equation in (1.1) represents the balance of forces in the system, while the second equation is the incompressibility of the fluid. The third equation is the heat exchange in the system.

The force within the fluid is the Cauchy stress tensor $\sigma$ given by the relation $\varvec{\sigma }=\varvec{\sigma }(\theta ,\varvec{u},p)$ given as:

$$\begin{aligned} \varvec{\sigma }=2\nu (\theta )\varvec{D}\varvec{u}-p\varvec{I}. \end{aligned}$$

Such that $\varvec{I}$ is the identity matrix in $M_d({\mathbb {R}})$ and $\varvec{D}\varvec{u}$ is the symmetric tensor defined by

$$\begin{aligned} \varvec{D}\varvec{u}=\frac{1}{2}(\nabla \varvec{u}+ \nabla \varvec{u} ^{\text {Tr}}). \end{aligned}$$

The system of equations (1.1) is supplemented by the boundary conditions on the velocity and temperature. For that purpose, we impose the following initial conditions

$$\begin{aligned} \varvec{u}(x,0)=\varvec{u}_0,\hspace{0.7cm}\theta (x,0)=\theta _0\hspace{1cm}\text {on}\quad \bar{\Omega }. \end{aligned}$$

(1.2)

We assume the Dirichlet conditions

$$\begin{aligned} \hspace{-0.6cm}\varvec{u}(\varvec{x},t)=\varvec{0} \hspace{0.5cm} \text {on}\quad \Gamma \times (0,T)\hspace{0.5cm}\text {and}\hspace{0.5cm} \theta (\varvec{x},t) = \theta _b \quad \text {on} \quad \partial \Omega \times (0,T), \end{aligned}$$

(1.3)

$\theta _b$ being given and non-negative.

On the part of the boundary S, the velocity is decomposed following its normal and tangential part. We assume the impermeability condition

$$\begin{aligned} u_N:=\varvec{u}\cdot \varvec{n}=\varvec{0}\hspace{0.5cm} \text {on}\quad S\times (0,T). \end{aligned}$$

(1.4)

Such that $\varvec{n}$ is the outward unit normal on the boundary $\partial \Omega$ and $u_N$ is the normal component of the velocity. Then the tangential component of the velocity is defined by $\varvec{u}_{\varvec{\tau }}=\varvec{u}-u_N\varvec{n},$ such that $\varvec{\tau }$ is the tangent vector orthogonal to $\varvec{n}$.

Additionally to (1.4), we also impose on S a threshold slip condition. The threshold slip condition is formulated through a positive function $g: S\rightarrow (0,\infty )$ which is called the barrier of threshold function and the tangential part of $\varvec{\sigma n}$ as follows:

$$\begin{aligned} \left. \begin{array}{l} \text{ If } \hspace{0.2cm} \left| (\varvec{\sigma n})_{\varvec{\tau }}\right| <g \hspace{0.2cm} \text{ then } \hspace{0.2cm} { \varvec{u}}_{\varvec{\tau }}={\varvec{0}}, \\ \text{ If } \hspace{0.2cm} \left| (\varvec{\sigma n})_{\varvec{\varvec{\tau }}}\right| =g \hspace{0.2cm} \text{ then } \hspace{0.2cm} \varvec{u}_{\varvec{\tau }} \ne \textbf{0} \hspace{0.2cm} \text{ and } \hspace{0.2cm} -(\varvec{\sigma n} )_{\varvec{\tau }}=g \displaystyle \frac{{\varvec{u}}_{\varvec{\tau }}}{\left| {\varvec{u}}_{\varvec{\tau }} \right| } \end{array} \right\} \hspace{0.2cm} \text {on}\hspace{0.2cm} S \times (0, T). \end{aligned}$$

(1.5)

Many works have been carried out for the system (1.1) with different boundary conditions and different discretization method. In [1] as well in [6], a similar model provided by the homogeneous Dirichlet conditions has been studied. And the authors treated the coupled system by using a spectral discretization. Furthermore, many works have examined the numerical analysis (see for instance [2, 7, 12]).

In this article, the novelty of our work compared to the existence ones is to propose an unsteady problem under non linear boundary condition by adding an explicit dependency to the temperature for the viscosity and the diffusion coefficient. One of the major challenges in this work is the presence of variational inequality in our problem. Its numerical solutions have been examined by many researchers, see for example, [9, 16, 19] for a first breakthrough for a systematic mathematical analysis of problems formulated as variational inequalities.

The point of departure of this study related to the numerical part is the work of J.K. and Djoko [8] where some new numerical methods for the Stokes and Navier–Stokes flow driven by nonlinear slip boundary conditions was discussed. To solve the Stokes system, they first reduced the related variational inequality into a saddle-point problem for a well chosen augmented Lagrangian. And to solve this saddle point problem they suggested an alternating direction method of multiplier together with finite element approximations. The solution of the Navier–Stokes was solved by operator splitting. The numerical experiments showed that these methods are scalable, i.e. the number of iterations required for convergence is virtually independent of mesh size.

In this article, the problem is discretized in time by Euler implicit scheme and in order to facilitate its resolution we have used the operator splitting technique. Indeed, it makes it possible to remove the complexity of a problem by reducing it to several simpler sub-problems.

The outline of the paper is as follows:

• In Sect. 2, we introduce some notations and functional spaces useful for the studies of the problem. Then, we study the variational formulation.

• Sect. 3 is devoted to introduce and study the descretized problem then perform the a priori corresponding error analysis for $d=2$.

• In Sect. 4, we introduce an operator-splitting method based on the Marchuk-Yanenko’s scheme.

• Some numerical experiments are presented in Sect. 5.

2 Analysis of the Continuous Problem

2.1 Variational Formulation

To write the system (1.1) in a variational form, we need some preliminaries. We use the standard definitions for the sobolev spaces [5] and for boldface characters we denote vector quantities

$$\begin{aligned} {\varvec{H}^1(\Omega )}=(H^1(\Omega ))^d,\quad \varvec{L}^2(\Omega )=(L^2(\Omega ))^d\quad \text {and}\quad L_+^2(S)=L^2(S)\cap \mathbb {R_+}.\end{aligned}$$

We claim that $H^{1/2}(\partial \Omega )$ is the space of trace for elements of $\varvec{H}^1(\Omega )$. We introduce the space

$$\begin{aligned} L^2_0(\Omega ) = \big \{ q \in L^2(\Omega ); \ \displaystyle {\int \limits _\Omega } q(x) dx =0 \big \}. \end{aligned}$$

(2.1)

And we define the following functional spaces

$$\begin{aligned} \varvec{V}(\Omega )=\big \{\varvec{v} \in \varvec{H}^1(\Omega ) \;\hspace{0.5cm}\varvec{v}_{{\displaystyle |}_{\displaystyle \Gamma } } =0,\hspace{0.5cm} \varvec{v}\cdot \varvec{n}_{{\displaystyle |}_{\displaystyle S}} =0 \big \}, \end{aligned}$$

(2.2)

And,

$$\begin{aligned} \varvec{V}_{\text {div}}(\Omega ) = \big \{ \varvec{v} \in \varvec{V}(\Omega )\;\hspace{0.5cm}\text {div}\,\varvec{v}=\varvec{0}\hspace{0.5cm} \text {on}\ \Omega \big \}. \end{aligned}$$

(2.3)

Lemma 2.1

For $d=2$, we have the relation

$$\begin{aligned} \Vert \varvec{v}\Vert ^2_{\varvec{L}^4(\Omega )} \le c \Vert \varvec{v}\Vert _{\varvec{L}^2(\Omega )}|\varvec{v}|_{\varvec{H}_0^1(\Omega )}. \end{aligned}$$

(2.4)

For the mathematical investigations of (1.1) we assume the following hypotheses:

Hypothesis H1. We assume that the viscosity $\nu (\theta )$ and the heat diffusion coefficient $\mu (\theta )$ are two functions which depend on the temperature $\theta$ and which satisfy:

$$\begin{aligned} \left\{ \begin{array}{lr} \forall s\in {\mathbb {R}},\hspace{0.5cm} 0<\nu _0\le \nu (s)\le \nu _1, \\ \forall s\in {\mathbb {R}},\hspace{0.5cm} 0<\mu _0\le \mu (s)\le \mu _1. \end{array} \right. \end{aligned}$$

(2.5)

And

$$\begin{aligned} \forall s,t\in {\mathbb {R}},\hspace{0.5cm}&|\nu (s)-\nu (t)|\le \nu ^* |s-t|,&\nonumber \\&|\mu (s)-\mu (t)|\le \mu ^* |s-t|.&\end{aligned}$$

(2.6)

where $\nu _0$, $\nu _1$, $\mu _0$, $\mu _1$, $\nu ^*$ and $\mu ^*$ are strictly positive constants.

Hypothesis H2. We assume that:

$$\begin{aligned} \varvec{f}\in L^\infty (0,T;\varvec{L}^{2}(\Omega )), \hspace{0.2cm} {k} \in L^2(0,T;H^{ -1}(\Omega )),\hspace{0.2cm} \theta _b\in L^2(0,T;H^{1/2}(\partial \Omega ))\hspace{0.2cm} \text {and}\hspace{0.2cm} g\in L^{2}_+(S). \end{aligned}$$

(2.7)

Hypothesis H3. The data $\theta _0$ belongs to $L^{2}(\Omega )$ and the data $\varvec{u}_0$ belongs to $\varvec{L}^2(\Omega )$ satisfying the following compatibility condition

$$\begin{aligned} \text {div}\, \varvec{u}_0=0 \hspace{0.5cm} \text {in}\ \Omega . \end{aligned}$$

(2.8)

We introduce the following functionals that will be used to write the weak form on the problem in abstract setting.

$$\begin{aligned} \begin{array}{ccccc} a_1 &{}: &{} \varvec{H}^1(\Omega )\times \varvec{H}^1(\Omega ) &{}\longrightarrow &{} \hspace{-4cm}{\mathbb {R}} \\ &{} &{} (\theta :\varvec{u},\varvec{v}) &{}\longmapsto &{} a_1(\theta :\varvec{u},\varvec{v})=\displaystyle {\int \limits _\Omega } 2\nu (\theta )\varvec{D}\varvec{u}:\varvec{D}\varvec{v}\ d\varvec{x}. \\ a_2 &{}: &{} H^1(\Omega )\times H^1(\Omega ) &{}\longrightarrow &{} \hspace{-3cm}{\mathbb {R}} \\ &{} &{} (\theta :\theta ,\rho ) &{}\longmapsto &{} a_2(\theta :\theta ,\rho )= \displaystyle {\int \limits _\Omega } \mu (\theta )\nabla \theta \cdot \nabla \rho \ d\varvec{x}. \\ b &{}: &{} \varvec{H}^1(\Omega ) \times L^2_0(\Omega ) &{}\longrightarrow &{} \hspace{-3cm}{\mathbb {R}} \\ &{} &{} (\varvec{u},q) &{}\longmapsto &{} b(\varvec{u},q)= - \displaystyle {\int \limits _\Omega } \text {div}\, \varvec{u}\ q \ d\varvec{x}. \\ d_1 &{}: &{} \varvec{H}^1(\Omega )\times \varvec{H}^1(\Omega )\times \varvec{H}^1(\Omega ) &{}\longrightarrow &{} \hspace{-3cm}{\mathbb {R}} \\ &{} &{} (\varvec{u},\varvec{u},\varvec{v}) &{}\longmapsto &{} d_1(\varvec{u},\varvec{u},\varvec{v})= \displaystyle {\int \limits _\Omega } (\varvec{u}\cdot \nabla )\varvec{u}\,\varvec{v}\, d\varvec{x}, \\ d_2 &{}: &{} \varvec{H}^1(\Omega )\times H^1(\Omega )\times H^1(\Omega ) &{}\longrightarrow &{} \hspace{-3cm}{\mathbb {R}} \\ &{} &{} (\varvec{v},\theta ,\rho ) &{}\longmapsto &{} d_2(\varvec{v},\theta ,\rho )= \displaystyle {\int \limits _\Omega } (\varvec{v}\cdot \nabla )\theta \rho \ d\varvec{x}, \\ J &{}: &{} \varvec{H}^1(\Omega ) &{}\longrightarrow &{} \hspace{-2cm} {\mathbb {R}} \\ &{} &{} \varvec{u} &{}\longmapsto &{} J(\varvec{u})= \displaystyle {\int \limits _S} g|\varvec{u}_{\varvec{\tau }}| \ {dS}. \\ \end{array} \end{aligned}$$

(2.9)

where dS being the measure on the surface S.

In order to simplify, we note by

$$\begin{aligned} \varvec{V}:=L^2(0,T;\varvec{V}(\Omega ))\cap L^\infty (0,T;\varvec{L}^2(\Omega )),\quad M:=L^2(0,T;L^2_0(\Omega ))\quad \text { and }\quad Q:=L^2(0,T;H^1(\Omega )). \end{aligned}$$

We consider the variational problem: For $\theta _b\in L^2(0,T;H^{1/2}(\partial \Omega ))$, $\varvec{f}\in L^\infty (0,T;\varvec{L}^{2}(\Omega )), \,{k}\in L^2(0,T;H^{-1}(\Omega )),\hspace{0.2cm}$

$$\begin{aligned} \hspace{-0.8cm}{\left\{ \begin{array}{ll} \text {Find}\ (\varvec{u},p,\theta ) \in {\varvec{V}\times M\times Q}\ \text {such that:} \\ \hspace{2cm} \varvec{u}(.,0)=\varvec{u}_0 \hspace{0.2cm}\text {and}\hspace{0.2cm} \theta (.,0)=\theta _0\hspace{0.2cm}\text {in}\hspace{0.2cm} \Omega ,\\ \hspace{2cm} \theta (.,t)=\theta _b(.,t)\ \text {on} \ \partial \Omega ,\quad \forall \ 0\le t\le T, \\ \forall \ (\varvec{v},q,\rho ) \in {\varvec{V}(\Omega )}\times L^2_0(\Omega )\times H^1_0(\Omega ), \\ \big (\partial _{t} \varvec{u}, \varvec{v}-\varvec{u}\big )+a_1(\theta :{\varvec{u}},{\varvec{v}-\varvec{u}})+b(\varvec{v}-\varvec{u},p)+d_1(\varvec{u},\varvec{u},\varvec{v}-\varvec{u})+J(\varvec{v})- J(\varvec{u})\ge \langle \varvec{f},\varvec{v}-\varvec{u}\rangle , \\ b(\varvec{u}, q)=0, \\ \big (\partial _{t} \theta , \rho \big )+ a_2(\theta :\theta ,\rho ) + d_2(\varvec{u},\theta ,\rho )=\langle {k},\rho \rangle , \end{array}\right. } \end{aligned}$$

(2.10)

Such that, $\langle .,.\rangle$ being the duality pairing between $H^{-1}(\Omega )$ and $H_0^{1}(\Omega )$.

Proposition 2.1

Problems (2.10) and (1.1–1.5) are equivalent. Indeed, any triplet $(\varvec{u},p,\theta ) \in {\varvec{V}\times M\times Q}$ is a solution of (1.1–1.5) in the sense of distribution if and only if it is a solution of (2.10).

The following standard results will be used for the analysis of problem (2.10) and its corresponding finite element discretization [5, 9, 21].

The following Poincaré-Friedrich’s inequality holds: there is a positive constant c depending on the domain $\Omega$ such that

$$\begin{aligned} \text {for all}\, \varvec{v} \in \varvec{V}(\Omega ),\hspace{0.5cm} \Vert \varvec{v}\Vert _{H^1(\Omega )}\le c|\varvec{v}|_{H^1(\Omega )}, \end{aligned}$$

(2.11)

which ensures that the norms $\Vert .\Vert _{H^1(\Omega )}$ and $|.|_{H^1(\Omega )}$are equivalent on $\varvec{V(\Omega )}$. We also recall Korn’s inequality [9]: there exists $c(\Omega )$ such that

$$\begin{aligned} \Vert D\varvec{v}\Vert _{L^2(\Omega )} \ge c(\Omega )\Vert \nabla \varvec{v}\Vert _{L^2(\Omega )}\quad \text {for all} \ \varvec{v}\in \varvec{V}(\Omega ). \end{aligned}$$

(2.12)

We deduce from (2.12) and (2.9) that $a_1(.:.,.)$ is continuous and elliptic on $\varvec{V(\Omega )}$. This means that for $(\varvec{v},\varvec{w})$ element of $\varvec{V(\Omega )}\times \varvec{V(\Omega )}$,

$$\begin{aligned} a_1(\theta :\varvec{v},\varvec{w})\le \nu _1 \Vert \varvec{v}\Vert _{\varvec{H}^1(\Omega )} \Vert \varvec{w}\Vert _{\varvec{H}^1(\Omega )}, \hspace{0.7cm} a_1(\theta :\varvec{v},\varvec{v})\ge c\nu _0 \Vert \varvec{v}\Vert ^2_{\varvec{H}^1(\Omega )}. \end{aligned}$$

(2.13)

From (2.11) and (2.9), we deduce that $a_2(.:.,.)$ is continuous and elliptic on $\varvec{H}^1_0(\Omega )$, this means that for $(\theta ,\rho )$ element of $H^1_0(\Omega )\times H^1_0(\Omega )$,

$$\begin{aligned} a_2(\theta :\theta ,\rho )\le c\mu _1 \Vert \theta \Vert _{H^1(\Omega )} \Vert \rho \Vert _{H^1(\Omega )}, \hspace{0.7cm} a_2(\theta :\rho ,\rho )\ge c\mu _0 \Vert \rho \Vert ^2_{{H}^1(\Omega )}. \end{aligned}$$

(2.14)

The space $\varvec{V}_{\text {div}}(\Omega )$ defined in (2.3) characterize the kernel of b(., .). One easily check that b(., .) is continuous. That is, for all $(\varvec{v},q)\in \varvec{H}^1(\Omega )\times L^2(\Omega )$

$$\begin{aligned} b(\varvec{v},q)\le \Vert \varvec{v}\Vert _{\varvec{H}^1(\Omega )}\Vert q\Vert _{ L^2_0(\Omega )}. \end{aligned}$$

(2.15)

The compatibility between velocity and pressure require the inf-sup condition for the study of (2.10), its proof can be seen in [3, 13]: there exist a constant $c>0$ such that

$$\begin{aligned} \forall \varvec{v}\in \varvec{V}(\Omega ),\hspace{0.7cm} \displaystyle \underset{\varvec{v}\in \varvec{V}(\Omega )}{\sup } \displaystyle \frac{b(\varvec{v},q)}{\Vert \varvec{v}\Vert _{\varvec{H}^1(\Omega )}}\ge c \Vert q\Vert _{L^2_0(\Omega )}. \end{aligned}$$

(2.16)

Lemma 2.2

[17, 20] (i) When $d=2$, for all $\varvec{u}, \varvec{v}, \varvec{w}\in \varvec{H}^1(\Omega )$ it holds that

$$\begin{aligned} |d_1(\varvec{u},\varvec{v},\varvec{w})|\le C\Vert \varvec{u}\Vert _{\varvec{L}^2(\Omega )}^{1/2}\Vert \varvec{u}\Vert _{\varvec{H}^1(\Omega )}^{1/2}\Vert \varvec{v}\Vert _{\varvec{H}^1(\Omega )}\Vert \varvec{w}\Vert _{\varvec{L}^2(\Omega )}^{1/2}\Vert \varvec{w}\Vert _{\varvec{H}^1(\Omega )}^{1/2} \end{aligned}$$

(2.17)

(ii) When $d=3$, for all $\varvec{u}, \varvec{v}, \varvec{w} \in \varvec{H}^1(\Omega )$ it holds that

$$\begin{aligned} |d_1(\varvec{u},\varvec{v},\varvec{w})|\le C\Vert \varvec{u}\Vert _{\varvec{L}^2(\Omega )}^{1/4}\Vert \varvec{u}\Vert _{\varvec{H}^1(\Omega )}^{3/4}\Vert \varvec{v}\Vert _{\varvec{H}^1(\Omega )}\Vert \varvec{w}\Vert _{\varvec{L}^2(\Omega )}^{1/4}\Vert \varvec{w}\Vert _{\varvec{H}^1(\Omega )}^{3/4} \end{aligned}$$

(2.18)

From (2.17) and (2.18), we get

$$\begin{aligned} |d_1(\varvec{u},\varvec{v},\varvec{w})|\le C\Vert \varvec{u}\Vert _{\varvec{H}^1(\Omega )}\Vert \varvec{v}\Vert _{\varvec{H}^1(\Omega )}\Vert \varvec{w}\Vert _{\varvec{H}^1(\Omega )}. \end{aligned}$$

(2.19)

Lemma 2.3

For all $\varvec{u}\in \varvec{V}_{\text {div}}(\Omega ), \varvec{v}\in \varvec{H}^1(\Omega )$ and $\eta ,\phi \in H^1(\Omega )$, we have (see [14])

$$\begin{aligned} \displaystyle {\int \limits _\Omega } (\varvec{u}\cdot \nabla )\varvec{v}. \varvec{v}\ d\varvec{x}=0,\hspace{0.5cm}\displaystyle {\int \limits _\Omega } (\varvec{u}\cdot \nabla )\phi \, \phi \ d\varvec{x}=0,\hspace{0.5cm}\displaystyle {\int \limits _\Omega } (\varvec{u}\cdot \nabla )\eta \,\phi \ d\varvec{x}=-\displaystyle {\int \limits _\Omega } (\varvec{u}\cdot \nabla )\phi \,\eta \ d\varvec{x}. \end{aligned}$$

(2.20)

From (2.20), the trilinear form $d_2(.,.,.)$ enjoys for all $(\varvec{v},\theta ,\rho )\in \varvec{V}_{\text {div}}(\Omega )\times {H}^1(\Omega )\times {H}^1(\Omega ),$

$$\begin{aligned} d_2(\varvec{v},\theta ,\rho )=-d_2(\varvec{v},\rho ,\theta )\quad \text {and}\quad d_2(\varvec{v},\rho ,\rho )=0. \end{aligned}$$

(2.21)

The functional J(.) is convexe, lower semi continuous on $\varvec{V}(\Omega )$ but not differentiable at zero.

We define the space

$$\begin{aligned} \varvec{\Lambda } = \big \{ \varvec{\alpha }\hspace{0.4cm} \text {such as }\ \varvec{\alpha }\in \varvec{L}^{\infty }( S), \hspace{0.4cm} |\varvec{\alpha }| \leqslant 1 \ \text {a.e on }\ S \big \}. \end{aligned}$$

(2.22)

It can be shown that the solution $(\varvec{u},p)$ of (2.10) is characterized by:

$$\begin{aligned} \hspace{-0.8cm}{\left\{ \begin{array}{ll} \text {Find} \ (\varvec{u},p,\theta ,\lambda ) \in {\varvec{V}\times M\times Q}\times \Lambda \ \text {such that:} \\ \hspace{2cm} \varvec{u}(.,0)=\varvec{u}_0 \hspace{0.2cm}\text {and}\hspace{0.2cm} \theta (.,0)=\theta _0\hspace{0.2cm}\text {in}\hspace{0.2cm} \Omega , \\ \hspace{2cm} \theta (.,t) = \theta _b(.,t)\ \text {on}\ \partial \Omega ,\quad \forall \ 0\le t\le T, \\ \forall \ (\varvec{v},q,\rho ) \in {\varvec{V}(\Omega )}\times L^2_0(\Omega )\times H^1_0(\Omega ), \\ \big (\partial _{t} \varvec{u}, \varvec{v}\big )+a_1(\theta :{\varvec{u}},{\varvec{v}})+d_1(\varvec{u},\varvec{u},\varvec{v})+b(\varvec{v},p)+\displaystyle \int \limits _S g\lambda \cdot \varvec{v}_{\tau }\ dS= \langle \varvec{f},\varvec{v}\rangle , \\ b(\varvec{u}, q)=0, \\ \lambda \cdot \varvec{u}_\tau =|\varvec{u}_\tau |, \\ \big (\partial _{t} \theta , \rho \big )+ a_2(\theta :\theta ,\rho ) + d_2(\varvec{u},\theta ,\rho )=\langle {k},\rho \rangle . \end{array}\right. } \end{aligned}$$

(2.23)

Remark 2.1

The existence of $\varvec{\lambda }$ is proved by using the Hahn-Banach theorem in (see [15], page 80, Theorem 5.3) and also by an approach based on a constructive regularization in (see [15], page 96, Theorem 6.3).

Next, we state the following result established in [13]: Let ${\mathcal {R}}$ denote a lifting operator, i.e., a continuous operator from $H^{1/2}(\partial \Omega )$ into $H^1(\Omega ).$ Since $\theta _b$ belongs to $L^2(0,T;H^{1/2}(\partial \Omega )),$ we denote by $\tilde{\theta _b}$ the function defined for a.e. t, $0 \le t \le T, \text {by}$

$$\begin{aligned}{\tilde{\theta }}_b(t)={\mathcal {R}} \theta _b(t).\end{aligned}$$

Clearly ${\tilde{\theta }}_b$ belongs to $L^2(0,T;H^1(\Omega ))$ and satisfies the following inequality

$$\begin{aligned} \forall t\in [0,T],\hspace{0.5cm} \Vert {\tilde{\theta }}_b\Vert _{L^2(0,T;H^1(\Omega ))} \le c\Vert \theta _b\Vert _{L^2(0,T;H^{1/2}(\partial \Omega ))}, \end{aligned}$$

(2.24)

where the positive constant c depends only on $\Omega$ and ${\mathcal {R}}$. By taking a reading on the temperature at the system (2.10) and setting $\theta ^* =\theta -{\tilde{\theta }}_b$, note that $\left. {\theta ^*}\right| _{\partial \Omega }=0$ and $(\varvec{u},p,\theta ^*)$ is the solution of the following variational problem:

$$\begin{aligned} \hspace{-0.8cm}{\left\{ \begin{array}{ll} \text {Find} \ (\varvec{u},p,\theta ^*) \in {\varvec{V}\times M\times Q_0}\ \text {such that}: \\ \varvec{u}(.,0)=\varvec{u}_0 \hspace{0.2cm}\text {et}\hspace{0.2cm} \theta ^*(.,0)=\theta _0-{\tilde{\theta }}_b(.,0)\hspace{0.2cm}\text {in}\hspace{0.2cm} \Omega , \\ \forall \ (\varvec{v},q,\rho ) \in {\varvec{V}(\Omega )}\times L^2_0(\Omega )\times H^1_0(\Omega ), \\ (\partial _{t} \varvec{u}, \varvec{v}-\varvec{u})+ a_1(\theta ^*+{\tilde{\theta }}_b:{\varvec{u}},{\varvec{v}-\varvec{u}})+ b(\varvec{v}-\varvec{u},p)+d_1(\varvec{u},\varvec{u},\varvec{v}-\varvec{u})+J(\varvec{v})-J(\varvec{u})\ge \langle \varvec{f},\varvec{v}-\varvec{u}\rangle , \hspace{-0.5cm} \\ b(\varvec{u}, q)=0, \\ (\partial _{t}\theta ^*, \rho ) + a_2(\theta ^* +{\tilde{\theta }}_b:\theta ^* +{\tilde{\theta }}_b,\rho ) + d_2(\varvec{u},\theta ^*,\rho )=\langle {k},\rho \rangle -( \partial _{t}{\tilde{\theta }}_b, \rho )-d_2(\varvec{u},{\tilde{\theta }}_b,\rho ). \end{array}\right. } \end{aligned}$$

(2.25)

Such that, $Q_0=L^2(0,T,H_0^1(\Omega ))$.

Moreover, the pair $(\varvec{u},\theta ^*)$ is a solution of the variational problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {Find} \ (\varvec{u},\theta ^*) \in {\varvec{V}_{\text {div}}\times M\times Q_0}\ \text {such that}: \\ \varvec{u}(.,0)=\varvec{u}_0 \hspace{0.2cm}\text {et}\hspace{0.2cm} \theta ^*(.,0)=\theta _0-{\tilde{\theta }}_b(.,0)\hspace{0.2cm}\text {in}\hspace{0.2cm} \Omega , \\ \forall \ (\varvec{v},\rho ) \in {\varvec{V}_{\text {div}}(\Omega )}\times H^1_0(\Omega ), \\ (\partial _{t} \varvec{u}, \varvec{v}-\varvec{u})+ a_1(\theta ^*+{\tilde{\theta }}_b:{\varvec{u}},{\varvec{v}-\varvec{u}})+d_1(\varvec{u},\varvec{u},\varvec{v}-\varvec{u})+J(\varvec{v})-J(\varvec{u})\ge \langle \varvec{f},\varvec{v}-\varvec{u}\rangle , \\ ( \partial _{t}\theta ^*,\rho ) + a_2(\theta ^* +{\tilde{\theta }}_b:\theta ^* +{\tilde{\theta }}_b,\rho ) + d_2(\varvec{u},\theta ^*,\rho )=\langle {k},\rho \rangle -( \partial _{t}{\tilde{\theta }}_b, \rho )- d_2(\varvec{u},{\tilde{\theta }}_b,\rho ). \end{array}\right. } \end{aligned}$$

(2.26)

Such that, $\varvec{V}_{\text {div}}=L^2(0,T;\varvec{V}_{\text {div}}(\Omega ))\cap L^\infty (0,T;\varvec{L}^2(\Omega ))$.

2.2 A Priori Estimates

Proposition 2.2

Assume hypothesis H1–H3, with $\varvec{u}_0\in \varvec{H}^1(\Omega ),$ such that (2.8) hold true. Then the following a priori estimate holds for any $t\in ]0,T[.$

$$\begin{aligned}{} & {} (1) \hspace{0.5cm} \Vert \varvec{u}\Vert _{L^2(0,T;H^1(\Omega ))} \le c\left( \Vert \varvec{u}_{0}\Vert _{\varvec{L}^2(\Omega )} + \Vert \varvec{f}\Vert _{L^2(0,T;\varvec{L}^{2}(\Omega ))} \right) . \end{aligned}$$

(2.27)

$$\begin{aligned}{} & {} (2) \hspace{0.5cm}\Vert \partial _t\varvec{u}\Vert _{L^2(0,T; \varvec{L}^2(\Omega ))}^2\le \Vert \varvec{f}\Vert _{L^2(0,T; \varvec{L}^2(\Omega ))}^2+c\Vert \varvec{u}_0\Vert _{\varvec{H}^1(\Omega )}^2+c\Vert g\Vert ^2_{L^\infty (S)}+c\Vert g\Vert _{L^\infty (S)}. \end{aligned}$$

(2.28)

$$\begin{aligned}{} & {} (3) \hspace{0.5cm} \nu _0\Vert \nabla \varvec{u}\Vert _{L^\infty (0,T; \varvec{L}^2(\Omega ))}^2\le \Vert \varvec{f}\Vert _{L^2(0,T; \varvec{L}^2(\Omega ))}^2+c\Vert \varvec{u}_0\Vert ^2_{\varvec{H}^1(\Omega )}+c\Vert g\Vert ^2_{L^\infty (S)}+c\Vert g\Vert _{L^\infty (S)}. \end{aligned}$$

(2.29)

$$\begin{aligned}{} & {} (4) \hspace{0.5cm}\Vert p\Vert _{L^2(0,T; L_0^2(\Omega ))}^2\le c\left( \Vert \varvec{f}\Vert _{L^2(0,T; \varvec{L}^2(\Omega ))}^2+\Vert \varvec{u}_0\Vert _{\varvec{H}^1(\Omega )}^2+\Vert g\Vert ^2_{L^\infty (S)}+\Vert g\Vert _{L^\infty (S)}\right) . \end{aligned}$$

(2.30)

$$\begin{aligned}{} & {} (5) \hspace{0.5cm} \Vert \theta ^* \Vert _{L^2(0,T;H^1(\Omega ))} \le c\bigg [ \Vert \theta _{0}\Vert _{L^2(\Omega )}+ \Vert {k}\Vert _{L^2(0,T;H^{-1}(\Omega ) )}+\Vert \partial _t{\tilde{\theta }}_b\Vert _{L^2(0,T;L^{2}(\Omega ) )}\nonumber \\{} & {} \quad \quad \qquad +\Vert {\tilde{\theta }}_{b}\Vert _{L^2(0,T;H^{1/2}(\partial \Omega ))}+\Vert {\tilde{\theta }}_{b}\Vert ^2_{H^1(0,T;H^{1/2}(\partial \Omega ))}+ \Vert \varvec{f}\Vert ^2_{L^2(0,T;\varvec{H}^{-1}(\Omega ))}+\Vert \varvec{u}_{0}\Vert ^2_{\varvec{L}(\Omega )} \bigg ]. \end{aligned}$$

(2.31)

Proof

(1) We take successively in (2.26) $\varvec{v}=\varvec{0}$ and $\varvec{v}=2\varvec{u}$. Comparing the two inequalities, we obtain

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Vert \varvec{u}\Vert ^2_{\varvec{L}^2(\Omega )}+a_1(\theta ^*+{\tilde{\theta }}_b:{\varvec{u}},\varvec{u})+ J(\varvec{u})= \langle \varvec{f},\varvec{u}\rangle , \end{aligned}$$

(2.32)

which after dropping the positive term J(.) and application of the proprieties of $a_1(.:,.,.)$ in (2.9) and Hölder’s inequality, we get

$$\begin{aligned} \frac{d}{dt} \Vert \varvec{u}\Vert ^2_{\varvec{L}^2(\Omega )} +2\nu _0 \Vert \varvec{u} \Vert ^2_{\varvec{H}^1(\Omega )} \le \frac{1}{\nu _0}\, \Vert \varvec{f}\Vert ^2_{\varvec{H}^{-1}(\Omega )}+\nu _0\Vert \varvec{u} \Vert ^2_{\varvec{H}^1(\Omega )}. \end{aligned}$$

(2.33)

It follows that,

$$\begin{aligned} \frac{d}{dt} \Vert \varvec{u} \Vert ^2_{\varvec{L}^2(\Omega )} +\nu _0\Vert \varvec{u} \Vert ^2_{\varvec{H}^1(\Omega )} \le \frac{1}{\nu _0}\, \Vert \varvec{f}\Vert ^2_{\varvec{H}^{-1}(\Omega )}. \end{aligned}$$

(2.34)

Integrating between 0 and T, we get

$$\begin{aligned} \Vert \varvec{u} (.,T)\Vert ^2_{\varvec{L}^2(\Omega )} +\nu _0\int \limits _{0}^{T} \Vert \varvec{u} (.,s)\Vert ^2_{\varvec{H}^1(\Omega )} ds \le \Vert \varvec{ u} (.,0)\Vert ^2_{\varvec{L}^2(\Omega )} +\frac{1}{\nu _0} \, \int \limits _{0}^{T} \Vert \varvec{f}(.,s)\Vert ^2_{\varvec{H}^{-1}(\Omega )} ds. \end{aligned}$$

(2.35)

Then

$$\begin{aligned} \Vert \varvec{u} \Vert _{L^2(0,T;H^1(\Omega ))} \le \frac{c}{\nu _0}\left( \Vert \varvec{u}_{0}\Vert _{\varvec{L}^2(\Omega )} + \Vert \varvec{f}\Vert _{L^2(0,T;\varvec{L}^{2}(\Omega ))} \right) , \end{aligned}$$

(2.36)

(2) Taking successively in (2.26) $\varvec{v}=\varvec{u}-\partial _t\varvec{u}$ and $\varvec{v}=\varvec{u}+\partial _t\varvec{u}$. Comparing the two inequalities,

$$\begin{aligned} \Vert \partial _t\varvec{u} \Vert ^2_{\varvec{L}^2(\Omega )}+a_1(\theta ^* +{\tilde{\theta }}_b:{\varvec{u} },\partial _t\varvec{u} )+J(\partial _t\varvec{u}) =\langle \varvec{f},\partial _t\varvec{u}\rangle , \end{aligned}$$

(2.37)

Then

$$\begin{aligned} \Vert \partial _t\varvec{u} \Vert ^2_{\varvec{L}^2(\Omega )} +\frac{1}{dt}\bigg [\frac{\nu _0}{2}a_1(.:{\varvec{u} },\varvec{u} )+ J (\varvec{u} )\bigg ]\le \frac{1}{2}\, \Vert \varvec{f}\Vert ^2_{\varvec{L}^{2}(\Omega )}+\frac{1}{2}\Vert \partial _t\varvec{u} \Vert ^2_{\varvec{L}^2(\Omega )}. \end{aligned}$$

(2.38)

So

$$\begin{aligned} \frac{1}{2}\Vert \partial _t\varvec{u} \Vert ^2_{\varvec{L}^2(\Omega )}+\frac{1}{dt}\bigg [\frac{\nu _0}{2}a_1(.:{\varvec{u} },\varvec{u} )+ J (\varvec{u} )\bigg ]\le \frac{1}{2}\, \Vert \varvec{f}\Vert ^2_{\varvec{L}^{2}(\Omega )}. \end{aligned}$$

(2.39)

Integration over the time interval (0, T) yields

$$\begin{aligned}{} & {} \frac{1}{2}\Vert \partial _t\varvec{u} \Vert _{L^2(0,T; \varvec{L}^2(\Omega ))}^2+\frac{\nu _0}{2} a_1(.:\varvec{u}(T), \varvec{u}(T))+J(\varvec{u}(T)) \nonumber \\{} & {} \quad \le \frac{1}{2}\Vert \varvec{f}\Vert _{L^2(0,T; \varvec{L}^2(\Omega ))}^2+\frac{\nu _0}{2} a_1(.:\varvec{u}_0,\varvec{u}_0)+J (\varvec{u}_0).\hspace{1cm} \end{aligned}$$

(2.40)

Owing to the positivity of $a_1(\cdot :\cdot ,\cdot )$ and $J(\cdot )$ defined in (2.9), we get

$$\begin{aligned} \Vert \partial _t\varvec{u} \Vert _{L^2(0,T; \varvec{L}^2(\Omega ))}^2\le \Vert \varvec{f}\Vert _{L^2(0,T; \varvec{L}^2(\Omega ))}^2+\nu _0\Vert \nabla \varvec{u}_0\Vert _{\varvec{L}^2(\Omega )}^2+2\int \limits _S g\,\left( |\varvec{u}_{\varvec{0,\tau }}|+1\right) \ dS. \end{aligned}$$

(2.41)

So

$$\begin{aligned} \Vert \partial _t\varvec{u} \Vert _{L^2(0,T; \varvec{L}^2(\Omega ))}^2\le \Vert \varvec{f}\Vert _{L^2(0,T; \varvec{L}^2(\Omega ))}^2+c\Vert \varvec{u}_0\Vert _{\varvec{H}^1(\Omega )}^2+c\Vert g\Vert ^2_{L^\infty (S)}+c\Vert g\Vert _{L^\infty (S)}. \end{aligned}$$

(2.42)

(3) Back to (2.40), we have

$$\begin{aligned} \nu _0\Vert \varvec{D}\varvec{u} \Vert _{L^\infty (0,T; \varvec{L}^2(\Omega ))}^2+2J(\varvec{u}(T))\le \Vert \varvec{f}\Vert _{L^2(0,T; \varvec{L}^2(\Omega ))}^2+\nu _0\Vert \nabla \varvec{u}_0\Vert ^2_{\varvec{L}^2(\Omega )}+2\int \limits _S g\,(|\varvec{u}_{\varvec{0,\tau }}|+1)\ dS. \qquad \end{aligned}$$

(2.43)

So

$$\begin{aligned} \nu _0\Vert \nabla \varvec{u} \Vert _{L^\infty (0,T; \varvec{L}^2(\Omega ))}^2\le \Vert \varvec{f}\Vert _{L^2(0,T; \varvec{L}^2(\Omega ))}^2+c\Vert \varvec{u}_0\Vert ^2_{\varvec{H}^1(\Omega )}+c\Vert g\Vert ^2_{L^\infty (S)}+c\Vert g\Vert _{L^\infty (S)}. \end{aligned}$$

(2.44)

(4) As for the pressure, we take successively in (2.25) $\varvec{v}=\varvec{0}$ and $\varvec{v}=2\varvec{u}$, such that $\varvec{u}\in \varvec{H}^1_0(\Omega )$. Comparing the two inequalities,

$$\begin{aligned} b(\varvec{u},p)= \langle \varvec{f},\varvec{u}\rangle - (\partial _{t} \varvec{u}, \varvec{u})- a_1(\theta ^*+{\tilde{\theta }}_b:{\varvec{u}},{\varvec{u}}), \end{aligned}$$

(2.45)

which with the inf-sup condition and the proprieties of $a_1(.:,.,.)$ in (2.13) gives

$$\begin{aligned} c\Vert p\Vert _{L^2_0(\Omega )}\le \displaystyle \underset{\varvec{v}\in \varvec{H}^1_0(\Omega )}{\sup } \displaystyle \frac{b(\varvec{u},p)}{\Vert \varvec{u}\Vert _{\varvec{H}^1(\Omega )}}\le \Vert \varvec{f}\Vert _{\varvec{H}^{-1}(\Omega )}+\Vert \partial _{{t}}\varvec{u}\Vert _{\varvec{L}^{2}(\Omega )}+2\nu _1\Vert \varvec{u}\Vert _{\varvec{H}^{1}(\Omega )}. \end{aligned}$$

(2.46)

(5) Concerning the temperature, by taking $\rho =\theta ^*$ in (2.26) we find

$$\begin{aligned} \big (\partial _{t}\theta ^*, \theta ^*)+ a_2(\theta ^* +{\tilde{\theta }}_{b,}:\theta ^* +{\tilde{\theta }}_{b,},\theta ^*) + d_2(\varvec{u},\theta ^*,\theta ^*)=\langle {k},\theta ^*\rangle -\big ( \partial _{t}{\tilde{\theta }}_{b}, \theta ^*)- d_2(\varvec{u},{\tilde{\theta }}_{b},\theta ^*). \end{aligned}$$

(2.47)

From (2.20) and application of the proprieties of $a_2(.:,.,.)$ in (2.14), one finds

$$\begin{aligned}(\partial _{t}\theta ^*,\theta ^* )+ (\mu (\theta ^* +{\tilde{\theta }}_b)\nabla \theta ^*,\nabla \theta ^* ) = \langle {k},\theta ^* \rangle -(\partial _{t}{\tilde{\theta }}_b\,,\theta ^*)-((\varvec{u} \cdot \nabla ) {\tilde{\theta }}_b,\theta ^* )-(\mu (\theta ^* +{\tilde{\theta }}_{b})\nabla {\tilde{\theta }}_{b},\nabla \theta ^* ). \end{aligned}$$

Using Hölder’s inequality on the right-hand side yields

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert \theta ^* \Vert ^2_{L^2(\Omega )}+\mu _0 c\Vert \theta ^ * \Vert ^2_{H^1(\Omega )}\le & {} c\bigg (\Vert {k}\Vert _{H^{-1}(\Omega )} \Vert \theta ^*\Vert _{H^ 1(\Omega )}+\Vert \partial _t{\tilde{\theta }}_{b}\Vert _{L^2(\Omega )}\Vert \theta ^*\Vert _{L^2(\Omega )} \nonumber \\{} & {} +\Vert \varvec{u}\Vert _{\varvec{H}^1(\Omega )}\Vert {\tilde{\theta }}_{b}\Vert _{H^1(\Omega )}\Vert \theta ^* \Vert _{H^1(\Omega )}+\mu _1\Vert {\tilde{\theta }}_{b}\Vert _{H^1(\Omega )}\Vert \theta ^*\Vert _{H^1(\Omega )}\bigg ).\qquad \end{aligned}$$

(2.48)

Young’s inequality yields the following bounds

$$\begin{aligned}{} & {} \Vert {k}\Vert _{H^{-1}(\Omega )} \Vert \theta ^*\Vert _{H^1(\Omega )} \le \frac{1}{2}\bigg [ \Vert {k}\Vert ^2_{H^{-1}(\Omega )}+\Vert \theta ^* \Vert ^2_{H^1(\Omega )} \bigg ], \end{aligned}$$

(2.49)

$$\begin{aligned}{} & {} \Vert \partial _t{\tilde{\theta }}_{b}\Vert _{L^2(\Omega )}\Vert \theta ^*\Vert _{L^2(\Omega )} \le \Vert \partial _t{\tilde{\theta }}_b\Vert _{L^2(\Omega )}\Vert \theta ^* \Vert _{H^1(\Omega )} \le \frac{1}{2}\bigg ( \Vert \partial _t{\tilde{\theta }}_{b}\Vert ^2_{L^2(\Omega )}+\Vert \theta ^* \Vert ^2_{H^1(\Omega )}\bigg ), \end{aligned}$$

(2.50)

$$\begin{aligned}{} & {} \Vert \varvec{u}\Vert _{\varvec{H}^1(\Omega )}\Vert {\tilde{\theta }}_{b}\Vert _{H^1(\Omega )}\Vert \theta ^*\Vert _ {H^1(\Omega )} \le \frac{c^\prime }{2}\left[ \Vert \varvec{u}\Vert ^2_{\varvec{H}^1(\Omega )}+\Vert {\tilde{\theta }}_{b}\Vert ^2_{H^1(\Omega )}\right] ^2 +\displaystyle \frac{\Vert \theta ^*\Vert ^2_{H^1(\Omega )}}{2}, \end{aligned}$$

(2.51)

and

$$\begin{aligned} \hspace{-2.7cm}\mu _1\Vert {\tilde{\theta }}_{b}\Vert _{H^1(\Omega )}\Vert \theta ^* \Vert _{H^1(\Omega )}\le & {} \frac{\mu _1}{2}\left[ \Vert {\tilde{\theta }}_{b}\Vert ^2_{H^1(\Omega )}+\Vert \theta ^* \Vert ^2_{H^1(\Omega )}\right] . \end{aligned}$$

(2.52)

So returning to (2.48), one gets

$$\begin{aligned} \frac{d}{dt} \Vert \theta ^* \Vert ^2_{L^2(\Omega )}+c\Vert \theta ^* \Vert ^2_{H^1(\Omega )}\le c\left( \Vert {k}\Vert ^2_{H^{-1}(\Omega )}+\Vert \partial _t{\tilde{\theta }}_{b}\Vert ^2_{L^2(\Omega )}+\Vert {\tilde{\theta }}_{b}\Vert ^2_{H^1(\Omega )} +\frac{c^\prime }{2} \left[ \Vert \varvec{u}\Vert ^2_{\varvec{H}^1(\Omega )}+\Vert {\tilde{\theta }}_{b}\Vert ^2_{H^1(\Omega )}\right] ^2 \right) . \end{aligned}$$

Integrating between 0 and T and recalling (2.24), we obtain

$$\begin{aligned}{} & {} \Vert \theta ^* (.,{T})\Vert ^2_{L^2(\Omega )}+c\int \limits _{0}^{{T}}\Vert \theta ^* (., s)\Vert ^2_{H^1(\Omega )} ds \le c\left( \Vert \theta ^* (.,0)\Vert ^2_{L^2(\Omega )}+ \int \limits _{ 0}^{{T}}\Vert {k}(.,s)\Vert ^2_{H^{-1}(\Omega )} ds \right. \\{} & {} \quad +\displaystyle \int \limits _{0}^{{T}}\Vert \partial _t{\tilde{\theta }}_{b}(.,s)\Vert ^2_{L^2(\Omega )} ds +\displaystyle \int \limits _{0}^{{T}}\Vert {\tilde{\theta }}_{b}(.,s){\Vert }^2_{H^{1/2}(\partial \Omega )}ds \\ \hspace{2cm}{} & {} \quad \left. +\displaystyle \frac{c^\prime }{2} \left[ \int \limits _{0}^{{T}}\Vert \varvec{u}(.,s)\Vert ^2_{\varvec{H}^1(\Omega )} ds+\int \limits _{0}^{{T}}\Vert {\tilde{\theta }}_{b}(.,s)\Vert ^2_{H^{1/2}(\partial \Omega )}ds\right] ^2\right) . \hspace{-2cm} \end{aligned}$$

Thus from (2.36), it results

$$\begin{aligned}{} & {} \Vert \theta ^* \Vert _{L^2(0,T;H^1(\Omega ))} \le c\bigg [ \Vert \theta _{0}\Vert _{L^2(\Omega )}+ \Vert {k}\Vert _{L^2(0,T;H^{-1}(\Omega ) )}+\Vert \partial _t{\tilde{\theta }}_b\Vert _{L^2(0,T;L^{2}(\Omega ) )}\nonumber \\{} & {} +\Vert {\tilde{\theta }}_{b}\Vert _{L^2(0,T;H^{1/2}(\partial \Omega ))}+\Vert {\tilde{\theta }}_{b}\Vert ^2_{H^1(0,T;H^{1/2}(\partial \Omega ))}+ \Vert \varvec{f}\Vert ^2_{L^2(0,T;\varvec{H}^{-1}(\Omega ))}+\Vert \varvec{u}_{0}\Vert ^2_{\varvec{L}^2(\Omega )} \bigg ]. \end{aligned}$$

(2.53)

Which completes the proof. $\square$

Theorem 2.1

Assume that the data $(\varvec{f},{k},\theta _b,g)$ satisfying (2.7) and the initial velocity $\varvec{u}_0$ belongs to $\varvec{H}^1(\Omega )$ and checks (2.8). That the initial temperature in $L^2(\Omega )$ and the temperature on the boundary $\theta _b$ in $H^{1 /2}(\partial \Omega )$. Then the problem (2.10) admits at least one solution $(\varvec{u},p,\theta )$ such that

$$\begin{aligned}\varvec{u}\in L^\infty (0,T;\varvec{V}(\Omega )),\hspace{0.5cm}p\in L^2(0,T;L_0^{2}(\Omega )),\hspace{0.5cm}\theta \in L^2(0,T;H_0^1(\Omega )),\end{aligned}$$

$$\begin{aligned} \frac{\partial \varvec{u}}{\partial t}\in L^2(0,T;\varvec{V}^{'}(\Omega ))\hspace{0.5cm} \text {if}\hspace{0.2cm} d=2, \end{aligned}$$

(2.54)

and

$$\begin{aligned} \frac{\partial \varvec{u}}{\partial t}\in L^{4/3}(0,T;\varvec{V}^{'}(\Omega ))\hspace{0.5cm} \text {if}\hspace{0.2cm} d=3. \end{aligned}$$

(2.55)

Proof

Step 1: Regularization:

Problem (2.10) is equivalent to

$$\begin{aligned} \hspace{-1.9cm}{\left\{ \begin{array}{ll} \text {Find} \ (\varvec{u},\theta ^*) \in \varvec{V}_{\text {div}} \times Q_0\ \text {such that:} \\ \varvec{u}(.,0)=\varvec{u}_0 \hspace{0.2cm}\text {and}\hspace{0.2cm} \theta ^*(.,0)=\theta _0-{\bar{\theta }}_b(.,0)\hspace{0.2cm}\text {in}\hspace{0.2cm} \Omega , \\ \forall \ (\varvec{v},\rho ) \in {\varvec{V}_{\text {div}}(\Omega )}\times H^1_0(\Omega ), \\ \big (\partial _{t} \varvec{u}, \varvec{v}-\varvec{u}\big )+a_1(\theta ^*+{\bar{\theta }}_b:{\varvec{u}},{\varvec{v}-\varvec{u}})+d_1(\varvec{u},\varvec{u},\varvec{v}-\varvec{u})+J(\varvec{v})-J(\varvec{u})\ge \langle \varvec{f},\varvec{v}-\varvec{u}\rangle , \\ \big (\partial _{t}\theta ^*, \rho \big )+ a_2(\theta ^* +{\bar{\theta }}_b:\theta ^* +{\bar{\theta }}_b,\rho ) + d_2(\varvec{u},\theta ^*,\rho ) = \langle k,\rho \rangle -\big ( \partial _{t}{\bar{\theta }}_b, \rho \big )- d_2(\varvec{u},{\bar{\theta }}_b,\rho ). \end{array}\right. } \end{aligned}$$

(2.56)

We introduce the functional

$$\begin{aligned}{} & {} J_{\varepsilon }: \varvec{V}(\Omega ) \longrightarrow {\mathbb {R}} \\{} & {} \varvec{v} \longrightarrow J_{\varepsilon }(\varvec{v})=\int \limits _S g\sqrt{|\varvec{v}_{\varvec{\tau }}|^{2}+{\varepsilon }^2}\ dS,\quad 0<\varepsilon<<1. \end{aligned}$$

(2.57)

We observe that

• (a) $J_{\varepsilon }$ is convex and differentiable, with Gateaux-derivative $J^{'}_{\varepsilon }:\varvec{V}(\Omega ) \longrightarrow \varvec{V}^{'}(\Omega )$ given by

  $$\begin{aligned} \langle J^{'}_{\varepsilon }(\varvec{u}),\varvec{v} \rangle =\int \limits _S g \dfrac{ \varvec{u}_{\tau }\cdot \varvec{v}_{\tau } }{\sqrt{|\varvec{u}_{\varvec{\tau }}|^{2}+{\varepsilon }^2}}\ dS. \end{aligned}$$

  (2.58)

  .

• (b) $J^{'}_{\varepsilon }$ is monotone, that is

  $$\begin{aligned} \forall \varvec{u},\varvec{v}\in \varvec{V}(\Omega ),\quad \langle J^{'}_{\varepsilon }(\varvec{u})-J^{'}_{\varepsilon }(\varvec{v}),\varvec{u}-\varvec{v} \rangle \ge 0. \end{aligned}$$

• (c) For all $\varvec{v}\in \varvec{V}(\Omega ),\ J_{\varepsilon }(\varvec{v}) \xrightarrow [\varepsilon \longrightarrow 0]{} J(\varvec{v}).$

  Indeed,

  $$\begin{aligned} J_{\varepsilon }(\varvec{v})- J(\varvec{v})= \int \limits _S g \dfrac{\varepsilon ^2}{\sqrt{|\varvec{v}_{\varvec{\tau }}|^{2}+{\varepsilon }^2}+|\varvec{v}_{\varvec{\tau }}|}\ dS. \end{aligned}$$

The regularized problem then takes the following form:

$$\begin{aligned} \hspace{-1.9cm}{\left\{ \begin{array}{ll} \text {Find} \ (\varvec{u}_{\varepsilon },\theta ^*_{\varepsilon }) \in \varvec{V}_{\text {div}} \times Q_0\ \text {such that:} \\ \varvec{u}_{\varepsilon }(.,0)=\varvec{u}_0 \hspace{0.2cm}\text {and}\hspace{0.2cm} \theta _{\varepsilon }^*(.,0)=\theta _0-{\bar{\theta }}_b(.,0)\hspace{0.2cm}\text {in}\hspace{0.2cm} \Omega , \\ \forall \ (\varvec{v},\rho ) \in {\varvec{V}_{\text {div}}(\Omega )}\times H^1_0(\Omega ), \\ \big (\partial _{t} \varvec{u}_{\varepsilon }, \varvec{v}\big )+a_1(\theta ^*_{\varepsilon }+{\bar{\theta }}_b:{\varvec{u}_{\varepsilon }},{\varvec{v}})+d_1(\varvec{u}_{\varepsilon },\varvec{u}_{\varepsilon },\varvec{v})+\langle J^{'}_{\varepsilon }(\varvec{u}_{\varepsilon }),\varvec{v} \rangle = \langle \varvec{f},\varvec{v }\rangle , \\ \big (\partial _{t}\theta ^*_{\varepsilon }, \rho \big )+ a_2(\theta ^*_{\varepsilon } +{\bar{\theta }}_b:\theta ^*_{\varepsilon } +{\bar{\theta }}_b,\rho ) + d_2(\varvec{u}_{\varepsilon },\theta ^*_{\varepsilon },\rho ) = \langle k,\rho \rangle -\big ( \partial _{t}{\bar{\theta }}_b, \rho \big )- d_2(\varvec{u}_{\varepsilon },{\bar{\theta }}_b,\rho ). \end{array}\right. } \end{aligned}$$

(2.59)

Step 2: Galerkin approximation:

The spaces $\varvec{V}_{\text {div}}(\Omega )$ and $H^1_0(\Omega )$ are separable, there exists an increasing sequence $(\varvec{V}^{m}_{\text {div}}(\Omega ))_m$ of finite dimensional subspaces of $\varvec{V}_{\text {div}}(\Omega )$ and an increasing sequence $(W^m(\Omega ))_m$ of finite-dimensional subspaces of $H^1_0(\Omega )$ such that $\underset{m\in {\mathbb {N}}}{\bigcup } \varvec{V}^m_{\text {div}}(\Omega )\times W^m(\Omega )$ is dense in $\varvec{V}_{\text {div}}(\Omega )\times H^1_0(\Omega )$. The Galerkin Problem is

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {Find} \ (\varvec{u}^{m}_{\varepsilon },\theta ^{*m}_{\varepsilon }) \in L^2(0,T;\varvec{V}^{m}_{\text {div}}(\Omega ))\cap L^\infty (0,T;\varvec{L}^2(\Omega ))\times L^2(0,T;W^m(\Omega ))\ \text {such that:} \\ \varvec{u}^m_{\varepsilon }(.,0)=\varvec{u}_0 \hspace{0.2cm}\text {and}\hspace{0.2cm} \theta ^{*m}_{\varepsilon }(.,0)=\theta _0-{\bar{\theta }}_b(.,0)\hspace{0.2cm}\text {in}\hspace{0.2cm} \Omega , \\ \forall \ (\varvec{v},\rho ) \in {\varvec{V}^{m}_{\text {div}}(\Omega )}\times W^m(\Omega ), \\ \big (\partial _{t} \varvec{u}^m_{\varepsilon }, \varvec{v}\big )+a_1(\theta ^{*m}_{\varepsilon }+{\bar{\theta }}_b:{\varvec{u}^m_{\varepsilon }},{\varvec{v}})+d_1(\varvec{u}^m_{\varepsilon },\varvec{u}^m_{\varepsilon },\varvec{v})+\langle J^{'}_{\varepsilon }(\varvec{u}^m_{\varepsilon }),\varvec{v} \rangle = \langle \varvec{f},\varvec{v}\rangle , \\ \big (\partial _{t}\theta ^{*m}_{\varepsilon }, \rho \big )+ a_2(\theta ^{*m}_{\varepsilon } +{\bar{\theta }}_b:\theta ^{*m}_{\varepsilon } +{\bar{\theta }}_b,\rho ) + d_2(\varvec{u}^{m}_{\varepsilon },\theta ^{*m}_{\varepsilon },\rho ) = \langle k,\rho \rangle -\big ( \partial _{t}{\bar{\theta }}_b, \rho \big )- d_2(\varvec{u}^m_{\varepsilon },{\bar{\theta }}_b,\rho ). \end{array}\right. } \end{aligned}$$

(2.60)

The mapping

$$\begin{aligned} (\varvec{w},z) \mapsto \left( \begin{array}{c} \varvec{f}+\text {div}\ (2\nu (z +{\bar{\theta }}_b)\varvec{D}\varvec{w})-(\varvec{w}\cdot \nabla )\varvec{w}-J^{'}_{\varepsilon }(\varvec{w}) \\ k-\partial _{t}{\bar{\theta }}_b- (\varvec{w} \cdot \nabla ){\bar{\theta }}_b-\text {div}\ (\mu (z +{\bar{\theta }}_b)\nabla (z +{\bar{\theta }}_b))-(\varvec{w} \cdot \nabla )z \end{array}\right) \end{aligned}$$

(2.61)

is locally Lipschitz in $\varvec{H}^1(\Omega )\times H^1(\Omega )$. It follows then from Cauchy-Lipschitz’s theorem that (2.60) has a unique solution $(\varvec{u}^{m}_{\varepsilon },\theta ^{*m}_{\varepsilon })$.

Step 3: A priori estimates and passage to the limit:

The a priori estimates obtained in proposition 2.2 will also hold in the discrete setting $\varvec{V}^m_{\text {div}}(\Omega )\times W^m(\Omega ).$ Hence one can pass to the limit exactly as in reference [1] with respect to m.

The result will give us now a sequence depending only on $\varepsilon$, that is

$$\begin{aligned} \hspace{-0.9cm}{\left\{ \begin{array}{ll} \text {Find} \ (\varvec{u}_{\varepsilon },\theta ^*_{\varepsilon }) \in \varvec{V}_{\text {div}} \times Q_0\ \text {such that:} \\ \varvec{u}_{\varepsilon }(.,0)=\varvec{u}_0 \hspace{0.2cm}\text {and}\hspace{0.2cm} \theta _{\varepsilon }^*(.,0)=\theta _0-{\bar{\theta }}_b(.,0)\hspace{0.2cm}\text {in}\hspace{0.2cm} \Omega , \\ \forall \ (\varvec{v},\rho ) \in {\varvec{V}_{\text {div}}(\Omega )}\times H^1_0(\Omega ), \\ \big (\partial _{t} \varvec{u}_{\varepsilon }, \varvec{v}-\varvec{u}_{\varepsilon }\big )+a_1(\theta ^*_{\varepsilon }+{\bar{\theta }}_b:{\varvec{u}_{\varepsilon }},{\varvec{v}-\varvec{u}_{\varepsilon }})+d_1(\varvec{u}_{\varepsilon },\varvec{u}_{\varepsilon },\varvec{v}-\varvec{u}_{\varepsilon })+J_{\varepsilon }(\varvec{v})-\langle \varvec{f},\varvec{v}-\varvec{u}_{\varepsilon }\rangle \ge J_{\varepsilon }(\varvec{u}_{\varepsilon }), \hspace{-0.5cm} \\ \big (\partial _{t}(\theta ^*_{\varepsilon }+{\bar{\theta }}_b), \rho \big )+ a_2(\theta ^*_{\varepsilon } +{\bar{\theta }}_b:\theta ^*_{\varepsilon } +{\bar{\theta }}_b,\rho ) + d_2(\varvec{u}_{\varepsilon },\theta ^*_{\varepsilon }+{\bar{\theta }}_b,\rho ) = \langle k,\rho \rangle . \end{array}\right. } \end{aligned}$$

(2.62)

which is equivalent to (2.59).

Again the a priori estimates obtained in Proposition 2.2 are valid. Hence one can pass to the limit as $\varepsilon \rightarrow 0$ with the same process in [10]. And by arguing as in [11] (see p 56–57), we ensure the existence of solutions claimed in Theorem 2.1. $\square$

3 Discretization

In this section, we propose a space-time discretization of the problem (2.10), derive and prove an a priori error estimation. To discretize the time derivative of velocity and temperature, we did call to the semi-implicit Euler method. We introduce a partition of the interval [0, T] into intervals $[t^{n-1},t^{n}],\ n\ge 1$ and we denote by

$$\begin{aligned} \Delta t=t^{n}-t^{n-1} \end{aligned}$$

the time discretization step such that $t^{n}=n\Delta t$. We approximate the derivative $\partial _t\varvec{u}(t^{n})$ and $\partial _t\theta (t^{n})$ by the differential quotient of order $\Delta t$,

$$\begin{aligned} \partial _t\varvec{u}^{n}\approx \frac{\varvec{u}^{n}-\varvec{u}^{n-1}}{\Delta t}\hspace{0.5cm}\text {and}\hspace{0.5cm}\partial _t\theta ^{n}\approx \frac{\theta ^{n}-\theta ^{n-1}}{\Delta t}. \end{aligned}$$

(3.1)

For the space discretization, we propose the finite element method. We assume that $\Omega$ is a polygon when $d = 2$ or polyhedron when $d = 3$, so it can be completely meshed. We denote by ${\mathcal {T}}_{h}$ a regular family of triangulations of $\Omega$ by closed triangles K. The elements ${\mathcal {T}}_{h}$ is a set of closed non degenerate triangles or tetrahedra, satisfying,

• For each h, $\Omega$ is the union of all elements of ${\mathcal {T}}_{h}$,

• The intersection of two distinct elements of ${\mathcal {T}}_{h}$ is either empty, a common vertex, or an entire common edge or face,

• The ratio of the diameter of an element K in ${\mathcal {T}}_{h}$ to the diameter of its inscribed circle or ball is bounded by a constant independent of h such that $h=\underset{K\in {\mathcal {T}}_h}{\max }\ \text {diam} K$ denotes the mesh size.

We define the space ${\mathcal {P}}_l(K)$ of polynomials of degree less than or equal to l on K. And we use “P2/P1” pair of finite element, that is

$$\begin{aligned} \varvec{V}_h(\Omega )= & {} \big \{\varvec{v}_h\in {\mathcal {C}}({\bar{\Omega }})^2\cap \varvec{V}(\Omega ),\ \text {for all}\ K\in {\mathcal {T}}_h,\ \varvec{v}_h|_K\in {\mathcal {P}}_2(K)^2\big \}, \\ L^2_{h}(\Omega )= & {} \big \{q_h\in L^2(\Omega )\cap {\mathcal {C}}({\bar{\Omega }}),\ \text {for all}\ K\in {\mathcal {T}}_h,\ q_h|_K\in {\mathcal {P}}_1(K) \big \}, \\ H^{1}_h(\Omega )= & {} \big \{ \rho _h\in H^1(\Omega )\cap {\mathcal {C}}({\bar{\Omega }}),\ \text {for all}\ K\in {\mathcal {T}}_h, \ \rho _h|_K\in {\mathcal {P}}_1(K) \big \}. \end{aligned}$$

We let

$$\begin{aligned} L^2_{0h}(\Omega )= L^2_{h}(\Omega )\cap L^2_{0}(\Omega ) \hspace{0.5cm}\text {and}\hspace{0.5cm} {H}^1_{0h}(\Omega )={H}^1_h(\Omega )\cap {H}^1_0(\Omega ). \end{aligned}$$

We introduce the interpolation operator $I_h$ such that, for any function $\varphi$ continue on ${\bar{\Omega }},\ I_h$ checks

$$\begin{aligned}I_h\varphi (\varvec{x},.)=\varphi (\varvec{x},.)\end{aligned}$$

We note $i_h^{\partial \Omega }$ the lagrange interpolation operator such that, for any function $\varphi$ continue on $\partial \Omega$, $i_h^{\partial \Omega }$ checks

$$\begin{aligned}i_h^{\partial \Omega }\varphi (\varvec{x},.)=\varphi (\varvec{x},.)\end{aligned}$$

There exists an approximation operator $P_h\in {\mathcal {L}}(\varvec{H}^1_0(\Omega ),\varvec{V}_h(\Omega ))$ and $Q_h\in {\mathcal {L}}(\varvec{H}^1(\Omega ),L^2_{0h}(\Omega ))$ such that (see for instance Takahito in [22] and V.Girault and P.Raviart in [14])

$$\begin{aligned} \forall \varvec{v}\in H^1_0(\Omega ),\quad \forall q_h\in L^2_h(\Omega ){,} \int \limits _{\Omega } q_h\text {div}(P_h(\varvec{v})-\varvec{v})\, d\varvec{x}=0. \end{aligned}$$

(3.2)

For $k=0$ or $k=1,$

$$\begin{aligned} \forall \varvec{v}\in [\varvec{H}^{1+k}(\Omega )\cap \varvec{H}_0^1(\Omega )],\quad \Vert P_h(\varvec{v})-\varvec{v}\Vert _{\varvec{L}^2(\Omega )} \le C_1h^{1+k}|\varvec{v}|_{\varvec{H}^{1+k}(\Omega )}. \end{aligned}$$

(3.3)

For all $r\ge 2,$ $k=0$ or $k=1,$

$$\begin{aligned} \forall \varvec{v}\in [\varvec{W}^{1+k,r}(\Omega )\cap \varvec{H}_0^1(\Omega )],\quad |P_h(\varvec{v})-\varvec{v}|_{\varvec{W}^{1,r}(\Omega )} \le C_2h^{k}|\varvec{v}|_{\varvec{W}^{1+k,r}(\Omega )}. \end{aligned}$$

(3.4)

$$\begin{aligned} \forall q\in H^1(\Omega ),\quad \Vert Q_hq-q\Vert _{L^2(\Omega )} \le C_3h|q|_{H^{1}(\Omega )}. \end{aligned}$$

(3.5)

In addition, there exists an approximation operator $R_h$ in ${\mathcal {L}}(W^{1,p}(\Omega ),H^1_h(\Omega ))$,(see for instance, Scott and Zhang [18]) such that for $m=0,1, l=0,1$ and for all $p\ge 2,$

$$\begin{aligned} \forall \rho \in {W}^{l+1,p}(\Omega ),\quad |R_h(\rho )-\rho |_{{W}^{m,p}(\Omega )} \le C_3h^{l+1-m}|\rho |_{{W}^{l+1,p}(\Omega )}. \end{aligned}$$

(3.6)

In order to introduce the discrete scheme, we define the following forms:

For all $\varvec{u}_h,\varvec{v}_h,\varvec{w}_h\in \varvec{V}_h(\Omega )$ and $\theta _h,\rho _h\in \varvec{H}_h^1(\Omega )$

$$\begin{aligned}{} & {} d_{1h}\left( \varvec{u}_h, \varvec{v}_h, \varvec{w}_h\right) =d_1\left( \varvec{u}_h, \varvec{v}_h, \varvec{w}_h\right) +\frac{1}{2}\left( \left( {\text {div}} \varvec{u}_h\right) \varvec{v}_h, \varvec{w}_h\right) , \\{} & {} d_{2h}\left( \varvec{u}_h, \theta _h, \rho _h\right) =d_2\left( \varvec{u}_h, \theta _h, \rho _h\right) +\frac{1}{2}\left( \left( {\text {div}} \varvec{u}_h\right) \theta _h, \rho _h\right) . \end{aligned}$$

The finite element problem reads;

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {Find} \, \varvec{u}_h^{n}\in \varvec{V}_h(\Omega ),\, p^{n}_h\in L^2_{0h}(\Omega )\, \text {and}\, \theta ^{n}\in \varvec{H}_h^1(\Omega )\ \text {such that:} \\ \varvec{u}^0_h=\varvec{I}_h\varvec{u}_{0} \hspace{0.2cm}\text {and}\hspace{0.2cm} \theta _h^0=I_h\theta _{0}\hspace{0.2cm}\text {in}\hspace{0.2cm} \Omega , \\ \theta ^n_h=i_h^{\partial \Omega }\theta ^n_b\hspace{0.2cm}\text {in}\ \partial \Omega , \quad \forall \ 1\le n\le N, \\ \forall \, (\varvec{v}_h,q_h,\rho _h) \in {\varvec{V}_h(\Omega )}\times L^2_{0h}(\Omega )\times \varvec{H}^1_{0h}(\Omega ), \\ \displaystyle \frac{1}{\Delta t}(\varvec{u}_h^{n}, \varvec{v}_h-\varvec{u}_h^{n})+a_1(\theta _h^{n-1}:{\varvec{u}_h^{n}},\varvec{v}_h-\varvec{u}_h^{n})+b(\varvec{v}_h-\varvec{u}_h^{n},p_h^{n}) \\ \hspace{1.5cm}+d_{1h}(\varvec{u}_h^{n-1},\varvec{u}^{n}_h,\varvec{v}_h-\varvec{u}_h^{n}) +J(\varvec{v}_h)-J(\varvec{u}_h^{n})\ge \langle \varvec{f}^{n},\varvec{v}_h-\varvec{u}_h^{n}\rangle -\displaystyle \frac{1}{\Delta t}(\varvec{u}_h^{n-1},\varvec{v}_h-\varvec{u}_h^{n}), \\ b(\varvec{u}_h^{n},q_h)=0, \\ \displaystyle \frac{1}{\Delta t}(\theta _h^{n},\rho _h)+a_2(\theta ^{n-1}_h:\theta ^{n}_h,\rho _h) +d_{2h}(\varvec{u}_h^{n},\theta _h^{n},\rho _h)= \langle {k}^n,\rho _h\rangle -\displaystyle \frac{1}{\Delta t}(\theta _h^{n-1},\rho _h). \end{array}\right. } \end{aligned}$$

(3.7)

where $\varvec{f}^{n}$ and $k^n$ are given as

$$\begin{aligned} \varvec{f}^n=\frac{1}{\Delta t}\int \limits _{t^{n-1}}^{t^n} \varvec{f}(s) ds \quad \text {and}\quad k^n=\frac{1}{\Delta t}\int \limits _{t^{n-1}}^{t^n} k(s) ds. \end{aligned}$$

(3.8)

with

Remark 3.1

The trilinear forms $d_{1\,h}(\cdot , \cdot , \cdot )$ and $d_{2\,h}(\cdot , \cdot , \cdot )$ enjoys due to the stability relations

$$\begin{aligned} d_{1h}\left( \varvec{u}_h, \varvec{v}_h, \varvec{v}_h\right) =d_{2h}\left( \varvec{u}_h, \theta _h, \theta _h\right) =0 \quad \forall \, (\varvec{u}_h,\varvec{v}_h)\in \varvec{V}_h(\Omega )^2, \theta _h\in \varvec{H}_h^1(\Omega ). \end{aligned}$$

$\triangleright$ We recall that the discrete version of inf-sup condition (2.16) holds: there exists $C>0$ independent of h such that

$$\begin{aligned} \sup _{\varvec{v}_h\in {\varvec{V}_h(\Omega )}} \displaystyle \frac{b(\varvec{v}_h, q_h)}{\Vert \varvec{v}_h\Vert _{\varvec{V}_h(\Omega )}} \geqslant C\Vert q_h\Vert _{L^2(\Omega )}\quad \forall q_h\in L^{2}_{0h}(\Omega ). \end{aligned}$$

(3.9)

Theorem 3.1

Under hypothesis H1–H3, at each time step n, for a given $\varvec{u}_h^{n-1}\in \varvec{V}_h(\Omega ), p^{n-1}_h\in L^2_{0\,h}(\Omega )$ and $\theta ^{n-1}_h\in \varvec{H}_h^1(\Omega )$, the variational problem (3.7) admits at least one solution $(\varvec{u}_h^n,p_h^n,\theta _h^n)$ which verifies, for $m=1,...,N$ the following bounds

$$\begin{aligned} \Vert \varvec{u}^{m}\Vert ^2_{\varvec{H}^1(\Omega )}+ 2\gamma \nu _0 C_\Omega \Delta t \sum _{n=1}^{m} \Vert \varvec{Du}^{n}\Vert ^2_{\varvec{L}^2(\Omega )} \le \Vert \varvec{u}_0\Vert ^2_{\varvec{H}^1(\Omega )} + \frac{2\gamma \Delta t}{\nu _0}\, \sum _{n=1}^{m}\Vert \theta ^{n-1}\Vert ^2_{\varvec{H}^{1}(\Omega )}. \end{aligned}$$

(3.10)

and

$$\begin{aligned} \Vert \theta ^{m}\Vert ^2_{H^1(\Omega )}+ C_\Omega \mu _0\Delta t \sum _{n=1}^{m} \Vert \theta ^{n}\Vert ^2_{{H}^1(\Omega )} \le \Vert \theta _0\Vert ^2_{H^1(\Omega )}+\frac{\Delta t}{C_\Omega \mu _0} \sum _{n=1}^{m}\Vert k\Vert ^2_{H^{-1}(\Omega )}. \end{aligned}$$

(3.11)

Similar to the continuous problem, the following characterization of the solution $(\varvec{u}, p,\theta )$ of the problem (2.23) holds. Then, problem (3.7) is equivalent to

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {Find} \, \varvec{u}_h^{n}\in \varvec{V}_h(\Omega ), p^{n}_h\in L^2_{0h}(\Omega ),\, \theta ^{n}_h\in \varvec{H}_{h}^1(\Omega ) \ \text {and}\ \varvec{\lambda }^{n}_h\in \varvec{\Lambda }_h \text {such that:} \\ \varvec{u}^0_h=\varvec{I}_h\varvec{u}_{0} \hspace{0.2cm}\text {and}\hspace{0.2cm} \theta _h^0=I_h\theta _{0}\hspace{0.2cm}\text {in}\hspace{0.2cm} \Omega , \\ \theta ^n_h=i_h^{\partial \Omega }\theta ^n_b\hspace{0.2cm}\text {in}\ \partial \Omega ,\quad \forall \ 1\le n\le N, \\ \forall \ (\varvec{v}_h,q_h,\rho _h) \in {\varvec{V}_h(\Omega )}\times L^2_{0h}(\Omega )\times \varvec{H}^1_{0h}(\Omega ), \\ \displaystyle \frac{1}{\Delta t}(\varvec{u}_h^{n}, \varvec{v}_h)+a_1(\theta _h^{n-1}:{\varvec{u}_h^{n}},\varvec{v}_h)+b(\varvec{v}_h,p_h^{n})+d_{1h}(\varvec{u}_h^{n-1},\varvec{u}^{n}_h,\varvec{v}_h) \\ \hspace{5cm} = \langle \varvec{f}^{n-1},\varvec{v}_h\rangle - \displaystyle \int _S g\varvec{\lambda }^{n-1}_h\cdot \varvec{v}_\tau \ dS-\displaystyle \frac{1}{\Delta t}(\varvec{u}_h^{n-1}, \varvec{v}_h), \\ b(\varvec{u}_h^{n},q_h)=0, \\ \varvec{u}^{n}_h\cdot \varvec{\lambda }^{n-1}_h=|\varvec{\lambda }^{n}_h|, \\ \displaystyle \frac{1}{\Delta t}(\theta _h^{n},\rho _h)+a_2(\theta ^{n-1}_h:\theta ^{n}_h,\rho _h) +d_{2h}(\varvec{u}_h^{n},\theta _h^{n},\rho _h)= \langle {k},\rho _h\rangle -\displaystyle \frac{1}{\Delta t}(\theta _h^{n-1},\rho _h). \end{array}\right. } \end{aligned}$$

And we let $\varvec{\Lambda }_h=\varvec{\Lambda }\cap \varvec{L}_h.$

Such that,

$$\begin{aligned} \varvec{L}_h=\big \{ \varvec{\alpha }_h|\varvec{\alpha }_h, \varvec{\alpha }_h=\sum _e \varvec{\alpha }_e {\mathcal {I}}_e, \varvec{\alpha }_e\in {\mathbb {R}}^2,\ {\forall }\, \text {edge}\ {e}\ \text {in}\ {S} \big \}, \end{aligned}$$

where ${\mathcal {I}}_e$ is the characteristic function of edge e.

3.1 A Priori Error Estimate

In this section, we establish the a priori estimates corresponding to the proposed numerical schemes. We begin by establishing the error estimates corresponding to the temperature, and then we will establish the corresponding to the velocity and the pressure.

In all the rest of the paper, we denote by $\varvec{u}^n=\varvec{u}(t^n), p^n=p(t^n)$ and $\theta ^n=\theta (t^n).$ And for the simplicity of the establishment of the a priori error estimates, we consider from now on $\theta _h^0 = R_h\theta (0)$ and $\varvec{u}^0_h=P_h\varvec{u}(0).$

Theorem 3.2

Let $(\varvec{u},p,\theta )$ be the solution of problem (2.10) and $(\varvec{u}^n_h,p^n_h,\theta ^n_h)$ be the solution of problem (3.7). Under hypothesis $H1-H3$, there exist two positive constants $C^\theta _1$ and $C^\theta _2$ independent on h and $\Delta t$ such that, for all $m \le N$,

$$\begin{aligned}{} & {} \frac{1}{2}\left\| \theta _h^m-R_h \theta ^m\right\| _{L^2(\Omega )}^2+\mu _0\sum _{n=1}^m \Delta t \left| \theta _h^n-R_h \theta ^n\right| _{H^1(\Omega )}^2 \nonumber \\{} & {} \qquad +\frac{1}{2} \sum _{n=1}^m\left\| \theta _h^n-R_h \theta ^n-\left( \theta _h^{n-1}-R_h \theta ^{n-1}\right) \right\| _{L^2(\Omega )}^2 \nonumber \\{} & {} \quad \le C^\theta _1\left( h^2+\Delta t^2\right) +C^\theta _2 \sum _{n=1}^m \Delta t \left| {\textbf{u}}_h^n-P_h {\textbf{u}}^n\right| ^2_{\varvec{H}^1(\Omega )} \end{aligned}$$

(3.12)

Proof

We consider the heat equation of the continuous variational problem (2.10), for all $\rho _h \in H^1_{0h}(\Omega )$:

$$\begin{aligned} \left( \frac{\partial \theta }{\partial t}, \rho _h\right) + a_2(\theta :\theta ,\rho _h) + d_2(\varvec{u},\theta ,\rho _h)= \langle k,\rho _h\rangle . \end{aligned}$$

(3.13)

Setting $\rho _h=\rho _h^n=\theta _h^n-R_h \theta ^{\text {n}}$ in (3.13) integrated over $[t^{n-1},t^n]$ by the definition of the operator $R_h$ gives

$$\begin{aligned} \left( \theta ^{n}-\theta ^{n-1}, \rho _h^n\right) +\left( \int \limits _{t^{n-1}}^{t^n} \mu (\theta ) \nabla \theta (t) dt, \nabla \rho _h^n\right) +\left( \int \limits _{t^{n-1}}^{t^n}({\textbf{u}}(t) \nabla ) \cdot \theta (t) d t, \rho _h^n\right) =\left( \int \limits _{t^{n-1}}^{t^n} k(t) d t,\rho _h^n\right) \qquad \end{aligned}$$

(3.14)

Then, we subtract (3.14) with the third equation of (3.7) after multiplying it by the time step $\Delta t$ and by using the definition of $k^n$ by (3.8), we obtain

$$\begin{aligned}{} & {} \left( \left( \theta _h^{\text {n}}-\theta _h^{n-1}\right) -\left( \theta ^n-\theta ^{\text {n}-1}\right) , \rho _h^n\right) + \left\{ \Delta t\, a_2(\theta ^{n-1}_h:\theta ^{n}_h,\rho ^n_h)-\left( \int \limits _{t^{n-1}}^{t^n} \mu (\theta ) \nabla \theta (t) dt, \nabla \rho _h^n \right) \right\} \nonumber \\{} & {} \quad +\left\{ \Delta t \,d_{2h}(\varvec{u}_h^{n},\theta _h^{n},\rho ^n_h)-\left( \int \limits _{t^{n-1}}^{t^n}({\textbf{u}}(t) \nabla ) \cdot \theta (t) d t, \rho _h^n\right) \right\} =0. \end{aligned}$$

(3.15)

We denote respectively by $b_1$, $b_2$ and $b_3$ the terms in the left-hand side of (3.15) as following:

$$\begin{aligned} b_1= & {} \left( \left( \theta _h^{\text {n}}-\theta _h^{n-1}\right) -\left( \theta ^n-\theta ^{\text {n}-1}\right) , \rho _h^n\right) , \\ b_2= & {} \Delta t\, a_2(\theta ^{n-1}_h:\theta ^{n}_h,\rho ^n_h)-\left( \int \limits _{t^{n-1}}^{t^n} \mu (\theta ) \nabla \theta (t) dt, \nabla \rho _h^n \right) , \end{aligned}$$

and

$$\begin{aligned} b_3=\Delta t \,d_{2h}(\varvec{u}_h^{n},\theta _h^{n},\rho ^n_h)-\left( \int \limits _{t^{n-1}}^{t^n}({\textbf{u}}(t) \nabla ) \cdot \theta (t) d t, \rho _h^n\right) . \end{aligned}$$

The first term $b_1$ can be bounded by inserting $\pm R_h \theta ^n$ and $\pm R_h \theta ^{n-1}$, we obtain

$$\begin{aligned}{} & {} \left( \left( \theta _h^n-\theta _h^{n-1}\right) -\left( \theta ^n-\theta ^{n-1}\right) , \rho _h^n\right) = \frac{1}{2}\left\| \rho _h^n\right\| _{L^2(\Omega )}^2-\frac{1}{2}\left\| \rho _h^{n-1}\right\| _{L^2(\Omega )}^2+\frac{1}{2}\left\| \rho _h^n-\rho _h^{n-1}\right\| _{L^2(\Omega )}^2 \nonumber \\{} & {} \quad +\left( \left( \theta ^{n-1}-R_h \theta ^{n-1}\right) -\left( \theta ^n-R_h \theta ^n\right) , \rho _h^n\right) . \end{aligned}$$

(3.16)

Noting that $\left( R_h\theta \right) ^{\prime }=R_h \theta ^{\prime }$, the last term of the previous equality can be bounded for any $\tilde{\varepsilon }_1>0$ as follows:

$$\begin{aligned} \left| \left( \left( \theta ^{n-1}-R_h \theta ^{n-1}\right) -\left( \theta ^n-R_h \theta ^n\right) , \rho _h^n\right) \right|= & {} \left| -\left( \int \limits _{t^{n-1}}^{t^n}\left( R_h \theta ^{\prime }-\theta ^{\prime }\right) (t) d t, \rho _h^n\right) \right| \\\le & {} \frac{c_1^2 h^2}{2 \tilde{\varepsilon }_1} \int \limits _{t^{n-1}}^{t^n}\left\| \theta ^{\prime }(t)\right\| _{H^1(\Omega )}^2 d t+\frac{\Delta t\, \tilde{\varepsilon }_1}{2}\left\| \rho _h^n\right\| _{L^2(\Omega )}^2 \\\le & {} \frac{c_1^2 h^2}{2 \tilde{\varepsilon }_1}\left\| \frac{\partial \theta }{\partial t}\right\| _{L^2\left( t^{n-1}, t^n; H^1(\Omega )\right) }^2+C_{\Omega } \frac{\Delta t\, \tilde{\varepsilon }_1}{2}\left| \rho _h^n\right| _{H_0^1(\Omega )}^2. \end{aligned}$$

where $C^1_{\Omega }$ is the square of the norm of the imbedding of ${H}_0^1(\Omega )$ into $L^2(\Omega )$.

The second term $b_2$ can be bounded, by inserting

$$\begin{aligned} \pm \left( \Delta t\, \mu \left( \theta _h^{n-1}\right) \nabla R_h \theta ^n, \nabla \rho _h^n\right) ,\pm \left( \int \limits _{t^{n-1}}^{t^n} \mu \left( \theta _h^{n-1}\right) \nabla R_h \theta (t), \nabla \rho _h^n\right) \text {and} \pm \left( \int \limits _{t^{n-1}}^{t^n} \mu (\theta (t)) \nabla R_h \theta (t) d t, \nabla \rho _h^n\right) . \end{aligned}$$

Then

$$\begin{aligned} b_2{} & {} = \left( \Delta t \, \mu \left( \theta _h^{n-1}\right) \nabla \left( \theta _h^n-R_h \theta ^n\right) , \nabla \rho _h^n\right) \\{} & {} \quad + \left( \Delta t \, \mu \left( \theta _h^{n-1}\right) \nabla R_h \theta ^n-\int \limits _{t^{n-1}}^{t^n} \mu \left( \theta _h^{n-1}\right) \nabla R_h \theta (t) d t, \nabla \rho _h^n\right) \\{} & {} \quad + \left( \int \limits _{t^{n-1}}^{t^n}\mu \left( \theta _h^{n-1}\right) \nabla R_h \theta (t) d t-\int \limits _{t^{n-1}}^{t^n} \mu (\theta (t)) \nabla R_h \theta (t) d t, \nabla \rho _h^n\right) \\{} & {} \quad + \left( \int \limits _{t^{n-1}}^{t^n} \mu (\theta (t)) \nabla R_h \theta (t) d t-\int \limits _{t^{n-1}}^{t^n} \mu (\theta (t)) \nabla \theta (t) d t, \nabla \rho _h^n\right) \\{} & {} = b_{2,1}+b_{2,2}+b_{2,3}+b_{2,4}. \end{aligned}$$

Thus,

$$\begin{aligned} b_{2,1}= & {} \left( \Delta t \, \mu \left( \theta _h^{n-1}\right) \nabla \rho _h^n, \nabla \rho _h^n\right) \ge \Delta t\, \mu _0 \left| \rho _h^n\right| ^2_{H^1(\Omega )}. \end{aligned}$$

For the term $b_{2,2}$,

$$\begin{aligned} \left| b_{2,2}\right|= & {} \left| \left( \int \limits _{t^{n-1}}^{t^n} \mu \left( \theta _h^{n-1}\right) \nabla R_h \theta ^n\, dt-\int \limits _{t^{n-1}}^{t^n} \mu \left( \theta _h^{n-1}\right) \nabla R_h \theta (t) d t, \nabla \rho _h^n\right) \right| \\= & {} \left| \left( \int \limits _{t^{n-1}}^{t^n} \mu \left( \theta _h^{n-1}\right) \int \limits _{t}^{t^n} \nabla R_h \theta ^\prime (s)\,ds\, dt, \nabla \rho _h^n\right) \right| , \end{aligned}$$

by applying Fubini’s theorem and using Hölder’s inequalities

$$\begin{aligned} \left| b_{2,2}\right| \le \mu _1 \int \limits _{t^{n-1}}^{t^n}\left| R_h \theta ^{\prime }(s)\right| _{H^1(\Omega )}\left| \rho _h^n\right| _{H^1(\Omega )}\left( s-t^{n-1}\right) ds. \end{aligned}$$

The Cauchy Schwarz’s inequality yields for any $\tilde{\varepsilon }_1>0$

$$\begin{aligned} \left| b_{2,2}\right| \le \frac{1}{6 \tilde{\varepsilon }_2}\mu _1^2 c^2_{2} \Delta t ^2\left\| \theta ^{\prime }\right\| _{L^2\left( t^{n-1}, t^n; H^1(\Omega )\right) }^2+\frac{\Delta t \, \tilde{\varepsilon }_2}{2}\left| \rho _h^n\right| _{H^1(\Omega )}^2. \end{aligned}$$

We now deal with the term $b_{2,3}:$

By inserting $\pm \left( \displaystyle \int \limits _{t^{n-1}}^{t^n} \mu \left( \theta ^{n-1}\right) \nabla R_h \theta (t) d t, \nabla \rho _h^n\right)$, then $\pm \left( \displaystyle \int \limits _{t^{n-1}}^{t^n}\left( \mu \left( \theta _h^{n-1}\right) -\mu \left( \theta ^{n-1}\right) \right) \nabla \theta (t) d t, \nabla \rho _h^n\right)$ and $\pm \left( \displaystyle \int \limits _{t^{n-1}}^{t^n}\left( \mu \left( \theta _h^{n-1}\right) -\mu \left( \theta (t)\right) \right) \nabla \theta (t) d t, \nabla \rho _h^n\right)$, we get

$$\begin{aligned} \left| b_{2,3}\right|{} & {} \le \left( \int \limits _{t^{n-1}}^{t^n}\left( \mu \left( \theta _h^{n-1}\right) -\mu \left( \theta ^{n-1}\right) \right) \nabla \left( R_h \theta (t)-\theta (t)\right) d t, \nabla \rho _h^n\right) \\{} & {} \quad + \left| \left( \int \limits _{t^{n-1}}^{t^n}\left( \mu \left( \theta _h^{n-1}\right) -\mu \left( \theta ^{n-1}\right) \right) \nabla \theta (t) d t, \nabla \rho _h^n\right) \right| \\{} & {} \quad + \left| \left( \int \limits _{t^{n-1}}^{t^n}\left( \mu \left( \theta ^{n-1}\right) -\mu (\theta (t))\right) \nabla \left( R_h \theta (t)-\theta (t)\right) d t, \nabla \rho _h^n\right) \right| \\{} & {} \quad + \left| \left( \int \limits _{t^{n-1}}^{t^n}\left( \mu \left( \theta ^{n-1}\right) -\mu (\theta (t))\right) \nabla \theta (t) d t, \nabla \rho _h^n\right) \right| \end{aligned}$$

Since $\mu$ satisfies the propreties (2.5) and (2.6) we get for any $\tilde{\varepsilon }_3>0$ and $\tilde{\varepsilon }_4>0$ the following:

$$\begin{aligned}{} & {} \left| b_{2,3}\right| \le \frac{1}{\tilde{\varepsilon }_3} 8 \mu _1^2 c_3^2 h^2\Vert \theta \Vert _{L^2\left( 0, T; H^2(\Omega ) \right) }^2+\frac{\Delta t\, \tilde{\varepsilon }_3}{2}\left| \rho _h^n\right| _{H^1(\Omega )}^2 \\{} & {} \quad + \frac{1}{2 \tilde{\varepsilon }_4}\left( \mu ^*\right) ^2 \Delta t \Vert \theta \Vert _{L^{\infty }\left( 0, T; W^{1,d}(\Omega )) \right. }^2\left\| \theta _h^{n-1}-\theta ^{n-1}\right\| _{L^4(\Omega )}^2+\frac{\Delta t \tilde{\varepsilon }_4}{2}\left| \rho _h^n\right| _{H^1(\Omega )}^2 \\{} & {} \quad + \mu ^* \int \limits _{\Omega } \int \limits _{t^{n-1}}^{t^n}\left| \theta \left( x, t^{n-1}\right) -\theta (x, t)|\cdot |\nabla \theta (x,t)|\cdot | \nabla \rho _h^n(x)\right| dt\, dx \end{aligned}$$

For the last term,

$$\begin{aligned}{} & {} \mu ^* \int \limits _{\Omega } \int \limits _{t^{n-1}}^{t^n}\left| \theta \left( x, t^{n-1}\right) -\theta (x, t)|\cdot |\nabla \theta (x,t)|\cdot | \nabla \rho _h^n(x)\right| dt\, dx \\{} & {} \quad \le \mu ^*(C_2^{\Omega })^{1/2}\Vert \theta \Vert _{L^{\infty }\left( 0, T; W^{1,4}(\Omega )\right) } \int \limits _{t^{n-1}}^{t^n}\left| \theta ^{\prime }(s)\right| _{H^1(\Omega )}\left| \rho _{h}^n\right| _{H^1(\Omega )}\left( t^n-s\right) ds. \end{aligned}$$

From Cauchy Schwarz’s inequality for any $\tilde{\varepsilon }_5>0$

$$\begin{aligned}{} & {} \mu ^* \int \limits _{\Omega } \int \limits _{t^{n-1}}^{t^n}\left| \theta ^{n-1}(x)-\theta (x, t)|\cdot |\nabla \theta (x,t)|\cdot | \nabla \rho _h^n(x)\right| dt\, dx \\{} & {} \quad \le \mu ^* \Delta t \, (C_2^{\Omega }\Delta t)^{\frac{1}{2}}\Vert \theta \Vert _{L^{\infty }\left( 0,T; W^{1,4}(\Omega )\right) }\left\| \theta ^{\prime }\right\| _{L^2\left( t^{n-1}, t^n;H^1(\Omega )\right) }\left| \rho _h^n\right| _{H^1(\Omega )}\\{} & {} \quad \le \frac{1}{2 \tilde{\varepsilon }_5}\left( \mu ^*\right) ^2 C_2^{\Omega } \Delta t ^2\Vert \theta \Vert _{L^{\infty }\left( 0,T; W^{1,4}(\Omega )\right) }^2\left\| \theta ^{\prime }\right\| _{L^2\left( t^{n-1}, t^n; H^1(\Omega )\right) }^2+\frac{\Delta t \tilde{\varepsilon _5}}{2}\left| \rho _h^n\right| _{H^1(\Omega )}^2 \end{aligned}$$

Such that, $C_2^{\Omega }$ is the norm of the imbedding of ${H}_0^1(\Omega )$ into $L^4(\Omega )$. It résult

$$\begin{aligned}{} & {} \left| b_{2,3}\right| \le \frac{1}{\tilde{\varepsilon }_3} 8 (\mu ^*)^2 c_3^2 h^2\Vert \theta \Vert _{L^2\left( t^{n-1}, t^n; H^2(\Omega )\right) }^2+\frac{\Delta t \tilde{\varepsilon }_3}{2}\left| \rho _h^n\right| _{H^1(\Omega )}^2+\frac{\Delta t \tilde{\varepsilon }_4}{2}\left| \rho _h^n\right| _{H^1(\Omega )}^2 \\{} & {} \quad + \frac{\left( \mu ^*\right) ^4}{16 \tilde{\varepsilon }_4^2}\Vert \theta \Vert _{L^{\infty }\left( 0,T; W^{1,4}(\Omega )\right) }^4 \frac{\Delta t}{2}\left\| \theta _h^{n-1}-\theta ^{n-1}\right\| _{L^2(\Omega )}^2+\frac{\Delta t}{2}\left| \theta _h^{n-1}-\theta ^{n-1}\right| _{H^1(\Omega )}^2 \\{} & {} \quad \frac{\Delta t \tilde{\varepsilon }_5}{2}\left| \rho _h^n\right| _{H^1(\Omega )}^2+\frac{1}{2 \tilde{\varepsilon }_5}\left( \mu ^*\right) ^2 C_2^{\Omega } \Delta t^2\Vert \theta \Vert _{L^{\infty }\left( 0, {\bar{T}}; W^{1,4}(\Omega )\right) }^2\left\| \theta ^{\prime }\right\| _{L^2\left( t^{n-1}, t^n; H^1(\Omega )\right) }^2. \end{aligned}$$

Furthermore, for the term $b_{2,4}$ can be bounded for every positive real number $\tilde{\varepsilon }_6>0$ as:

$$\begin{aligned}{} & {} \left| b_{2,4}\right| \le \mu _1 \int \limits _{t^{n-1}}^{t^n}\left\| \nabla \left( R_h\theta -\theta \right) (t)\right\| _{L^2(\Omega )}\left| \rho _h^n\right| _{H^1(\Omega )} \, dt \\&\quad&\le \frac{\left( \mu _1\right) ^2 c_4^2 h^2}{2 \tilde{\varepsilon }_6}\Vert \theta \Vert _{L^2\left( t^{n-1}, t^n; H^2(\Omega )\right) }^2+\frac{\Delta t\, \tilde{\varepsilon }_6}{2}\left| \rho _h^n\right| _{H^1(\Omega )}^2. \end{aligned}$$

Finally, the third term $b_3$

$$\begin{aligned} b_3=\Delta t \,d_2\left( \varvec{u}^n_h, \theta ^n_h, \rho ^n_h\right) +\frac{\Delta t }{2}\left( \left( {\text {div}} \varvec{u}^n_h\right) \theta ^n_h, \rho ^n_h\right) -\left( \int \limits _{t^{n-1}}^{t^n}({\textbf{u}}(t) \nabla ) \cdot \theta (t) d t, \rho _h^n\right) . \end{aligned}$$

can be bounded by inserting

$$\begin{aligned} \pm \Delta t\, d_2(\varvec{u}_h^n, R_h \theta ^{n},\rho ^n_h)\quad \text {and}\quad \pm \frac{\Delta t }{2}\left( \left( {\text {div}} \varvec{u}^n_h\right) R_h\theta ^n, \rho ^n_h\right) . \end{aligned}$$

$b_3$ becomes

$$\begin{aligned} b_3=\Delta t \,d_2\left( \varvec{u}^n_h, R_h\theta ^n, \rho ^n_h\right) +\frac{\Delta t}{2}\left( \left( {\text {div}}\varvec{u}^n_h\right) R_h\theta ^n,\rho ^n_h\right) -\left( \int \limits _{t^{n-1}}^{t^n}({\textbf{u}}(t) \nabla ) \cdot \theta (t) d t, \rho _h^n\right) . \end{aligned}$$

Furthermore, by inserting $\pm \Delta t\left( P_h \varvec{u}^n \nabla R_h\theta ^n, \rho _h^n\right) ,\quad \pm \displaystyle \frac{\Delta t}{2}\left( {\text {div}}\left( P_h \varvec{u}^n\right) R_h \theta ^n, \rho _h^n\right) ,$

$\pm \displaystyle \int \limits _{t^{n-1}}^{t^n}\left( \left( P_h \varvec{u}^n \nabla \right) \cdot \theta (t), \rho _h^n\right) d t$ and $\pm \displaystyle \frac{1}{2} \int \limits _{t^{n-1}}^{t^n}\left( {\text {div}} P_h\varvec{u}^n\theta (t), \rho _h^n\right) d t$, we obtain

$$\begin{aligned}{} & {} *b_3 = \Delta t\left( \left( \left( {\textbf{u}}_h^n-P_h {\textbf{u}}^n\right) \nabla \right) \cdot R_h \theta ^n, \rho _h^n\right) +\int \limits _{t^{n-1}}^{t^n}\left( \left( P_h {\textbf{u}}^n \nabla \right) \cdot \left( R_h\theta ^n-\theta (t)\right) , \rho _h^n\right) d t \nonumber \\{} & {} \quad + \int \limits _{t^{n-1}}^{t^n}\left( \left( \left( P_h \varvec{u}^n-\varvec{u}(t)\right) \nabla \right) \cdot \theta (t), \rho _h^n\right) d t+\frac{\Delta t}{2}\left( {\text {div}}\left( \varvec{u}^n_h-P_h \varvec{u}^n\right) R_h \theta ^n, \rho _h^n\right) \nonumber \\{} & {} \quad + \frac{1}{2} \int \limits _{t^{n-1}}^{t^n}\left( {\text {div}}\left( P_h \varvec{u}^n\right) \left( R_h \theta ^n-\theta (t)\right) , \rho _h^n\right) d t+\frac{1}{2} \int \limits _{t^{n-1}}^{t^n}\left( {\text {div}}\left( P_h\varvec{u}^n-{\textbf{u}}(t)\right) \theta (t), \rho _h^n\right) d t. \end{aligned}$$

(3.17)

We denote by $b_{3, 1}, \ldots , b_{3, 6}$ the terms of the right hand side of (3.17) which can be bounded as following:

The term $b_{3,1}$ can be bounded thanks to Holder’s inequality for every positive real number $\tilde{\varepsilon }_7>0$ as:

$$\begin{aligned} \left| b_{3,1}\right| \le \frac{\Delta t}{2 \tilde{\varepsilon }_7} c_5^2\left( C^\Omega _2\right) ^2\Vert \theta \Vert _{L^{\infty }(0,T,H^1(\Omega ))}^2\left| \varvec{u}_h^n-P_h \varvec{u}^n\right| _{\varvec{H}^1(\Omega )} +\frac{\tilde{\varepsilon }_7 \Delta t}{2}\left| \rho _h^n\right| _{H^1(\Omega )}^2. \end{aligned}$$

The term $b_{3,2}$ can be bounded for every positive real number $\tilde{\varepsilon }_8>0$ and $\tilde{\varepsilon }_9>0$ as:

$$\begin{aligned} \left| b_{3,2}\right|\le & {} \frac{1}{2}c_6^2\left( C^\Omega _2\right) ^2 \Vert \varvec{u}\Vert _{L^{\infty }\left( 0, T; H^1(\Omega )\right) }^2 \left( \frac{c_5^2\Delta t^2}{3 \tilde{\varepsilon }_8}\left\| \theta ^{\prime }\right\| ^2_{L^\infty (0,T; H^1(\Omega ))}+\frac{c_1^2 h^2}{2\tilde{\varepsilon }_9} \left\| \theta \right\| _{L^2(t^{n-1},t^n; H^2(\Omega ))}^2\right) \\{} & {} +\frac{\Delta t\left( \tilde{\varepsilon }_8+\tilde{\varepsilon }_9\right) }{2}\left| \rho _h^n\right| _{H^1(\Omega )}^2. \end{aligned}$$

For the term $b_{3,3}$, we insert $\pm \displaystyle \int \limits _{t^{n-1}}^{t^n}\left( \left( P_h \varvec{u}^n \nabla R_h\theta (t), \rho _h^n\right) d t\right.$. It results for every positive real number $\tilde{\varepsilon }_{10}>0$ and $\tilde{\varepsilon }_{11}>0$:

$$\begin{aligned} \left| b_{3,3}\right|\le & {} \displaystyle \frac{1}{6\tilde{\varepsilon }_{10}}\left( C_2^\Omega \right) ^2c^2_6 \Delta t^2\Vert \theta \Vert _{L^{\infty }\left( 0, T; H^1(\Omega )\right) }^2\left\| \varvec{u}^{\prime }\right\| _{L^2\left( t^{n-1}, t^n; \varvec{H}^1(\Omega )\right) }^2 \hspace{4cm} \\{} & {} +\displaystyle \frac{1}{2\tilde{\varepsilon }_{11}}\left( C_2^\Omega \right) ^2 c_5^2 h^2\Vert \theta \Vert _{L^{\infty }\left( 0,T; H^1(\Omega )\right) }^2\Vert \varvec{u}\Vert _{L^2\left( t^{n-1},t^n;\varvec{H}^2(\Omega )\right) }^2+\displaystyle \frac{\Delta t \left( \tilde{\varepsilon }_{10}+\tilde{\varepsilon }_{11}\right) }{2}\left| \rho _h^{n}\right| _{H^1(\Omega )}^2. \\ \end{aligned}$$

The term $b_{3,4}$ can be bounded for every positive real number $\tilde{\varepsilon }_{12}>0$ as:

$$\begin{aligned} \left| b_{3,4}\right| \le \frac{c_7}{2\tilde{\varepsilon }_{12}} \Delta t\left( C_2^\Omega \right) ^2\Vert \theta \Vert _{L^{\infty }\left( 0,T; H^1(\Omega )\right) }^2\left| \varvec{u}_h^n-P_h \varvec{u}^n\right| _{\varvec{H}^1(\Omega )}^2+\frac{\Delta t}{2} \tilde{\varepsilon }_{12} \left| \rho _h^n\right| _{H^1(\Omega )}^2. \end{aligned}$$

The terms $b_{3, 5}$ and $b_{3, 6}$ can be treated exactly like $b_{3, 2}$ and $b_{3, 3}$ and we get for every positive real numbers $\tilde{\varepsilon }_{13}>0$, $\tilde{\varepsilon }_{14}>0$, $\tilde{\varepsilon }_{15}>0$ and $\tilde{\varepsilon }_{16}>0$,

$$\begin{aligned} \left| b_{3,5}\right|\le & {} \frac{c_{8}(C_2^\Omega )^2}{2}\Vert \varvec{u}\Vert _{L^{\infty }\left( 0,T; \varvec{H}^1(\Omega )\right) }^2\left( \frac{\Delta t^2}{\tilde{\varepsilon }_{13}}\left\| \theta ^{\prime }\right\| _{L^2\left( 0,T; H^1(\Omega )\right) }^2+\frac{h^2}{ \tilde{\varepsilon }_{14}}\Vert \varvec{u}\Vert _{L^{\infty }\left( 0,T;\varvec{H}^1(\Omega )\right) }^2\right) \\{} & {} +\frac{\Delta t\left( \tilde{\varepsilon }_{13}+\tilde{\varepsilon }_{14}\right) }{2} \left| \rho _h^n\right| _{H^1(\Omega )}^2, \end{aligned}$$

and

$$\begin{aligned} \left| b_{3,6}\right|\le & {} \frac{c_{9}(C_2^\Omega )^2}{2}\Vert \theta \Vert _{L^{\infty }\left( 0,T; H^1(\Omega )\right) }^2\left( \frac{\Delta t^2}{\tilde{\varepsilon }_{15}}\left\| {\textbf{u}}^{\prime }\right\| _{L^2\left( 0,T;\varvec{H}^1(\Omega )\right) }^2+\frac{h^2}{\tilde{\varepsilon }_{16}}\Vert {\textbf{u}}\Vert _{L^2\left( 0,T; \varvec{H}^2(\Omega )\right) }\right) \\{} & {} +\frac{\Delta t\left( \tilde{\varepsilon }_{15}+\tilde{\varepsilon }_{16}\right) }{2} \left| \rho _h^n\right| _{H^1(\Omega )}^2. \end{aligned}$$

Finally, by using the obtained bounds and summing over n from 1 to $m \le N$, we get

$$\begin{aligned}{} & {} \frac{1}{2}\left\| \rho _h^m\right\| _{L^2(\Omega )}^2+\frac{1}{2} \sum _{n=1}^m\left\| \rho _h^n-\rho _h^{n-1}\right\| _{L^2(\Omega )}^2+\mu _0\sum _{n=1}^m \Delta t\left| \rho _h^n\right| _{H^1(\Omega )}^2 \\{} & {} \quad \le \bar{\xi }_1\left( h^2+\Delta t^2\right) +\bar{\xi }_2 \sum _{n=1}^m \Delta t\left| {\textbf{u}}_h^n-P_{h} {\textbf{u}}^n\right| _{\varvec{H}^1(\Omega )}^2 \\{} & {} \qquad +\bar{\xi }_3 \sum _{n=1}^m \Delta t\left| \rho _h^n\right| _{H^1(\Omega )}^2 \end{aligned}$$

where $\bar{\xi }_1, \bar{\xi }_2$, and $\bar{\xi }_3$ depending of $\tilde{\varepsilon }_{i}, i=1 \ldots 16.$ After a suitable choice of $\tilde{\varepsilon }_{i},$ we conclude the following bound

$$\begin{aligned}{} & {} \frac{1}{2}\left\| \rho _h^m\right\| _{L^2(\Omega )}^2+\mu _0\sum _{n=1}^m \Delta t \left| \rho _h^n\right| _{H^1(\Omega )}^2 +\frac{1}{2} \sum _{n=1}^m\left\| \rho _h^n-\rho _h^{n-1}\right\| _{L^2(\Omega )}^2 \nonumber \\{} & {} \quad \le C^\theta _1\left( h^2+\Delta t^2\right) +C^\theta _2 \sum _{n=1}^m \Delta t \left| {\textbf{u}}_h^n-P_h {\textbf{u}}^n\right| ^2_{\varvec{H}^1(\Omega )}. \end{aligned}$$

(3.18)

$\square$

To establish the a priori error estimates corresponding to the numerical scheme (3.7), we use Lemma 2.1 which is valid only for $d = 2$.

Theorem 3.3

Let $(\varvec{u},p,\theta )$ be the solution of problem (2.10) and $(\varvec{u}^n_h,p^n_h,\theta ^n_h)$ be the solution of problem (3.7). Under hypothesis $H1-H3$, there exist three positive constants $C^{\varvec{u}}_1,$ $C^{\varvec{u}}_2$ and $C^{\varvec{u}}_3$ independ on h and $\Delta t$ such that, for all $m \le N$,

$$\begin{aligned}{} & {} \frac{1}{2}\left\| \varvec{u}_h^m-P_h\varvec{u}^m\right\| _{L^2(\Omega )}^2+\mu _0\sum _{\text {n}=1}^m \Delta t\left| \varvec{u}_h^n-P_h\varvec{u}^n\right| _{H^1(\Omega )}^2 \nonumber \\{} & {} \qquad +\frac{1}{2} \sum _{m=1}^m\left\| (\varvec{u}_h^n-P_h\varvec{u}^n)-(\varvec{u}_h^{n-1}-P_h\varvec{u}^{n-1})\right\| _{L^2(\Omega )}^2 \nonumber \\{} & {} \quad \le C^{\varvec{u}}_{1}\left( h^2+\Delta t^2\right) +C^{\varvec{u}}_2 \sum _{n=1}^m \Delta t\left| \theta _h^n-\theta (t^n)\right| _{H^1(\Omega )}^2+C^{\varvec{u}}_3 \sum _{n=1}^m \Delta t\left\| \theta _h^{n-1}-\theta (t^{n-1})\right\| _{L^2(\Omega )}^2 \end{aligned}$$

(3.19)

Proof

We consider the fluid inequality of the continuous variational problem (2.10). Let $\varvec{w}\in \varvec{H}^1_0(\Omega )$, we take $\varvec{v}-\varvec{u}=\pm \varvec{w}$ in (2.10) and obtain

$$\begin{aligned} \left( \frac{\partial \varvec{u}}{\partial t}, \varvec{w}\right) +a_1(\theta :{\varvec{u}},{\varvec{w}})+b(\varvec{w},p)+d_1(\varvec{u},\varvec{u},\varvec{w})=\langle \varvec{f},\varvec{w}\rangle , \end{aligned}$$

(3.20)

We denote by $\varvec{H}^1_{0h}(\Omega )$ a conforming finite element discretization of $\varvec{H}^1_0(\Omega )$. In a similar way we have from (3.7), for all $\varvec{w}_h\in \varvec{H}^1_{0\,h}(\Omega )$

$$\begin{aligned} \displaystyle \frac{1}{\Delta t}(\varvec{u}_h^{n}, \varvec{w}_h)+a_1(\theta _h^{n-1}:{\varvec{u}_h^{n}},\varvec{w}_h)+b(\varvec{w}_h,p_h^{n}) +d_{1h}(\varvec{u}^{n-1}_h,\varvec{u}^n_h,\varvec{w}_h) = \langle \varvec{f}^n,\varvec{w}_h\rangle -\displaystyle \frac{1}{\Delta t}(\varvec{u}_h^{n-1}, \varvec{w}_h), \qquad \end{aligned}$$

(3.21)

Setting $\varvec{w}_h=\varvec{w}_h^n=\varvec{u}_h^n-P_h\varvec{u}^n,$ in (3.20) integrated over $[t^{n-1},t^n]$ by the definition of the operator $P_h$ gives

$$\begin{aligned} (\varvec{u}^n-\varvec{u}^{n-1},\varvec{w}_h^n)+\left( \int \limits _{t^{n-1}}^{t^n} \nu (\theta )\nabla \varvec{u}\,dt,\nabla \varvec{w}_h^n\right) - \left( \int \limits _{t^{n-1}}^{t^n}p(t)\, dt, \text {div}\,\varvec{w}_h^n\right) =\left( \int \limits _{t^{n-1}}^{t^n}\varvec{f}(t)\, dt,\varvec{w}_h^n\right) \hspace{0.2cm} \end{aligned}$$

(3.22)

Then, we subtract (3.22) with (3.21) after multiplying it by the time step $\Delta t$, we obtain

$$\begin{aligned}{} & {} \left( (\varvec{u}_h^n-\varvec{u}_h^{n-1})-(\varvec{u}^n-\varvec{u}^{n-1}),\varvec{w}_h^n\right) +\Delta t\, a_1(\theta _h^{n-1}:{\varvec{u}_h^{n}},\varvec{w}_h) \\{} & {} \quad -\left( \int \limits _{t^{n-1}}^{t^n} \nu (\theta )\nabla \varvec{u}\,dt,\nabla \varvec{w}_h^n\right) +\Delta t\, b(\varvec{w}_h,p_h^{n})-\left( \int \limits _{t^{n-1}}^{t^n}p(t)\, dt, \text {div}\,\varvec{w}_h^n\right) \\{} & {} \quad +\Delta t\,d_{1}(\varvec{u}^{n-1}_h,\varvec{u}^n_h,\varvec{w}^n_h)+\frac{1}{2}\left( \Delta t{\text {div}}(\varvec{u}^{n-1}_h)\varvec{u}^{n}_h,\varvec{w}^n_h \right) -\left( \int \limits _{t^{n-1}}^{t^n}(\varvec{u}(t)\nabla )\cdot \varvec{u}(t)\, dt,\varvec{w}^n_h \right) =0 \end{aligned}$$

We denote respectively by $A_1$, $A_2$, $A_3$ and $A_4$ the following terms

$$\begin{aligned}{} & {} {A_1} = \left( (\varvec{u}_h^n-\varvec{u}_h^{n-1})-(\varvec{u}^n-\varvec{u}^{n-1}),\varvec{w}_h^n\right) , \\{} & {} {A_2} = \Delta t\, a_1(\theta _h^{n-1}:{\varvec{u}_h^{n}},\varvec{w}_h)-\left( \int \limits _{t^{n-1}}^{t^n} \nu (\theta )\nabla \varvec{u}\,dt,\nabla \varvec{w}_h^n\right) , \\{} & {} {A_3} = \Delta t\, b(\varvec{w}_h,p_h^{n})-\left( \int \limits _{t^{n-1}}^{t^n}p(t)\, dt, \text {div}\,\varvec{w}_h^n\right) , \end{aligned}$$

and

$$\begin{aligned} {A_4}=\Delta t\,d_{1}(\varvec{u}^{n-1}_h,\varvec{u}^n_h,\varvec{w}^n_h)+\frac{1}{2}\left( \Delta \,t{\text {div}}(\varvec{u}^{n-1}_h)\varvec{u}^{n}_h,\varvec{w}^n_h \right) -\left( \int \limits _{t^{n-1}}^{t^n}(\varvec{u}(t)\nabla )\cdot \varvec{u}(t)\, dt,\varvec{w}^n_h \right) . \end{aligned}$$

The term $A_1$ can be expressed as

$$\begin{aligned}{} & {} \left( \left( \varvec{u}_h^n-\varvec{u}_h^{n-1}\right) -\left( \varvec{u}^n-\varvec{u}^{n-1}\right) , \varvec{w}_h^n\right) = \frac{1}{2}\left\| \varvec{w}_h^n\right\| _{L^2(\Omega )}^2-\frac{1}{2}\left\| \varvec{w}_h^{n-1}\right\| _{L^2(\Omega )}^2+\frac{1}{2}\left\| \varvec{w}_h^n-\varvec{w}_h^{n-1}\right\| _{L^2(\Omega )}^2 \hspace{1cm} \\{} & {} \quad +\left( \left( \varvec{u}^{n-1}-P_h \varvec{u}^{n-1}\right) -\left( \varvec{u}^n-P_h \varvec{u}^n\right) , \varvec{w}_h^n\right) . \end{aligned}$$

And similar to $b_1$, for any $\varepsilon _1>0$:

$$\begin{aligned} \left| \left( \left( \varvec{u}^{n-1}-P_h \varvec{u}^{n-1}\right) -\left( \varvec{u}^n-P_h \varvec{u}^n\right) , \varvec{w}_h^n\right) \right| \le \frac{c_1^2 h^2}{2 {\varepsilon }_1}\left\| \frac{\partial \varvec{u}}{\partial t}\right\| _{L^2\left( t^{n-1}, t^n; \varvec{H}^1(\Omega )\right) }^2+C_{\Omega } \frac{\Delta t\, {\varepsilon }_1}{2}\left| \varvec{w}_h^n\right| _{H^1(\Omega )}^2. \end{aligned}$$

To bound the term $A_2$, we follow the same approach in $b_2$ to get:

$$\begin{aligned}{} & {} \left| {A_2}\right| \le \frac{\left( {\nu }_1\right) ^2}{6 \varepsilon _2} \Delta t^2\,\left\| \varvec{u}^{\prime }\right\| _{L^2\left( 0, T; \varvec{H}^1(\Omega )\right) }^2+\frac{\Delta t\, \varepsilon _2}{2}\Vert \varvec{w}_h^n\Vert _{\varvec{H}^1(\Omega )}^2 +\frac{c_1}{\varepsilon _3} {\nu }_1^2 h^2\Vert \varvec{u}\Vert _{L^2\left( 0,T; \varvec{H}^2(\Omega )\right) }^2+\frac{\Delta t\, \varepsilon _3}{2}\left| \varvec{w}_h^n\right| _{\varvec{H}^1(\Omega )}^2 \\{} & {} \quad +\frac{c_2}{\varepsilon _4}\left( \nu ^*\right) ^2 C^\Omega _2\Vert \varvec{u}\Vert _{L^{\infty }\left( 0,T; W^{1,4}(\Omega )\right) }^2 \Delta t\,\left( \frac{1}{2 }\left\| \theta _h^{n-1}-\theta \left( t^{n-1}\right) \right\| _{L^2(\Omega )}^2+\frac{1}{2} \left| \theta _h^{n-1}-\theta \left( t^{n-1}\right) \right| _{H^1(\Omega )}^2\right) \\{} & {} \quad +\frac{\Delta t\, \varepsilon _4}{2}\left| \varvec{w}_h^n\right| _{\varvec{H}^1(\Omega )}^2+\frac{c_3\left( \nu ^*C^\Omega _2\right) ^2}{\varepsilon _5}\Vert \varvec{u}\Vert _{L^\infty \left( 0,T;\varvec{W}^{1,4}(\Omega )\right) }^2\left\| \theta ^{\prime }\right\| _{L^2\left( t^{n-1},t^n; H^1(\Omega )\right) }^2+\frac{\Delta t\, \varepsilon _5}{2}\left| \varvec{w}_h^{\text {n}}\right| _{\varvec{H}^1(\Omega )}^2. \end{aligned}$$

For the term $A_3$, since $Q_hp\in L^2_{0\,h}(\Omega )$ and $p_h^n\in L^2_{0\,h}(\Omega )$ then

$$\begin{aligned} \left( Q_hp(t), {\text {div}}(\varvec{w}_h^{n})\right) =\left( p_h^n, {\text {div}}(\varvec{w}_h^{n})\right) =0. \end{aligned}$$

Hence,

$$\begin{aligned} |{A_3}| = \left| \int \limits _{t^{n-1}}^{t^n}\left( Q_h p(t)-p(t), {\text {div}}\left( \varvec{w}_h^{n}\right) \right) d t\right| \le \frac{c_4 h^2}{\varepsilon _6}\Vert p\Vert _{L^2\left( 0, T; H^1(\Omega )\right) }^2+\frac{\varepsilon _6 \Delta t\,}{2}\left| \varvec{w}_h^n\right| _{\varvec{H}^1(\Omega )} \end{aligned}$$

The term $A_4$ can be bounded by inserting $\pm \Delta t\left( \varvec{u}_h^n\nabla P_h\varvec{u}^n,\varvec{w}_h^n\right)$ and $\pm \Delta t\left( \displaystyle \frac{1}{2}\text {div}\,\varvec{u}_h^{n-1}P_h\varvec{u}^n,\varvec{w}_h^n\right)$ then insering $\pm \left( \displaystyle \int \limits _{t^{n-1}}^{t^n}P_h \varvec{u}^{n-1}\nabla \cdot P_h \varvec{u}^n,\varvec{w}_h^n\right)$ and $\pm \displaystyle \frac{1}{2}\left( \int \limits _{t^{n-1}}^{t^n}P_h \varvec{u}^{n-1} \nabla \cdot P_h \varvec{u}^n, \varvec{w}_h^n\right)$. We have

$$\begin{aligned}{} & {} {A_4} = \Delta t\left( \left( \left( \varvec{u}_h^{n-1}-P_h \varvec{u}\left( t^{n-1}\right) \right) \nabla \right) \cdot P_h \varvec{u}\left( t^n\right) , \varvec{w}_h^n\right) +\int \limits _{t^{n-1}}^{t^n}\left( \left( P_h \varvec{u}\left( t^{n-1}\right) \nabla \left( P_h \varvec{u}\left( t^n\right) -\varvec{u}(t)\right) , \varvec{w}_h^n\right) \right) d t \\{} & {} \quad + \int \limits _{t_{n-1}}^{t^n}\left( \left( \left( P_h \varvec{u}\left( t^{n-1}\right) -\varvec{u}(t)\right) \nabla \varvec{u}(t), \varvec{w}_h^n\right) d t+\frac{\Delta t}{2}\left( \text {div}\left( \varvec{u}_h^{n-1}-P_h \varvec{u}\left( t^{n-1}\right) \right) P_h \varvec{u}\left( t^n\right) , \varvec{w}_h^n\right) \right. \\{} & {} \quad + \frac{1}{2} \int \limits _{t^{n-1}}^{t^n}\left( \text {div}\left( P_h \varvec{u}\left( t^{n-1}\right) \right) \left( P_h \varvec{u}\left( t^n\right) -\varvec{u}(t)\right) , \varvec{w}_h^n\right) d t+\frac{1}{2} \int \limits _{t^{n-1}}^{t^n}\left( \text {div}\left( P_h \varvec{u}\left( t^{n-1}\right) -\varvec{u}(t)\right) \varvec{u}(t), \varvec{w}_h^n\right) d t \end{aligned}$$

The terms $A_{4,2}, A_{4,3}, A_{4,5}$ and $A_{4,6}$ can be treated exactly by following the steps of the corresponding terms in the proof of Theorem 3.2. Then, we have to treat only the sum of the first term $A_{4,1}$ and the fourth term $A_{4,4}$ as following:

$$\begin{aligned}{} & {} {A_{4,1}+A_{4,4}} = \Delta t\left( \left( \left( \varvec{u}_h^{n-1}-P_h \varvec{u}\left( t^{n-1}\right) \right) \nabla \right) \cdot P_h \varvec{u}\left( t^n\right) , \varvec{w}_h^n\right) +\frac{\Delta t}{2}\left( \text {div}\left( \varvec{u}_h^{n-1}-P_h \varvec{u}\left( t^{n-1}\right) \right) P_h \varvec{u}\left( t^n\right) , \varvec{w}_h^n\right) \\{} & {} \quad = \frac{\Delta t}{2}\left( \left( \left( \varvec{u}_h^{n-1}-P_h \varvec{u}\left( t^{n-1}\right) \right) \nabla \right) \cdot P_h \varvec{u}\left( t^n\right) , \varvec{w}_h^{{n}}\right) -\frac{\Delta t}{2}\left( \left( \left( \varvec{u}_h^{n-1}-P_h \varvec{u}\left( t^{n-1}\right) \right) \nabla \right) \cdot \varvec{w}_h^{n}, P_h \varvec{u}\left( t^n\right) \right) . \end{aligned}$$

Then, for every positive real number $\varepsilon _7$,

$$\begin{aligned}{} & {} \left| {A_{4,1}+A_{4,4}}\right| \\{} & {} \quad \le c_5 \Delta t\left\| \varvec{u}_h^{n-1}-P_h \varvec{u}\left( t^{n-1}\right) \right\| _{\varvec{L}^4(\Omega )}\left| \varvec{w}_h^n\right| _{\varvec{H}^1(\Omega )}\left| \varvec{u}\left( t^n\right) \right| _{\varvec{H}^1(\Omega )} \\{} & {} \quad \le c_5\Vert \varvec{u}\Vert _{L^ \infty \left( 0,T; \varvec{H}^1(\Omega )\right) }\\{} & {} \quad \left[ \frac{1}{2 \varepsilon _7}\left( \frac{\Delta t}{2} \left| \varvec{u}_h^{n-1}-P_h \varvec{u}\left( t^{n-1}\right) \right| _{\varvec{H}^1(\Omega )}^2+\frac{\Delta t}{2} \left\| \varvec{u}_h^{n-1}-P_h \varvec{u}\left( t^{n-1}\right) \right\| ^2_{\varvec{L}^2(\Omega )}\right) +\frac{\varepsilon _7}{2} \left| \varvec{w}_h^n\right| ^2_{\varvec{H}^1(\Omega )}\right] \end{aligned}$$

By collecting the above bounds and summing over n from 1 to $m \le N$, we get

$$\begin{aligned}{} & {} \frac{1}{2}\left\| \varvec{w}_h^m\right\| _{\varvec{L}^2(\Omega )}^2+\frac{1}{2} \sum _{n=1}^m\left\| \varvec{w}_h^n-\varvec{w}_h^{n-1}\right\| _{\varvec{L}^2(\Omega )}^2+\nu _0 \sum _{n=1}^m \Delta t\left| \varvec{w}_h^n\right| _{\varvec{H}^1(\Omega )}^2 \\{} & {} \quad \le \xi _1\left( h^2+\Delta t^2\right) +\xi _2 \sum _{n=1}^m \frac{\Delta t}{2}\left\| \theta _h^{n-1}-\theta \left( t^{n-1}\right) \right\| _{L^2(\Omega )}^2 \\{} & {} \qquad +\xi _3\sum _{n=1}^m \frac{\Delta t}{2}\left| \theta _h^n-\theta \left( t^n\right) \right| _{H^1(\Omega )}^2 +\xi _4 \sum _{n=1}^m \Delta t\left\| \varvec{w}_h^n\right\| _{\varvec{L}^2(\Omega )}^2+\xi _5 \sum _{n=1}^m \Delta t\left\| \varvec{w}_h^n\right\| _{\varvec{H}^1(\Omega )}^2 \end{aligned}$$

where $\xi _1 \xi _2, \xi _3, \xi _4$ and $\xi _5$ depending of $\varepsilon _i, i=1..7$. The last term can be bounded following the relation

$$\begin{aligned} \sum _{n=1}^m\left\| \varvec{w}_h^n\right\| _{\varvec{H}^1(\Omega )} \le \sum _{n=1}^m\left\| \varvec{w}_h^n-\varvec{w}_h^{m-1}\right\| _{\varvec{H}^1(\Omega )}+\sum _{n=1}^m \left\| \varvec{w}_h^{n-1}\right\| _{\varvec{H}^1(\Omega )}, \end{aligned}$$

After a suitable choice of $\varepsilon _i, i=1..7$ and $\xi _i, i=1..5$ and applyication of Gronwall’s lemma, we get

$$\begin{aligned}{} & {} \frac{1}{2}\left\| \varvec{w}_h^m\right\| _{\varvec{L}^2(\Omega )}^2+\frac{1}{2} \sum _{n=1}^m\left\| \varvec{w}_h^n-\varvec{w}_h^{n-1}\right\| _{\varvec{L}^2(\Omega )}^2+\frac{\nu _0}{2} \sum _{n=1}^m \Delta t\left| \varvec{w}_h^n\right| _{\varvec{H}^1(\Omega )}^2 \hspace{5cm} \\{} & {} \quad \le C^{\varvec{u}}_1\left( h^2+\Delta t^2\right) +C^{\varvec{u}}_2 \sum _{n=1}^m \frac{\Delta t}{2}\left\| \theta _h^{n-1}-\theta \left( t^{n-1}\right) \right\| _{L^2(\Omega )}^2 +C^{\varvec{u}}_3\sum _{n=1}^m \frac{\Delta t}{2}\left| \theta _h^n-\theta \left( t^n\right) \right| _{H^1(\Omega )}^2. \end{aligned}$$

$\square$

Theorem 3.4

Under the assumptions of Theorems 3.2 and 3.3, and by choosing $\theta \in L^\infty (0,T,H^2(\Omega )),$ there exists constants $k_0$, ${\tilde{C}}^{\varvec{u}}$ and ${\tilde{C}}^{\theta }$, independent of h and $\Delta t$, such that for $\Delta t <k_0$

$$\begin{aligned}{} & {} \underset{1\le n\le N}{\sup }\left\| \theta _h^n-\theta (t^n)\right\| _{L^2(\Omega )}^2+\mu _0\sum _{n=1}^N \Delta t \left| \theta _h^n-\theta (t^n)\right| _{H^1(\Omega )}^2 \le {\tilde{C}}^\theta \left( h^2+\Delta t^2\right) . \end{aligned}$$

(3.23)

$$\begin{aligned}{} & {} \quad \underset{1\le n\le N}{\sup }\left\| \varvec{u}_h^n-\varvec{u}(t^n)\right\| _{\varvec{L}^2(\Omega )}^2+\nu _1\sum _{n=1}^N \Delta t \left| \varvec{u}_h^n-\varvec{u}(t^n)\right| _{\varvec{H}^1(\Omega )}^2 \le {\tilde{C}}^{\varvec{u}}\left( h^2+\Delta t^2\right) . \end{aligned}$$

(3.24)

Proof

We insert $\pm R_h\theta ^{n-1}$ in (3.19), use Theorem 3.2, choose $\theta \in L^\infty (0,T,H^2(\Omega )),$ and apply the properties of the operator $R_h$ to get the following bound

$$\begin{aligned}{} & {} \frac{1}{2}\left\| \rho _h^m\right\| _{L^2(\Omega )}^2+\mu _0\sum _{\text {n}=1}^m \Delta t\left| \rho _h^n\right| _{H^1(\Omega )}^2+\frac{1}{2} \sum _{n=1}^m\left\| \rho _h^n-\rho _h^{n-1}\right\| _{L^2(\Omega )}^2 \\{} & {} \quad \le C_1\left( h^2+\Delta t^2\right) +C_2 \sum _{n=1}^m \Delta t\left| \rho _h^{n-1}\right| _{H^1(\Omega )}^2 + C_3 \sum _{n=1}^m \Delta t\left\| \rho _h^{n-1}\right\| _{L^2(\Omega )}^2. \end{aligned}$$

with $\rho _h^n=\theta _h^n-R_h \theta ^{\text {n}}$ and $C_1$, $C_2$ and $C_3$ are constants independent of h and $\Delta t.$ From the continuous injection from $H^1(\Omega )$ into $L^2(\Omega )$, we obtain

$$\begin{aligned} \frac{1}{2}\left\| \rho _h^m\right\| _{L^2(\Omega )}^2+\mu _0\sum _{\text {n}=1}^m \Delta t\left| \rho _h^n\right| _{H^1(\Omega )}^2 \le C_1\left( h^2+\Delta t^2\right) + C^{'}_2 \sum _{n=1}^m \Delta t\left\| \rho _h^{n-1}\right\| _{L^2(\Omega )}^2. \end{aligned}$$

It follows from the discrete Gronwall Lemma the following inequality

$$\begin{aligned} \frac{1}{2}\left\| \rho _h^m\right\| _{L^2(\Omega )}^2+\mu _0\sum _{\text {n}=1}^m \Delta t\left| \rho _h^n\right| _{H^1(\Omega )}^2 \le C_1\left( h^2+\Delta t^2\right) . \end{aligned}$$

To get the relation (3.23), it suffices to apply

$$\begin{aligned} |\theta _h^n-\theta (t^n)|_{H^1(\Omega )}\le |\theta _h^n-R_h\theta ^n|_{H^1(\Omega )}+|R_h\theta ^n-\theta (t^n)|_{H^1(\Omega )} \end{aligned}$$

Finally, the bound (3.24) can be directly deduced from Theorem 3.3. $\square$

4 Operator Splitting

The operator splitting is a decomposition technique that consists, as its name suggests, in treating each of the operators of a partial differential equation separately, in order to facilitate its resolution. Indeed, it makes it possible to remove the complexity of a problem by reducing it to several simpler sub-problems. We recall that (2.10) can be decomposed as variational equation, an inequality and a variational equation for the heat. The two variational equations are written as the following initial value problem:

Find $\phi$ and $\psi$ such that

$$\begin{aligned} \left\{ \begin{array}{l} \phi (0)=\phi _0, \psi (0)=\psi _0. \\ \displaystyle \frac{d \phi }{d t}+\sum _{i=1}^q {\mathcal {A}}_i \phi =0 \quad \text {on}(0,T), \\ \displaystyle \frac{d \psi }{d t}+\sum _{i=1}^q {\mathcal {B}}_i \psi =0 \quad \text {on}(0,T), \end{array}\right. \end{aligned}$$

(4.1)

Let $\phi _n \approx \phi (t_n)$ and $\psi _n \approx \psi (t_n)$. Then Marchuk-Yanenko’s method to solve (2.10) borrows mainly from [15]:

$$\begin{aligned} \phi ^0=\phi _0,\quad \psi ^0=\psi _0 \end{aligned}$$

(4.2)

for $n \ge 1$, knowing $\phi _n$, $\psi _n$ and for $i=1,2,3, \ldots , q$ we compute $\phi ^{n+1}$ via

$$\begin{aligned} \frac{\phi ^{n+i /q}-\phi ^{n+(i-1) / q}}{\Delta t}+{\mathcal {A}}_i\left( \phi ^{n+i / q}, t^{n+1}\right) =0. \end{aligned}$$

(4.3)

Then compute $\psi ^{n+1}$ via

$$\begin{aligned} \frac{\psi ^{n+i}-\psi ^{n+i-1}}{\Delta t}+{\mathcal {B}}_i\left( \psi ^{n+i},t^{n+1}\right) =0. \end{aligned}$$

(4.4)

For $n\le 0$, with $t^{n+\alpha }=(n+\alpha )\Delta t$ and for non negative $\gamma _1$ and $\gamma _2,$ such that $\gamma _1+\gamma _2=1,$ we compute $\varvec{u}^{n+1/2}, (\varvec{u}^{n+1},p^{n+1})$ then $\theta ^{n+1}$ as in the following Algorithm:

Theorem 4.1

Suppose that $\varvec{u}_0 \in \varvec{L}^2(\Omega )$; then the scheme (4.5)-(4.8) is unconditionally stable in the following sense: for $n=0,1, \ldots , m$ there exists a positive constant $C_{\Omega }$ such that

$$\begin{aligned}{} & {} \Vert \varvec{u}^m\Vert _{\varvec{L}^2(\Omega )}^2 \le \Vert \varvec{u}_0\Vert _{\varvec{L}^2(\Omega )}^2+\displaystyle \frac{C_{\Omega } t_m}{\gamma _1 \nu _0}\Vert \varvec{f}\Vert _{\varvec{L}^2(\Omega )}^2, \end{aligned}$$

(4.9)

$$\begin{aligned}{} & {} \Delta t \sum _{n=0}^{m-1}\Vert (\varvec{Du}^{n+1/2})\Vert _{\varvec{L}^2(\Omega )}^2 \le \frac{1}{3 \nu _0 \gamma _1}\bigg (\Vert \varvec{u}_0\Vert _{\varvec{L}^2(\Omega )}^2+\frac{C_{\Omega } t_m}{\gamma _1 \nu _0}\Vert \varvec{f}\Vert _{\varvec{L}^2(\Omega )}^2\bigg ) \end{aligned}$$

(4.10)

$$\begin{aligned}{} & {} \quad \Delta t \sum _{n=0}^{m-1} \Vert (\varvec{Du}^{n+1})\Vert _{\varvec{L}^2(\Omega )}^2 \le \frac{1}{4 \nu _0 \gamma _2}\bigg (\Vert \varvec{u}_0\Vert _{\varvec{L}^2(\Omega )}^2+\frac{C_{\Omega } t_m}{\gamma _1 \nu _0}\Vert \varvec{f}\Vert _{\varvec{L}^2(\Omega )}^2\bigg ) \end{aligned}$$

(4.11)

$$\begin{aligned}{} & {} \quad \sum _{n=0}^{m-1}\Vert \varvec{u}^{n+1 / 2}-\varvec{u}^n\Vert _{\varvec{L}^2(\Omega )}^2 \le \Vert \varvec{u}_0\Vert _{\varvec{L}^2(\Omega )}^2+\frac{C_{\Omega } t_m}{\gamma _1 \nu _0}\Vert \varvec{f}\Vert _{\varvec{L}^2(\Omega )}^2 \end{aligned}$$

(4.12)

$$\begin{aligned}{} & {} \quad \sum _{n=0}^{m-1}\Vert \varvec{u}^{n+1}-\varvec{u}^{n+1/2}\Vert _{\varvec{L}^2(\Omega )}^2 \le \Vert \varvec{u}_0\Vert _{\varvec{L}^2(\Omega )}^2+\frac{C_{\Omega } t_m}{\gamma _1 \nu _0}\Vert \varvec{f}\Vert _{\varvec{L}^2(\Omega )}^2 \end{aligned}$$

(4.13)

$$\begin{aligned}{} & {} \quad \Delta t \sum _{n=0}^{m-1} J(\varvec{u}^{n+1})\le \frac{1}{2}\bigg (\Vert \varvec{u}_0\Vert _{\varvec{L}^2(\Omega )}^2+\frac{C_{\Omega } t_m}{\gamma _1 \nu _0}\Vert \varvec{f}\Vert _{\varvec{L}^2(\Omega )}^2\bigg ) \end{aligned}$$

(4.14)

$$\begin{aligned}{} & {} \quad \Vert \varvec{u}^{m+1/2}\Vert _{\varvec{L}^2(\Omega )}^2 \le \Vert \varvec{u}_0\Vert _{\varvec{L}^2(\Omega )}^2+\frac{C_{\Omega } t_m}{\gamma _1 \nu _0}\Vert \varvec{f}\Vert _{\varvec{L}^2(\Omega )}^2 \end{aligned}$$

(4.15)

Proof

Let $\varvec{v}=\varvec{u}^{n+1/2}$ in (4.5), one has

$$\begin{aligned} 2(\varvec{u}^{n+1/2}-\varvec{u}^{n}, \varvec{u}^{n+1/2})+ 2\gamma _1\Delta t\ a_1(\theta ^n:{\varvec{u}^{n+1/2}},\varvec{u}^{n+1/2})= 2\Delta t\langle \varvec{f},\varvec{v}\rangle . \end{aligned}$$

(4.16)

The second term on the right hand side of (4.16) can be treated with Cauchy–Schwarz, Korn and Young inequalities as follows

$$\begin{aligned} 2\Delta t\langle \varvec{f},\varvec{v}\rangle \le \frac{ C_\Omega \Delta t}{\nu _0\gamma _1} \Vert \varvec{f}\Vert _{\varvec{L}^2(\Omega )}^2 + \nu _0\gamma _1\Delta t\Vert \varvec{Du}^{n+1/2}\Vert _{\varvec{L}^2(\Omega )}^2 \end{aligned}$$

(4.17)

Throughout

$$\begin{aligned} (\varvec{u}^{n+1/2}-\varvec{u}^{n}, \varvec{u}^{n+1/2})= \frac{1}{2}\big (\Vert \varvec{u}^{n+1/2}\Vert ^2_{\varvec{L}^2(\Omega )}-\Vert \varvec{u}^{n}\Vert ^2_{\varvec{L}^2(\Omega )}+\Vert \varvec{u}^{n+1/2}-\varvec{u}^{n}\Vert ^2_{\varvec{L}^2(\Omega )}\big ) \end{aligned}$$

(4.18)

Dropping (4.17) and (4.18) in (4.16), we find that

$$\begin{aligned} \Vert \varvec{u}^{n+1/2}\Vert ^2_{\varvec{L}^2(\Omega )}-\Vert \varvec{u}^{n}\Vert ^2_{\varvec{L}^2(\Omega )}+\Vert \varvec{u}^{n+1/2}-\varvec{u}^{n}\Vert ^2_{\varvec{L}^2(\Omega )}+ 3\gamma _1\nu _0 \Delta t \Vert \varvec{Du}^{n+1/2}\Vert ^2_{\varvec{L}^2(\Omega )}\le \frac{ C_\Omega \Delta t}{\nu _0\gamma _1} \Vert \varvec{f}\Vert _{\varvec{L}^2(\Omega )}^2 \nonumber \\ \end{aligned}$$

(4.19)

Next, we take successively $\varvec{v}=\varvec{0}$ and $\varvec{v}=2\varvec{u}^{n+1}$ in (4.6), and after comparison of the resulting relations, we find

$$\begin{aligned} \frac{1}{\Delta t}(\varvec{u}^{n+1}-\varvec{u}^{n+1/2}, \varvec{u}^{n+1})+\gamma _2a_1(\theta ^n:\varvec{u}^{n+1},\varvec{u}^{n+1})+J(\varvec{u}^{n+1})= 0 \end{aligned}$$

(4.20)

Applying 4.18 we find

$$\begin{aligned} \Vert \varvec{u}^{n+1}\Vert ^2_{\varvec{L}^2(\Omega )}-\Vert \varvec{u}^{n+1/2}\Vert ^2_{\varvec{L}^2(\Omega )}+\Vert \varvec{u}^{n+1}-\varvec{u}^{n+1/2}\Vert ^2_{\varvec{L}^2(\Omega )} +4\gamma _2\nu _0 \Delta t \Vert \varvec{Du}^{n+1}\Vert ^2_{\varvec{L}^2(\Omega )}+2\Delta t J(\varvec{u}^{n+1})\le 0 \nonumber \\ \end{aligned}$$

(4.21)

We do (4.19)+(4.21) for $n = 0, 1, 2,...,m-1$, we obtain

$$\begin{aligned}{} & {} \Vert \varvec{u}^{m}\Vert ^2_{\varvec{L}^2(\Omega )}+ 3\gamma _1\nu _0\Delta t\sum _{n=0}^{m-1} \Vert \varvec{Du}^{n+1/2}\Vert ^2_{\varvec{L}^2(\Omega )}+ 4\gamma _2\nu _0 \Delta t \sum _{n=0}^{m-1} \Vert \varvec{Du}^{n+1}\Vert ^2_{\varvec{L}^2(\Omega )} \nonumber \\{} & {} \quad +\sum _{n=0}^{m-1}\Vert \varvec{u}^{n+1/2}-\varvec{u}^{n}\Vert ^2_{\varvec{L}^2(\Omega )}+\sum _{n=0}^{m-1}\Vert \varvec{u}^{n+1}-\varvec{u}^{n+1/2}\Vert ^2_{\varvec{L}^2(\Omega )}+2\Delta t\sum _{n=0}^{m-1} J(\varvec{u}^{n+1}) \nonumber \\{} & {} \quad \le \Vert \varvec{u}_0\Vert ^2_{\varvec{L}^2(\Omega )} + \frac{C_\Omega t_m}{\gamma _1\nu _0}\Vert \varvec{f}\Vert ^2_{\varvec{L}^{2}(\Omega )} \, \end{aligned}$$

(4.22)

$\square$

From which we obtain (4.9)-(4.14).

By adding the relation (4.19) for $n = 0, 1, 2,...,m$ and dropping some positive terms, we obtain

$$\begin{aligned} \sum _{n=0}^{m}\Vert \varvec{u}^{n+1/2}\Vert ^2_{\varvec{L}^2(\Omega )}-\sum _{n=0}^{m}\Vert \varvec{u}^{n}\Vert ^2_{\varvec{L}^2(\Omega )}\le \frac{ C_\Omega t_m}{\nu _0\gamma _1} \Vert \varvec{f}\Vert _{\varvec{L}^2(\Omega )}^2 \end{aligned}$$

(4.23)

By adding the relation (4.21) for $n = 0, 1, 2,...,m-1$ and dropping some positive terms, we obtain

$$\begin{aligned} \sum _{n=0}^{m-1}\Vert \varvec{u}^{n+1}\Vert ^2_{\varvec{L}^2(\Omega )}-\sum _{n=0}^{m-1}\Vert \varvec{u}^{n+1/2}\Vert ^2_{\varvec{L}^2(\Omega )}\le 0 \end{aligned}$$

(4.24)

Adding (4.23) and (4.24) gives (4.15).

Theorem 4.2

Suppose that $\varvec{u}_0 \in \varvec{L}^2(\Omega )$; then we have the following heat stability results: for $n=0,1, \ldots , m$ there exists a positive constant $C_{\Omega }$ such that

$$\begin{aligned}{} & {} \Vert \theta ^{m}\Vert ^2_{H^1(\Omega )} \le \Vert \theta _0\Vert ^2_{H^1(\Omega )}+\frac{t_m}{C_\Omega \mu _0}\Vert {k}\Vert ^2_{H^{-1}(\Omega )}. \end{aligned}$$

(4.25)

$$\begin{aligned}{} & {} \quad \sum _{n=0}^{m-1} \Vert \theta ^{n+1}-\theta ^{n}\Vert ^2_{H^1(\Omega )} \le \Vert \theta _0\Vert ^2_{H^1(\Omega )}+\frac{t_m}{C_\Omega \mu _0}\Vert {k}\Vert ^2_{H^{-1}(\Omega )}. \end{aligned}$$

(4.26)

$$\begin{aligned}{} & {} \quad \Delta t \sum _{n=0}^{m-1} \Vert \theta ^{n+1}\Vert ^2_{{H}^1(\Omega )} \le \frac{1}{C_\Omega \mu _0}\Vert \theta _0\Vert ^2_{H^1(\Omega )}+t_m\Vert {k}\Vert ^2_{H^{-1}(\Omega )}. \end{aligned}$$

(4.27)

Let $\rho =\theta ^{n+1}$ in (4.8), one has

$$\begin{aligned} \frac{1}{\Delta t}(\theta ^{n+1}-\theta ^{n},\theta ^{n+1})+a_2(\theta ^n:\theta ^{n+1},\theta ^{n+1})= \langle {k},\theta ^{n+1}\rangle . \end{aligned}$$

(4.28)

We have

$$\begin{aligned} \langle {k},\theta ^{n+1}\rangle \le \frac{1}{2} \bigg [\frac{1}{C_\Omega \mu _0}\Vert h\Vert ^2_{H^{-1}(\Omega )}+C_\Omega \mu _0\Vert \theta ^{n+1}\Vert ^2_{H^1(\Omega )} \bigg ]. \end{aligned}$$

(4.29)

Thanks to Cauchy-Schwarz and Young’s inequalities and the proprieties of $a_2(.:,.,.)$ in (2.14),

$$\begin{aligned} \Vert \theta ^{n+1}\Vert ^2_{H^1(\Omega )}-\Vert \theta ^{n}\Vert ^2_{H^1(\Omega )}+\Vert \theta ^{n+1}-\theta ^{n}\Vert ^2_{H^1(\Omega )}+C_\Omega \mu _0\Delta t \Vert \theta ^{n+1}\Vert ^2_{{H}^1(\Omega )}\le \frac{\Delta t}{C_\Omega \mu _0} \Vert {k}\Vert ^2_{H^{-1}(\Omega )}. \end{aligned}$$

(4.30)

For $n = 0, 1, 2,...,m-1,$ we obtain

$$\begin{aligned} \Vert \theta ^{m}\Vert ^2_{H^1(\Omega )}+ \sum _{n=0}^{m-1} \Vert \theta ^{n+1}-\theta ^{n}\Vert ^2_{H^1(\Omega )}+ C_\Omega \mu _0\Delta t \sum _{n=0}^{m-1} \Vert \theta ^{n+1}\Vert ^2_{{H}^1(\Omega )} \le \Vert \theta _0\Vert ^2_{H^1(\Omega )}+\frac{t_m}{C_\Omega \mu _0}\Vert {k}\Vert ^2_{H^{-1}(\Omega )}. \nonumber \\ \end{aligned}$$

(4.31)

This completes the proof of the theorem.

5 Numerical Results and Discussion

All computations were performed using Matlab on DELL i7 with 16 GB RAM. The test problems used are designed to illustrate the behavior of the algorithm more than to model an actual phenomenon. For the discretization, w use the the MINI-element ($P_1$-Bubble/$P_1$) for the velocity-pressure pair and the $P_1$ element for the temperature. The algorithm described will be tested computationally. We stop the computations when the following condition is satisfied

$$\begin{aligned} \displaystyle \frac{\left\| \varvec{u}^{n}-\varvec{u}^{n-1}\right\| _{\varvec{L^2(\Omega )}}^{2}+\left\| \theta ^{n}-\theta ^{n-1}\right\| _{L^2(\Omega )}^{2}}{\left\| \varvec{u}^{n-1}\right\| _{\varvec{L^2(\Omega )}}^{2}+\left\| \theta ^{n-1}\right\| _{L^2(\Omega )}^{2}} < {Tol}\equiv 10^{-6}. \end{aligned}$$

5.1 Test Case with Stationary Solution

We consider the unit square $\Omega =(0,1)^{2}$ and we assume that its boundary consists of two portions $\Gamma$ and S defined as follows

$$\begin{aligned} \begin{aligned}&\Gamma =\{0\} \times (0,1) \cup (0,1) \times \{0\} \cup \{1\} \times (0,1)\\&S=(0,1) \times \{1\}. \end{aligned} \end{aligned}$$

We consider

$$\begin{aligned}\nu (\theta )=\frac{1}{2} e^{-\theta }+\frac{1}{4} \quad \text { for which one has } \quad \frac{1}{4} \le \nu (\theta ) \le \frac{3}{4}. \end{aligned}$$

and

$$\begin{aligned}\mu (\theta )=\frac{1}{2}+1.7\theta -6.46\times 10^{-3}\theta ^2\end{aligned}$$

In order to obtain a stationary solution of the problem

$$\begin{aligned} \left\{ \begin{array}{lrcl} \displaystyle \frac{\partial \varvec{u}}{\partial t}-\text {div}\ (2\nu (\theta )\varvec{D}\varvec{u})+(\varvec{u}\cdot \nabla )\varvec{u} + \nabla p = \varvec{f} \ {} &{}\text {in} \quad \Omega \times [0,T],&{} \\ \displaystyle \frac{\partial \theta }{\partial t}-\text {div}\ (\mu (\theta )\nabla \theta ) \,+(\varvec{u} . \ \nabla )\theta = k\ {} &{}\text {in} \quad \Omega \times [0,T],&{} \end{array} \right. \end{aligned}$$

(5.1)

the right-hand side is adjusted accordingly, i.e.

$$\begin{aligned} f_1(x,y)= & {} 80x^2(1-x)^2-20(2+12x^2-12x)y(1-2y)+2(2y-1), \end{aligned}$$

(5.2)

$$\begin{aligned} f_2(x,y)= & {} 20(12x-6)y^2(1-y)^2+20x(1-2x)(1-x)(2+12y^2-12y), \end{aligned}$$

(5.3)

$$\begin{aligned} k(x,y)= & {} 2x(1-x)+2y(1-y). \end{aligned}$$

(5.4)

We take the time step $\Delta t=0.01$. Figures 1, 2 show the velocity fields and the heat distribution for $g=0.5$ and $g=1.75$. One can notice in Fig. 1 (left) that the friction occurs on S for $g=0.5$.

Fig. 1

Velocity fields for $g=0.5$ and $g=1.75$ with $h=1/16$

Fig. 2

Heat diffusion for $g=0.5$ and $g=1.75$ with $h=1/16$

Since we do not have the exact solution, we assume that the solution obtained for $h=1/512$ is the reference solution. For the velocity-pressure pair, we compute the following error

$$\begin{aligned} e_h(\varvec{u},p)=\Vert \varvec{u}_{\text {ref}}-\varvec{u}_{h}\Vert _{\varvec{H}^1(\Omega )}+\Vert p_{\text {ref}}-p_{h}\Vert _{L^2(\Omega )}. \end{aligned}$$

For the MINI-element or the stabilized $P_1/P_1$ element, $e_h(\varvec{u},p)$ converges linearly, see e.g. [4]. We report, in the Tables 1, 2, the errors and convergence rates for both stick/slip cases. From the obtained results we can conclude that the convergence rates are almost independent from the stick bound g. We also notice that the convergence rates are linear for all components.

Table 1 Convergence rates for $(\varvec{u},p,\theta )$ with $g=0.5$

Table 2 Convergence rates for $(\varvec{u},p,\theta )$ with $g=1.75$

5.2 Test Case with Time-Dependent Solution

In order to verify time convergence rates, we consider a test problem with the same data as in the previous section except the right-hand side (now time-dependent)

$$\begin{aligned} f_1(x,y,t)= & {} t^2f_1(x,y), \\ f_2(x,y,t)= & {} t^2f_2(x,y), \\ k(x,y,t)= & {} tk(x,y), \end{aligned}$$

where $(f_1(x,y),\ f_2(x,y),\ k(x,y))$ is the right-hand side (5.2)-(5.4). We compute a reference solution using $\Delta t=2^{-10}$ as time step and $h=1/256$ as mesh size. To check the results of Theorem 3.4 we compute the following errors at $T=0.5$.

$$\begin{aligned}{} & {} e_h(\varvec{u})=\sup _{1\le n\le N} \parallel \varvec{u}^n_h-\varvec{u}_{ref}\parallel _{L^2(\Omega )} +\nu _1\Delta t\sum _{n=1}^N|\varvec{u}^n_h-\varvec{u}_{ref}|_{H^1(\Omega )}, \end{aligned}$$

(5.5)

$$\begin{aligned}{} & {} e_h({\theta })=\sup _{1\le n\le N} \parallel \theta ^n_h-\theta _{ref}\parallel _{L^2(\Omega )} +\mu _0\Delta t\sum _{n=1}^N|\theta ^n_h-\theta _{ref}|_{H^1(\Omega )}. \end{aligned}$$

(5.6)

where $(\varvec{u}_{ref},\theta _{ref})$ is a reference (computed) solution. In (5.5–5.6), $\nu _1=1/4$ and $\mu _0=1/2$, assuming $\theta \ge 0$. Figure 3 shows the velocity fields for $g=0.1$ and $g=0.5$ at $T=0.5$. We notice that the slip occurs for $g=0.1$.

Fig. 3

Velocity fields for $g=0.1$ (left) and $g=0.5$ (right) at $T=0.5$

To investigate experimentally the convergence rates for $e_h(\varvec{u})$ and $e_h({\theta })$, we compute a reference solution with $h=1/256$ and $\Delta t=2^{-10}$. We report in Tables 3, 4$e_h(\varvec{u})$, $e_h({\theta })$ and convergence rates for various time-steps. We notice again that the convergence rates are almost independent of the stick/slip bound g. Moreover, $e_h(\varvec{u})$ and $e_h({\theta })$ converge linearly as predicted by Thorem 3.4.

Table 3 Convergence rates at $T=0.5$ with $g=0.1$, $\Delta t=2^{-1-\ell }$ and $h=1/256$

Table 4 Convergence rates at $T=0.5$ with $g=0.5$, $\Delta t=2^{-1-\ell }$ and $h=1/256$

6 Concusion

The purpose of this work was to deal with the elements finite approximation of the time-dependent Navier–Stokes system coupled with the heat equation and governed by the nonlinear Tresca boundary conditions where both the viscosity and conductivity depend on the temperature. We present optimal error estimates for velocity, pressure and temperature. We have formulated and established the convergence of the Marchuk-Yanenko’s algorithm associated to the finite element equations. And finally established some numerical simulations which confirm the theoretical estimate. The extension of this work is in process, we aim to proceed for experiments on realistic examples, such as a lake heated by the sun.