We can finally prove the last inequality. Thanks to the triangular inequality:

$$ (3.26) \qquad \begin{aligned} \nu \|\mathbf{u} - \mathbf{u}_h\|_0 &+ h(\nu \|\nabla(\mathbf{u} - \mathbf{u}_h)\|_0 + \|p - p_h\|_0) \le \nu \|\mathbf{e}_h\|_0 + h(\nu \|\nabla(\mathbf{e}_h)\|_0 + \|\eta_h\|_0) \\ &+ \nu \|\mathbf{u} - R_h\|_0 + h(\nu \|\nabla(\mathbf{u} - R_h)\|_0 + \|p - Q_h\|_0), \end{aligned} $$

where

$$ \nu \|\mathbf{e}_h\|_0 + h(\nu \|\nabla(\mathbf{e}_h)\|_0 + \|\eta_h\|_0) \le ch + O(h^2) $$

comes from the inequalities (3.23) and (3.25).

The other term verifies (3.10), thus

$$ \nu \|\mathbf{u} - R_h\|_0 + h(\nu \|\nabla(\mathbf{u} - R_h)\|_0 + \|p - Q_h\|_0) \le ch^2. $$

Putting these two estimates in (3.26), we finally obtain

$$ \nu \|\mathbf{u} - \mathbf{u}_h\|_0 + h(\nu \|\nabla(\mathbf{u} - \mathbf{u}_h)\|_0 + \|p - p_h\|_0) \le ch + O(h^2). $$

<figure><img src="image_14.png" /></figure>

**Remark 3.6.** We only have a first order estimate in h, compared to second order in "more classic" Navier Stokes problems (see for instance [16],[13]).

We can add another control inequality

**Theorem 3.7.**

$$ (3.27) \qquad \|\nabla(\mathbf{u} - \mathbf{u}_h)\|_0 + \|p - p_h\|_0 \le ch + O(h^2) $$

which comes straightforwardly from the previous inequations.

3.4. **Stability of the method and error analysis.** We define the linearized numerical scheme corresponding to the problem (3.1) with:

$$ (3.28) \qquad \left\{ \begin{array}{ll} (\mathbf{u}_i^n \cdot \nabla)\mathbf{u}_i^{n-1} - \nu_i \Delta \mathbf{u}_i^n + \nabla p_i^n = \mathbf{f}_i & \text{in } \Omega_i \\ \nabla \cdot \mathbf{u}_i^n = 0, & \text{in } \Omega_i \\ \nu_i \frac{\partial \mathbf{u}_i^n}{\partial \mathbf{n}_i} = -C_D(\mathbf{u}_{i,h}^n - \mathbf{u}_{j,h}^n) |\mathbf{u}_{i,h}^{n-1} - \mathbf{u}_{j,h}^{n-1}|, & \text{on } \Gamma_{\text{Int}} \\ \nu_i \frac{\partial \mathbf{u}_i^n}{\partial \mathbf{n}_i} = -c_{d,i}(\mathbf{u}_{i,h}^n - \mathbf{V}_i) & \text{on } \Gamma_i, \\ \mathbf{u}_i^n \cdot \mathbf{n}_i = 0 & \text{on } \Gamma_{\text{Int}} \cup \Gamma_i, \\ \mathbf{u}_i^n = 0 & \text{on } \Gamma_{i,s}, \end{array} \right. $$

for $n \ge 1$.

The weak formulation associated with (3.28) can be seen as an Oseen iterative method (see [13]):

We start from the couple $(\mathbf{u}_h^0, p_h^0) \in W_h \times X_h$ defined by the discrete Stokes problem on $\Omega = \Omega_1 \times \Omega_2$

$$ (3.29) \qquad a(\mathbf{u}_h^0, \mathbf{v}_h) - d(\mathbf{v}_h, p_h^0) + d(\mathbf{u}_h^0, q_h) = (\mathbf{f}, \mathbf{v}_h), $$

for all $(\mathbf{v}_h, q_h) \in W_h \times X_h$.

**Remark 3.8.** The two Stokes problems for the two different fluids can be solved separately on $\Omega_1$ and $\Omega_2$ since there is no boundary condition involving the two fluids at this point.
The iterative method at order $n$ reads:

$$ (3.30) \qquad a(\mathbf{u}_h^n, \mathbf{v}_h) - d(\mathbf{v}_h, p_h^n) + d(\mathbf{u}_h^n, q_h) + b(\mathbf{u}_h^{n-1}, \mathbf{u}_h^n, \mathbf{v}_h) + \langle \tilde{G}(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}), \mathbf{v}_h \rangle \\ \qquad + \langle H(\mathbf{u}_h^n), \mathbf{v}_h \rangle = (\mathbf{f}, \mathbf{v}_h), $$

for all $(\mathbf{v}_h, q_h) \in W_h \times X_h$, where the map $\tilde{G}$ is the discrete version of $G$ and verifies

$$ (3.31) \qquad \langle \tilde{G}(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}), \mathbf{v}_h \rangle = \int_{\Gamma_I} (\mathbf{u}_{1,h}^n - \mathbf{u}_{2,h}^n) |\mathbf{u}_{1,h}^{n-1} - \mathbf{u}_{2,h}^{n-1}| (\mathbf{v}_{1,h} - \mathbf{v}_{2,h}). $$

This leads to the stability result:

**Theorem 3.9.** The iterative method is unconditionally stable, meaning that the couple $(\mathbf{u}_h^n, p_h^n)$ defined by the scheme (3.30) satisfies

$$ (3.32) \qquad \nu \|\nabla \mathbf{u}_h^n\|_0 \le \|\mathbf{f}\|_{-1}, $$

and

$$ (3.33) \qquad \|A_h \mathbf{u}_h^n\|_0 \le c_2, $$

for all $n \ge 0$, where $c_2$ depends on $\nu_1, \nu_2, \|\mathbf{f}\|_0$ (and on $\Omega$ through different Sobolev embedding theorems).

*Proof.* The result is verified for $n = 0$, by definition of $\mathbf{u}_h^0$ as a solution of a standard Stokes problem. We assume that the first inequality is true at the step $n$ and we will prove it for the step $n + 1$. Taking $\mathbf{v}_h = \mathbf{u}_h^{n+1}$, $q_h = p_h^{n+1} \in W_h \times X_h$ in (3.30) at the step $n + 1$, we get

$$ a(\mathbf{u}_h^{n+1}, \mathbf{u}_h^{n+1}) + \langle \tilde{G}(\mathbf{u}_h^{n+1}, \mathbf{u}_h^n), \mathbf{u}_h^{n+1} \rangle + \langle H(\mathbf{u}_h^{n+1}), \mathbf{u}_h^{n+1} \rangle = (\mathbf{f}, \mathbf{u}_h^{n+1}). $$

Besides, both $\tilde{G}$ and $H$-terms satisfy

$$ \langle \tilde{G}(\mathbf{u}_h^{n+1}, \mathbf{u}_h^n), \mathbf{u}_h^{n+1} \rangle \ge 0, $$

and

$$ \langle H(\mathbf{u}_h^{n+1}), \mathbf{u}_h^{n+1} \rangle \ge 0. $$

Hence, we get the estimate

$$ \nu \|\nabla \mathbf{u}_h^{n+1}\|_0^2 \le \|\mathbf{f}\|_{-1} \|\nabla \mathbf{u}_h^{n+1}\|_0, $$

and the first claim is proved.

The second claim will be proved without iteration argument. Taking $\mathbf{v}_h = A_h \mathbf{u}_h^{n+1}$ and $q_h = 0$ in (3.30) at the step $n + 1$, recalling that $A_h$ satisfy the inequality (3.17) and cancel the $G$ and $H$-terms, we get by Cauchy-Schwarz and transport trilinear term (3.13) inequalities:

$$ \nu \|A_h \mathbf{u}_h^{n+1}\|_0^2 \le \|\mathbf{f}\|_0 \|A_h \mathbf{u}_h^{n+1}\|_0 + \frac{1}{2} c_0 \gamma_0^{-1/4} \|\nabla \mathbf{u}_h^n\|_0 \|\nabla \mathbf{u}_h^{n+1}\|_0^{1/2} \|A_h \mathbf{u}_h^{n+1}\|_0^{3/2}, $$

and from the Young inequality:

$$ \nu \|A_h \mathbf{u}_h^{n+1}\|_0^2 \le \|\mathbf{f}\|_0 \|A_h \mathbf{u}_h^{n+1}\|_0 + \frac{1}{4\nu} c_0^2 \gamma_0^{-1/2} \|\nabla \mathbf{u}_h^n\|_0^2 \|\nabla \mathbf{u}_h^{n+1}\|_0 \|A_h \mathbf{u}_h^{n+1}\|_0 + \frac{\nu}{2} \|A_h \mathbf{u}_h^{n+1}\|_0^2. $$

Thanks to (3.32), we obtain

$$ \frac{\nu}{2} \|A_h \mathbf{u}_h^{n+1}\|_0 \le \|\mathbf{f}\|_0 + \frac{1}{4\nu^4} c_0^2 \gamma_0^{-1/2} \|\mathbf{f}\|_{-1}^3, $$

for all $n \ge 0$, which gives the stability estimate (3.33) and concludes the proof. $\square$
The iterative method at order $n$ reads:

$$ (3.30) \qquad a(\mathbf{u}_h^n, \mathbf{v}_h) - d(\mathbf{v}_h, p_h^n) + d(\mathbf{u}_h^n, q_h) + b(\mathbf{u}_h^{n-1}, \mathbf{u}_h^n, \mathbf{v}_h) + \langle \tilde{G}(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}), \mathbf{v}_h \rangle \\ \qquad + \langle H(\mathbf{u}_h^n), \mathbf{v}_h \rangle = (\mathbf{f}, \mathbf{v}_h), $$

for all $(\mathbf{v}_h, q_h) \in W_h \times X_h$, where the map $\tilde{G}$ is the discrete version of $G$ and verifies

$$ (3.31) \qquad \langle \tilde{G}(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}), \mathbf{v}_h \rangle = \int_{\Gamma_I} (\mathbf{u}_{1,h}^n - \mathbf{u}_{2,h}^n) |\mathbf{u}_{1,h}^{n-1} - \mathbf{u}_{2,h}^{n-1}| (\mathbf{v}_{1,h} - \mathbf{v}_{2,h}). $$

This leads to the stability result:

**Theorem 3.9.** The iterative method is unconditionally stable, meaning that the couple $(\mathbf{u}_h^n, p_h^n)$ defined by the scheme (3.30) satisfies

$$ (3.32) \qquad \nu \|\nabla \mathbf{u}_h^n\|_0 \le \|\mathbf{f}\|_{-1}, $$

and

$$ (3.33) \qquad \|A_h \mathbf{u}_h^n\|_0 \le c_2, $$

for all $n \ge 0$, where $c_2$ depends on $\nu_1, \nu_2, \|\mathbf{f}\|_0$ (and on $\Omega$ through different Sobolev embedding theorems).

*Proof.* The result is verified for $n=0$, by definition of $\mathbf{u}_h^0$ as a solution of a standard Stokes problem. We assume that the first inequality is true at the step $n$ and we will prove it for the step $n+1$. Taking $\mathbf{v}_h = \mathbf{u}_h^{n+1}$, $q_h = p_h^{n+1} \in W_h \times X_h$ in (3.30) at the step $n+1$, we get

$$ a(\mathbf{u}_h^{n+1}, \mathbf{u}_h^{n+1}) + \langle \tilde{G}(\mathbf{u}_h^{n+1}, \mathbf{u}_h^n), \mathbf{u}_h^{n+1} \rangle + \langle H(\mathbf{u}_h^{n+1}), \mathbf{u}_h^{n+1} \rangle = (\mathbf{f}, \mathbf{u}_h^{n+1}). $$

Besides, both $\tilde{G}$ and $H$-terms satisfy

$$ \langle \tilde{G}(\mathbf{u}_h^{n+1}, \mathbf{u}_h^n), \mathbf{u}_h^{n+1} \rangle \ge 0, $$

and

$$ \langle H(\mathbf{u}_h^{n+1}), \mathbf{u}_h^{n+1} \rangle \ge 0. $$

Hence, we get the estimate

$$ \nu \|\nabla \mathbf{u}_h^{n+1}\|_0^2 \le \|\mathbf{f}\|_{-1} \|\nabla \mathbf{u}_h^{n+1}\|_0, $$

and the first claim is proved.

The second claim will be proved without iteration argument. Taking $\mathbf{v}_h = A_h \mathbf{u}_h^{n+1}$ and $q_h = 0$ in (3.30) at the step $n+1$, recalling that $A_h$ satisfy the inequality (3.17) and cancel the $G$ and $H$-terms, we get by Cauchy-Schwarz and transport trilinear term (3.13) inequalities:

$$ \nu \|A_h \mathbf{u}_h^{n+1}\|_0^2 \le \|\mathbf{f}\|_0 \|A_h \mathbf{u}_h^{n+1}\|_0 + \frac{1}{2} c_0 \gamma_0^{-1/4} \|\nabla \mathbf{u}_h^n\|_0 \|\nabla \mathbf{u}_h^{n+1}\|_0^{1/2} \|A_h \mathbf{u}_h^{n+1}\|_0^{3/2}, $$

and from the Young inequality:

$$ \nu \|A_h \mathbf{u}_h^{n+1}\|_0^2 \le \|\mathbf{f}\|_0 \|A_h \mathbf{u}_h^{n+1}\|_0 + \frac{1}{4\nu} c_0^2 \gamma_0^{-1/2} \|\nabla \mathbf{u}_h^n\|_0^2 \|\nabla \mathbf{u}_h^{n+1}\|_0 \|A_h \mathbf{u}_h^{n+1}\|_0 + \frac{\nu}{2} \|A_h \mathbf{u}_h^{n+1}\|_0^2. $$

Thanks to (3.32), we obtain

$$ \frac{\nu}{2} \|A_h \mathbf{u}_h^{n+1}\|_0 \le \|\mathbf{f}\|_0 + \frac{1}{4\nu^4} c_0^2 \gamma_0^{-1/2} \|\mathbf{f}\|_{-1}^3, $$

for all $n \ge 0$, which gives the stability estimate (3.33) and concludes the proof. $\square$
A corollary from the stability theorem 3.9 and the a priori estimates from the theoretical section is the boundedness result:

**Lemma 3.10.** There exists $C > 0$ depending only on the data such that

$$ (3.34) \qquad \|\mathbf{u}_h\|_{W_h} \le C \quad \text{and} \quad \|\mathbf{u}_h^n\|_{W_h} \le C, $$

for all $n \ge 0$.

We define the errors between the Galerkin theoretical velocities and the Galerkin approximated velocities obtained from the Oseen scheme $\mathbf{e}_h^n = \mathbf{u}_h^n - \mathbf{u}_h$ and the pressure errors $\eta_h^n = p_h^n - p_h$.

This leads to the error convergence result:

**Theorem 3.11.** Provided the condition on the data

$$ (3.35) \qquad \min(\nu_1, \nu_2, c_{d,1}, c_{d,2}) - CC_D \ge 0 \quad \text{and} \quad \frac{C(C_b + C_D)}{\min(\nu_1, \nu_2, c_{d,1}, c_{d,2}) - CC_D} < 1, $$

the method is converging and we have the error estimate:

$$ \|\mathbf{e}_h^n\|_{W_h} \le \frac{2}{\nu} \left( \frac{C(C_b + C_D)}{\min(\nu_1, \nu_2, c_{d,1}, c_{d,2}) - CC_D} \right)^{n+1} \|\mathbf{f}\|_{-1}. $$

*Proof.* We consider the difference at the n-th step, between the equations satisfied by $(\mathbf{u}_h, p_h)$ and by $(\mathbf{u}_h^n, p_h^n)$, which gives the relation

$$ (3.36) \qquad a(\mathbf{e}_h^n, \mathbf{v}_h) + b(\mathbf{e}_h^{n-1}, \mathbf{u}_h, \mathbf{v}_h) + b(\mathbf{u}_h^{n-1}, \mathbf{e}_h^n, \mathbf{v}_h) + g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{v}_h) - g(\mathbf{u}_h, \mathbf{u}_h, \mathbf{v}_h) $$

$$ (3.37) \qquad a(\mathbf{e}_h^n, \mathbf{v}_h) + b(\mathbf{e}_h^{n-1}, \mathbf{u}_h, \mathbf{v}_h) + b(\mathbf{u}_h^{n-1}, \mathbf{e}_h^n, \mathbf{v}_h) + g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{v}_h) - g(\mathbf{u}_h, \mathbf{u}_h, \mathbf{v}_h) $$

$$ = h(\mathbf{e}_h^n, \mathbf{v}_h) - d(\mathbf{v}_h, \eta_h^n) + d(\mathbf{e}_h^n, q_h) = 0, $$

for $n \ge 1$ and $(\mathbf{v}_h, q_h) \in W_h \times Q_h$.

Consider the difference $g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{v}_h) - g(\mathbf{u}_h, \mathbf{v}_h)$, we have

$$ g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{v}_h) - g(\mathbf{u}_h, \mathbf{v}_h) = C_D \int_{\Gamma_I} \left( (\mathbf{u}_{1,h}^n - \mathbf{u}_{2,h}^n) |\mathbf{u}_{1,h}^{n-1} - \mathbf{u}_{2,h}^{n-1}| - (\mathbf{u}_{1,h} - \mathbf{u}_{2,h}) |\mathbf{u}_{1,h} - \mathbf{u}_{2,h}| \right) (\mathbf{v}_{1,h} - \mathbf{v}_{2,h}), $$

and denoting $\mathbf{U}_h^n := \mathbf{u}_{1,h}^n - \mathbf{u}_{2,h}^n$ and $\mathbf{U}_h := \mathbf{u}_{1,h} - \mathbf{u}_{2,h}$, $\mathbf{V}_h = \mathbf{v}_{1,h} - \mathbf{v}_{2,h}$, we get

$$ \begin{align*} g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{v}_h) - g(\mathbf{u}_h, \mathbf{v}_h) &= C_D \int_{\Gamma_I} (\mathbf{U}_h^n |\mathbf{U}_h^{n-1}| - \mathbf{U}_h |\mathbf{U}_h|) \mathbf{V}_h \\ &= C_D \int_{\Gamma_I} (\mathbf{U}_h^n |\mathbf{U}_h^{n-1}| - \mathbf{U}_h^n |\mathbf{U}_h| + \mathbf{U}_h^n |\mathbf{U}_h| - \mathbf{U}_h |\mathbf{U}_h|) \mathbf{V}_h \\ &= C_D \int_{\Gamma_I} (\mathbf{U}_h^n (|\mathbf{U}_h^{n-1}| - |\mathbf{U}_h|) + |\mathbf{U}_h| (\mathbf{U}_h^n - \mathbf{U}_h)) \mathbf{V}_h. \end{align*} $$

We get by the triangular inequality

$$ \begin{align*} \|g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{v}_h) - g(\mathbf{u}_h, \mathbf{v}_h)\|_{W_h} &\le C_D (\|\mathbf{U}_h^n\|_{W_h} \|\mathbf{e}_h^{n-1}\|_{W_h} + \|\mathbf{U}_h\|_{W_h} \|\mathbf{e}_h^n\|_{W_h}) \|\mathbf{V}_h\|_{W_h} \\ &= C_D (\|\mathbf{u}_h^n\|_{W_h} \|\mathbf{e}_h^{n-1}\|_{W_h} + \|\mathbf{u}_h\|_{W_h} \|\mathbf{e}_h^n\|_{W_h}) \|\mathbf{v}_h\|_{W_h} \\ &\le CC_D (\|\mathbf{e}_h^{n-1}\|_{W_h} + \|\mathbf{e}_h^n\|_{W_h}) \|\mathbf{v}_h\|_{W_h}, \end{align*} $$

using the stability result 3.10..

Choosing $\mathbf{v}_h = \mathbf{e}_h^n$ and $q_h = \eta_h^n$ in (3.36), we get

$$ a(\mathbf{e}_h^n, \mathbf{e}_h^n) + b(\mathbf{e}_h^{n-1}, \mathbf{u}_h, \mathbf{e}_h^n) + b(\mathbf{u}_h^{n-1}, \mathbf{e}_h^n, \mathbf{e}_h^n) + g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{e}_h^n) - g(\mathbf{u}_h, \mathbf{u}_h, \mathbf{e}_h^n) + h(\mathbf{e}_h^n, \mathbf{e}_h^n) = 0. $$
A corollary from the stability theorem 3.9 and the a priori estimates from the theoretical section is the boundedness result:

**Lemma 3.10.** There exists $C > 0$ depending only on the data such that

$$ (3.34) \qquad \|\mathbf{u}_h\|_{W_h} \le C \quad \text{and} \quad \|\mathbf{u}_h^n\|_{W_h} \le C, $$

for all $n \ge 0$.

We define the errors between the Galerkin theoretical velocities and the Galerkin approximated velocities obtained from the Oseen scheme $\mathbf{e}_h^n = \mathbf{u}_h^n - \mathbf{u}_h$ and the pressure errors $\eta_h^n = p_h^n - p_h$.

This leads to the error convergence result:

**Theorem 3.11.** Provided the condition on the data

$$ (3.35) \qquad \min(\nu_1, \nu_2, c_{d,1}, c_{d,2}) - CC_D \ge 0 \quad \text{and} \quad \frac{C(C_b + C_D)}{\min(\nu_1, \nu_2, c_{d,1}, c_{d,2}) - CC_D} < 1, $$

the method is converging and we have the error estimate:

$$ \|\mathbf{e}_h^n\|_{W_h} \le \frac{2}{\nu} \left( \frac{C(C_b + C_D)}{\min(\nu_1, \nu_2, c_{d,1}, c_{d,2}) - CC_D} \right)^{n+1} \|\mathbf{f}\|_{-1}. $$

*Proof.* We consider the difference at the n-th step, between the equations satisfied by $(\mathbf{u}_h, p_h)$ and by $(\mathbf{u}_h^n, p_h^n)$, which gives the relation

$$ (3.36) \qquad a(\mathbf{e}_h^n, \mathbf{v}_h) + b(\mathbf{e}_h^{n-1}, \mathbf{u}_h, \mathbf{v}_h) + b(\mathbf{u}_h^{n-1}, \mathbf{e}_h^n, \mathbf{v}_h) + g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{v}_h) - g(\mathbf{u}_h, \mathbf{u}_h, \mathbf{v}_h) $$

$$ (3.37) \qquad a(\mathbf{e}_h^n, \mathbf{v}_h) + b(\mathbf{e}_h^{n-1}, \mathbf{u}_h, \mathbf{v}_h) + b(\mathbf{u}_h^{n-1}, \mathbf{e}_h^n, \mathbf{v}_h) + g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{v}_h) - g(\mathbf{u}_h, \mathbf{u}_h, \mathbf{v}_h) $$

$$ (3.37) \qquad +h(\mathbf{e}_h^n, \mathbf{v}_h) - d(\mathbf{v}_h, \eta_h^n) + d(\mathbf{e}_h^n, q_h) = 0, $$

for $n \ge 1$ and $(\mathbf{v}_h, q_h) \in W_h \times Q_h$.

Consider the difference $g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{v}_h) - g(\mathbf{u}_h, \mathbf{v}_h)$, we have

$$ g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{v}_h) - g(\mathbf{u}_h, \mathbf{v}_h) = C_D \int_{\Gamma_I} \left( (\mathbf{u}_{1,h}^n - \mathbf{u}_{2,h}^n) |\mathbf{u}_{1,h}^{n-1} - \mathbf{u}_{2,h}^{n-1}| - (\mathbf{u}_{1,h} - \mathbf{u}_{2,h}) |\mathbf{u}_{1,h} - \mathbf{u}_{2,h}| \right) (\mathbf{v}_{1,h} - \mathbf{v}_{2,h}), $$

and denoting $\mathbf{U}_h^n := \mathbf{u}_{1,h}^n - \mathbf{u}_{2,h}^n$ and $\mathbf{U}_h := \mathbf{u}_{1,h} - \mathbf{u}_{2,h}$, $\mathbf{V}_h = \mathbf{v}_{1,h} - \mathbf{v}_{2,h}$, we get

$$ \begin{aligned} g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{v}_h) - g(\mathbf{u}_h, \mathbf{v}_h) &= C_D \int_{\Gamma_I} (\mathbf{U}_h^n |\mathbf{U}_h^{n-1}| - \mathbf{U}_h |\mathbf{U}_h|) \mathbf{V}_h \\ &= C_D \int_{\Gamma_I} (\mathbf{U}_h^n |\mathbf{U}_h^{n-1}| - \mathbf{U}_h^n |\mathbf{U}_h| + \mathbf{U}_h^n |\mathbf{U}_h| - \mathbf{U}_h |\mathbf{U}_h|) \mathbf{V}_h \\ &= C_D \int_{\Gamma_I} (\mathbf{U}_h^n (|\mathbf{U}_h^{n-1}| - |\mathbf{U}_h|) + |\mathbf{U}_h| (\mathbf{U}_h^n - \mathbf{U}_h)) \mathbf{V}_h. \end{aligned} $$

We get by the triangular inequality

$$ \begin{aligned} \|g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{v}_h) - g(\mathbf{u}_h, \mathbf{v}_h)\|_{W_h} &\le C_D (\|\mathbf{U}_h^n\|_{W_h} \|\mathbf{e}_h^{n-1}\|_{W_h} + \|\mathbf{U}_h\|_{W_h} \|\mathbf{e}_h^n\|_{W_h}) \|\mathbf{V}_h\|_{W_h} \\ &= C_D (\|\mathbf{u}_h^n\|_{W_h} \|\mathbf{e}_h^{n-1}\|_{W_h} + \|\mathbf{u}_h\|_{W_h} \|\mathbf{e}_h^n\|_{W_h}) \|\mathbf{v}_h\|_{W_h} \\ &\le CC_D (\|\mathbf{e}_h^{n-1}\|_{W_h} + \|\mathbf{e}_h^n\|_{W_h}) \|\mathbf{v}_h\|_{W_h}, \end{aligned} $$

using the stability result 3.10.

Choosing $\mathbf{v}_h = \mathbf{e}_h^n$ and $q_h = \eta_h^n$ in (3.36), we get

$$ a(\mathbf{e}_h^n, \mathbf{e}_h^n) + b(\mathbf{e}_h^{n-1}, \mathbf{u}_h, \mathbf{e}_h^n) + b(\mathbf{u}_h^{n-1}, \mathbf{e}_h^n, \mathbf{e}_h^n) + g(\mathbf{u}_h^n, \mathbf{u}_h^{n-1}, \mathbf{e}_h^n) - g(\mathbf{u}_h, \mathbf{u}_h, \mathbf{e}_h^n) + h(\mathbf{e}_h^n, \mathbf{e}_h^n) = 0. $$
Using the properties of the different operations, this yields to

$$
\min(\nu_1, \nu_2, c_{d,1}, c_{d,2}) \|\mathbf{e}_h^n\|_{W_h}^2 \le C_b \|\mathbf{u}_h\|_{W_h} \|\mathbf{e}_h^n\|_{W_h} \|\mathbf{e}_h^{n-1}\|_{W_h} + CC_D (\|\mathbf{e}_h^{n-1}\|_{W_h} + \|\mathbf{e}_h^n\|_{W_h}) \|\mathbf{e}_h\|_{W_h},
$$
hence

$$
(\min(\nu_1, \nu_2, c_{d,1}, c_{d,2}) - CC_D) \|\mathbf{e}_h^n\|_{W_h} \le (CC_b + CC_D) \|\mathbf{e}_h^{n-1}\|_{W_h}.
$$

Using again the stability result 3.10, we conclude thanks to the recursivity principle.

<figure><img src="image_15.png" /></figure>

**Remark 3.12.** The major issue of this convergence result is the fact that it is difficult to have access to the constants C and C_b involved. Nevertheless, choosing small values of C_D relatively to the range of values of v_1, v_2, c_{d,1} and c_{d,2} guarantees both conditions. We will see in the numerical simulations an example of this criteria being not fulfilled.

3.5. **Numerical simulations.** Finally, we are numerically simulating with Freefem++ the problem (3.28) thanks to the Oseen linearisation method.

We consider P2 finite elements for the velocities and P1 finite elements for the pressure. The figure 6 shows the mesh we have used in Freefem++.

We also decide to consider different viscosities: $\nu_1 = 0.4$ and $\nu_2 = 0.08$ and horizontal periodic boundary conditions, where $\Omega_1 = 1 \times 0.6$ and $\Omega_2 = 1 \times 0.4$, The convergence criteria we consider is

$$
(3.38) \qquad \frac{\|\mathbf{u}^n - \mathbf{u}^{n-1}\|_{H^1}}{\|\mathbf{u}^n\|_{H^1}} < \varepsilon_{\text{tol}},
$$

where the tolerance error $\varepsilon_{\text{tol}} = 10^{-3}$. Note that the stopping criteria is based on the relative error. The source $\mathbf{f}$ has been chosen constant equal to 1 everywhere in the domains and the constants $c_{d,1}$ and $c_{d,2}$ equal to 1. The velocity $\mathbf{V}$ has been chosen such that the air velocity at the top $\mathbf{V}_1 = 1$ and the water velocity at the bottom $\mathbf{V} = 0.1$.

The table 1 shows the number of iterations until the stopping criteria (3.38) is fulfilled, for different values of C_D.

We have proved that there is convergence if the condition (3.35) is fulfilled; we don't know what is happening when it is not fulfilled.

In this configuration, we have seen that there was quite a fast convergence when $C_D < 2$, but when $C_D \to 2.1$, the simulation is "exploding" and the numerical results are not what we expect. The figure 7 shows the simulation of the flows after 144 iterations for $C_D = 1$ and the figure 8, the case $C_D = 2.5$ after only 3 iterations.
Using the properties of the different operations, this yields to

$$ \min(\nu_1, \nu_2, c_{d,1}, c_{d,2}) \|\mathbf{e}_h^n\|_{W_h}^2 \le C_b \|\mathbf{u}_h\|_{W_h} \|\mathbf{e}_h^n\|_{W_h} \|\mathbf{e}_h^{n-1}\|_{W_h} + CC_D (\|\mathbf{e}_h^{n-1}\|_{W_h} + \|\mathbf{e}_h^n\|_{W_h}) \|\mathbf{e}_h\|_{W_h}, $$
hence

$$ (\min(\nu_1, \nu_2, c_{d,1}, c_{d,2}) - CC_D) \|\mathbf{e}_h^n\|_{W_h} \le (CC_b + CC_D) \|\mathbf{e}_h^{n-1}\|_{W_h}. $$

Using again the stability result 3.10, we conclude thanks to the recursivity principle.

□

**Remark 3.12.** The major issue of this convergence result is the fact that it is difficult to have access to the constants C and C_b involved. Nevertheless, choosing small values of C_D relatively to the range of values of ν_1, ν_2, c_{d,1} and c_{d,2} guarantees both conditions. We will see in the numerical simulations an example of this criteria being not fulfilled.

3.5. **Numerical simulations.** Finally, we are numerically simulating with Freefem++ the problem (3.28) thanks to the Oseen linearisation method.

We consider P2 finite elements for the velocities and P1 finite elements for the pressure. The figure 6 shows the mesh we have used in Freefem++.

We also decide to consider different viscosities: ν_1 = 0.4 and ν_2 = 0.08 and horizontal periodic boundary conditions, where Ω_1 = 1 × 0.6 and Ω_2 = 1 × 0.4, The convergence criteria we consider is

$$ (3.38) \qquad \frac{\|\mathbf{u}^n - \mathbf{u}^{n-1}\|_{H^1}}{\|\mathbf{u}^n\|_{H^1}} < \varepsilon_{tol}, $$

where the tolerance error ε_{tol} = 10⁻³. Note that the stopping criteria is based on the relative error. The source f has been chosen constant equal to 1 everywhere in the domains and the constants c_{d,1} and c_{d,2} equal to 1. The velocity V has been chosen such that the air velocity at the top V_1 = 1 and the water velocity at the bottom V = 0.1.

The table 1 shows the number of iterations until the stopping criteria (3.38) is fulfilled, for different values of C_D.

We have proved that there is convergence if the condition (3.35) is fulfilled; we don't know what is happening when it is not fulfilled.

In this configuration, we have seen that there was quite a fast convergence when C_D < 2, but when C_D → 2.1, the simulation is "exploding" and the numerical results are not what we expect. The figure 7 shows the simulation of the flows after 144 iterations for C_D = 1 and the figure 8, the case C_D = 2.5 after only 3 iterations.
<figure><img src="image_16.png" /><figcaption>FIGURE 6. Mesh for the domain $\Omega = \Omega_1 \times \Omega_2$</figcaption></figure>

<table><caption>TABLE 1. Convergence rate for different values of C<sub>D</sub></caption><thead><tr><th>Value of C<sub>D</sub></th><th>First value of n</th><th>||u<sub>1</sub><sup>n</sup>||<sub>L<sup>2</sup></sub></th><th>||u<sub>2</sub><sup>n</sup>||<sub>L<sup>2</sup></sub></th><th>||u<sub>1</sub><sup>n</sup>||<sub>H<sup>1</sup></sub></th><th>||u<sub>2</sub><sup>n</sup>||<sub>H<sup>1</sup></sub></th></tr></thead><tbody><tr><td>0.1</td><td>160</td><td>5.48849</td><td>16.1656</td><td>4.42758</td><td>15.9186</td></tr><tr><td>0.2</td><td>152</td><td>5.48479</td><td>16.1702</td><td>4.38152</td><td>16.0235</td></tr><tr><td>0.5</td><td>146</td><td>5.47764</td><td>16.1937</td><td>4.32994</td><td>16.3304</td></tr><tr><td>1</td><td>144</td><td>5.46818</td><td>16.2515</td><td>4.29199</td><td>16.7047</td></tr><tr><td>2</td><td>461</td><td>5.45242</td><td>16.4263</td><td>4.25692</td><td>17.2604</td></tr><tr><td>2.01</td><td>558</td><td>5.45227</td><td>16.4292</td><td>4.2564</td><td>17.2688</td></tr><tr><td>2.02</td><td>722</td><td>5.45214</td><td>16.4313</td><td>4.25593</td><td>17.2738</td></tr><tr><td>2.03</td><td>1325</td><td>5.45208</td><td>16.4322</td><td>4.25607</td><td>17.2753</td></tr><tr><td>2.05</td><td>8250</td><td>5.45192</td><td>16.4341</td><td>4.25618</td><td>17.2771</td></tr><tr><td>2.1</td><td>+&infin;</td><td></td><td></td><td></td><td></td></tr></tbody></table>

REFERENCES

[1] D. N. Arnold, F. Brezzi and F. Fortin, A stable finite element for the Stokes equations, Calcolo 21, 337-344, 1984
<figure><img src="image_15.png" /><figcaption>FIGURE 6. Mesh for the domain $\Omega = \Omega_1 \times \Omega_2$</figcaption></figure>

<table><caption>TABLE 1. Convergence rate for different values of C<sub>D</sub></caption><thead><tr><th>Value of C<sub>D</sub></th><th>First value of n</th><th>||u<sub>1</sub><sup>n</sup>||<sub>L<sup>2</sup></sub></th><th>||u<sub>2</sub><sup>n</sup>||<sub>L<sup>2</sup></sub></th><th>||u<sub>1</sub><sup>n</sup>||<sub>H<sup>1</sup></sub></th><th>||u<sub>2</sub><sup>n</sup>||<sub>H<sup>1</sup></sub></th></tr></thead><tbody><tr><td>0.1</td><td>160</td><td>5.48849</td><td>16.1656</td><td>4.42758</td><td>15.9186</td></tr><tr><td>0.2</td><td>152</td><td>5.48479</td><td>16.1702</td><td>4.38152</td><td>16.0235</td></tr><tr><td>0.5</td><td>146</td><td>5.47764</td><td>16.1937</td><td>4.32994</td><td>16.3304</td></tr><tr><td>1</td><td>144</td><td>5.46818</td><td>16.2515</td><td>4.29199</td><td>16.7047</td></tr><tr><td>2</td><td>461</td><td>5.45242</td><td>16.4263</td><td>4.25692</td><td>17.2604</td></tr><tr><td>2.01</td><td>558</td><td>5.45227</td><td>16.4292</td><td>4.2564</td><td>17.2688</td></tr><tr><td>2.02</td><td>722</td><td>5.45214</td><td>16.4313</td><td>4.25593</td><td>17.2738</td></tr><tr><td>2.03</td><td>1325</td><td>5.45208</td><td>16.4322</td><td>4.25607</td><td>17.2753</td></tr><tr><td>2.05</td><td>8250</td><td>5.45192</td><td>16.4341</td><td>4.25618</td><td>17.2771</td></tr><tr><td>2.1</td><td>+&infin;</td><td></td><td></td><td></td><td></td></tr></tbody></table>

REFERENCES

[1] D. N. Arnold, F. Brezzi and F. Fortin, A stable finite element for the Stokes equations, Calcolo 21, 337-344, 1984
<figure><img src="image_17.png" /><figcaption>FIGURE 7. Two fluids with $\nu_1 = 0.4$, $\nu_2 = 0.08$, $C_D = 1, c_{d,1} = c_{d,2} = 1$</figcaption></figure>

[2] M. Bercovier, O.Pironneau Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numerische Mathematik 33, 1979.

[3] C. Bernardi, T. Chacón Rebollo, R. Lewandowski and F. Murat, A model for two coupled turbulent fluids. I. Analysis of the system Stud. Math. Appl., 2002.

[4] M. Bulicek, J. Malek, K. R Rajagopal, Navier's slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J. 56(1), 51-85, 2006.

[5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universi-text, Springer, 2011.

[6] T. Chacón Rebollo and R. Lewandowski. *Mathematical and numerical foundations of turbulence models and applications*. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2014.

[7] P. G. Ciarlet, The finite element method for elliptic problems, volume Vol. 4 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.

[8] J. Connors, Convergence analysis and computational testing of the finite element discretization of the Navier-Stokes alpha model Numerical methods for partial differential equations, 2009
<figure><img src="image_16.png" /><figcaption>FIGURE 7. Two fluids with $\nu_1 = 0.4$, $\nu_2 = 0.08$, $C_D = 1, c_{d,1} = c_{d,2} = 1$</figcaption></figure>

[2] M. Bercovier, O. Pironneau. Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numerische Mathematik 33, 1979.

[3] C. Bernardi, T. Chacón Rebollo, R. Lewandowski and F. Murat, A model for two coupled turbulent fluids. I. Analysis of the system Stud. Math. Appl., 2002.

[4] M. Bulicek, J. Malek, K. R Rajagopal, Navier's slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J. 56(1), 51-85, 2006.

[5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universi-text, Springer, 2011.

[6] T. Chacón Rebollo and R. Lewandowski. *Mathematical and numerical foundations of turbulence models and applications*. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2014.

[7] P. G. Ciarlet, The finite element method for elliptic problems, volume Vol. 4 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.

[8] J. Connors, Convergence analysis and computational testing of the finite element discretization of the Navier-Stokes alpha model Numerical methods for partial differential equations, 2009
<figure><img src="image_18.png" /><figcaption>FIGURE 8. Two fluids with $\nu_1 = 0.4$, $\nu_2 = 0.08$, $C_D = 2.5$, $c_{d,1} = c_{d,2} = 1$</figcaption></figure>

[9] V. Duchêne, *On the rigid-lid approximation for two shallow layers of immiscible fluids with small density contrast*, J. Nonlinear Sci., Vol. 24, No. 4, 2014, pp. 579–632.

[10] P. Gauckler, Etudes théoriques et pratiques sur l'écoulement et le mouvement des eaux, Compte rendu de l'Académie des sciences, 1867.

[11] V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations, volume 5 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986. Theory and algorithms.

[12] J.G. Haywood, R. Rannacher, Finite-element approximations of the nonstationary Navier-Stokes problem. Part II: stability of the solution and error estimates uniform in time, SIAM J. Numer. Anal. 23, 750-777, 1986.

[13] Y. He, J. Li, Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg, 198, 1351-1359, 2009.

[14] Y. He, A. Wang, A simplified two-level method for the steady Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 197, 1568-1576, 2008.

[15] F. Hecht, New development in Freeem++, Journal of Numerical Mathematics, 2012

[16] F. Jauberteau, C. Rosier, R. Temam, A nonlinear Galerkin method for the Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, Vol 80, 245-260, 1990.
<figure><img src="image_17.png" /><figcaption>FIGURE 8. Two fluids with $\nu_1 = 0.4$, $\nu_2 = 0.08$, $C_D = 2.5$, $c_{d,1} = c_{d,2} = 1$</figcaption></figure>

[9] V. Duchêne, *On the rigid-lid approximation for two shallow layers of immiscible fluids with small density contrast*, J. Nonlinear Sci., Vol. 24, No. 4, 2014, pp. 579–632.

[10] P. Gauckler, Etudes théoriques et pratiques sur l'écoulement et le mouvement des eaux, Compte rendu de l'Académie des sciences, 1867.

[11] V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations, volume 5 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986. Theory and algorithms.

[12] J.G. Haywood, R. Rannacher, Finite-element approximations of the nonstationary Navier-Stokes problem. Part II: stability of the solution and error estimates uniform in time, SIAM J. Numer. Anal. 23, 750-777, 1986.

[13] Y. He, J. Li, Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg, 198, 1351-1359, 2009.

[14] Y. He, A. Wang, A simplified two-level method for the steady Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 197, 1568-1576, 2008.

[15] F. Hecht, New development in Freeem++, Journal of Numerical Mathematics, 2012

[16] F. Jauberteau, C. Rosier, R. Temam, A nonlinear Galerkin method for the Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, Vol 80, 245-260, 1990.
[17] W. Layton, A Two-Level Discretization Method for the Navier-Stokes Equations, Computers Maths. Applic., Vol 26, 33-38, 1993.

[18] J-L. Lions, R. Temam, S. Hong Wang, *Mathematical theory for the coupled atmosphere-ocean models. (CAO III)*, *J. Math. Pures Appl. (9)*, Vol. 74, No. 2, 1995, pp. 105-163.

[19] Jacques-Louis Lions, Roger Temam, Shou Hong Wang, *Modèles et analyse mathématiques du système océan/atmosphère. II. Couplage*, C. R. Acad. Sci. Paris Sér. I Math., Vol. 316, No. 2, 1993, pp. 211-215.

[20] Jacques-Louis Lions, Roger Temam, Shou Hong Wang, *Problème à frontière libre pour les modèles couplés de l'océan et de l'atmosphère*, C. R. Acad. Sci. Paris Sér. I Math., Vol. 318, No. 12, 1994, pp. 1165-1171.

[21] F. Legeais and R. Lewandowski, Continuous boundary condition at the interface for two coupled fluids, Appl. Math. Lett., Volume 135, 2023.

[22] R.Lewandowski, *Analyse mathématique et océanographie, Recherches en Mathématiques Appliquées* [Research in Applied Mathematics], Vol. 39, 1997, xx+281 pages.

[23] R. Manning, On the flow of water in open channels an pipes, Transactions of the Institution of civil engineers of Ireland, 1891.

[24] C. Pelletier, *Mathematical study of the air-sea coupling problem including turbulent scale effects*. Thesis, University of Grenoble, France, UCLouvain, Louvain-la-Neuve, Belgium, 2018.

[25] Charles Pelletier, Florian Lemarié, Eric Blayo, *Sensitivity analysis and metamodels for the bulk parametrization of turbulent air-sea fluxes*, Quarterly Journal of the Royal Meteorological Society, Vol. 144, No. 712, 2017.

[26] R. Temam, Navier-Stokes equations, theory and numerical analysis, North-Holland, Amsterdam, 1984

[27] Sophie Thery, Charles Pelletier, Florian Lemarié, and Eric Blayo, *Analysis of Schwarz wave-form relaxation for the coupled Ekman boundary layer problem with continuously variable coefficients*. Numer. Algorithms, 89(3):1145-1181, 2022.

[28] Xuejun Xu, C. O. Chow, and S. H. Lui, On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations. M2AN Math. Model. Numer. Anal., 39(6):1251-1269, 2005.

[29] H. Xu, Y. He, Some iterative element methods for steady Navier-Stokes equations with different viscosities, Journal of Computational Physics, Volume 232, 136-152, 2013

(F. Legeais) UNIV RENNES, IRMAR, UMR CNRS 6625, AND ODYSSEY TEAM, INRIA RENNES, FRANCE

Email address: francois.legeais@univ-rennes1.fr
[17] W. Layton, A Two-Level Discretization Method for the Navier-Stokes Equations, Computers Maths. Applic., Vol 26, 33-38, 1993.

[18] J-L. Lions, R. Temam, S. Hong Wang, *Mathematical theory for the coupled atmosphere-ocean models. (CAO III)*, *J. Math. Pures Appl. (9)*, Vol. 74, No. 2, 1995, pp. 105-163.

[19] Jacques-Louis Lions, Roger Temam, Shou Hong Wang, *Modèles et analyse mathématiques du système océan/atmosphère. II. Couplage*, C. R. Acad. Sci. Paris Sér. I Math., Vol. 316, No. 2, 1993, pp. 211-215.

[20] Jacques-Louis Lions, Roger Temam, Shou Hong Wang, *Problème à frontière libre pour les modèles couplés de l'océan et de l'atmosphère*, C. R. Acad. Sci. Paris Sér. I Math., Vol. 318, No. 12, 1994, pp. 1165-1171.

[21] F. Legeais and R. Lewandowski, Continuous boundary condition at the interface for two coupled fluids, Appl. Math. Lett., Volume 135, 2023.

[22] R.Lewandowski, *Analyse mathématique et océanographie, Recherches en Mathématiques Appliquées* [Research in Applied Mathematics], Vol. 39, 1997, xx+281 pages.

[23] R. Manning, On the flow of water in open channels an pipes, Transactions of the Institution of civil engineers of Ireland, 1891.

[24] C. Pelletier, *Mathematical study of the air-sea coupling problem including turbulent scale effects*. Thesis, University of Grenoble, France, UCLouvain, Louvain-la-Neuve, Belgium, 2018.

[25] Charles Pelletier, Florian Lemarié, Eric Blayo, *Sensitivity analysis and metamodels for the bulk parametrization of turbulent air-sea fluxes*, Quarterly Journal of the Royal Meteorological Society, Vol. 144, No. 712, 2017.

[26] R. Temam, Navier-Stokes equations, theory and numerical analysis, North-Holland, Amsterdam, 1984

[27] Sophie Thery, Charles Pelletier, Florian Lemarié, and Eric Blayo, *Analysis of Schwarz wave-form relaxation for the coupled Ekman boundary layer problem with continuously variable coefficients*. Numer. Algorithms, 89(3):1145-1181, 2022.

[28] Xuejun Xu, C. O. Chow, and S. H. Lui, On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations. M2AN Math. Model. Numer. Anal., 39(6):1251-1269, 2005.

[29] H. Xu, Y. He, Some iterative element methods for steady Navier-Stokes equations with different viscosities, Journal of Computational Physics, Volume 232, 136-152, 2013

(F. Legeais) UNIV RENNES, IRMAR, UMR CNRS 6625, AND ODYSSEY TEAM, INRIA RENNES, FRANCE

Email address: francois.legeais@univ-rennes1.fr