# A three-dimensional model for two coupled turbulent fluids: numerical analysis of a finite element approximation

A three-dimensional model for two coupled turbulent fluids: numerical analysis of a finite... Abstract This article deals with the numerical analysis of a coupled two-fluid Reynolds-averaged Navier–Stokes (RANS) turbulence model, such as atmosphere–ocean flow. Each fluid is modeled by the coupled steady Stokes equations with the equation for the turbulent kinetic energy (TKE). In this model, the eddy viscosities for velocity and TKE depend on the TKE, the production (source) term for the TKEs is only in $$L^1$$ and the boundary condition for the TKEs on the interface between the two flows depends quadratically on the difference of velocities. To overcome the lack of regularity, we approximate the initial system by a regularized system, in which the eddy viscosities and source terms for the TKEs are regularized by convolution. We perform a full finite element discretization of the regularized model, combined with a decoupled iterative linearization procedure. We prove that the discrete scheme converges to the continuous scheme for large enough eddy viscosities in natural norms. Finally, we present some numerical tests where we study the accuracy of the procedure, and simulate a realistic flow in which an imposed wind in the upper atmosphere generates an upwelling in the oceanic flow. 1. Introduction This article deals with the numerical analysis of the approximation of Reynolds-averaged Navier–Stokes (RANS) turbulence models and, more specifically, on coupled two-fluid RANS models. Typically, this model addresses the coupled atmosphere–ocean system, whose accurate numerical simulation is crucial to analyse the main issues related to climate change. From the practical point of view, RANS models are rather diffusive and provide overall predictions of many flows of engineering interest (cf. Davidson, 2013). However, from the mathematical point of view, RANS models are more singular problems than the Navier–Stokes equations. The main mathematical difficulties in the analysis of RANS models are that the source term (the production term) for the turbulent kinetic energy (TKE) has just $$L^1({\it{\Omega}})$$ regularity, and that the equation for the TKE does not make sense in $$H^{-1}({\it{\Omega}})$$. Its solution must be understood in the renormalized—or entropy—sense (cf. Boccardo & Gallou, 1992; Bénilan et al., 1995; Boccardo et al., 1996a,b; Betta et al., 2002, 2003). The numerical analysis of the finite element approximation of elliptic equations with right-hand side in $$L^1$$ has been performed for linear diffusion equations (cf. Casado-Díaz et al., 2007). The main reason is that the extension of the basic estimates of renormalized solutions (the Boccardo–Gallouet estimates) only hold if the numerical scheme satisfies a discrete maximum principle. For convection–diffusion equations, there exist a few finite element schemes that satisfy this principle, based on the multidimensional unwinding techniques (cf. Tabata, 1978; Codina, 1993; Burman & Ern, 2005). However, the extension of the numerical analysis of these schemes to the solution of equations with $$L^1$$ right-hand side has not been performed yet. For finite volume discretizations, there exist more schemes satisfying the maximum principle, whose analysis has been performed in some cases. Because of these difficulties, the mathematical and numerical analyses of RANS models performed up to date has been applied to simplified equations. In Bernardi et al. (2002) study, a model for two-coupled turbulent fluids and its numerical approximation, where the turbulent kinetic energy (TKE) equation only contains eddy diffusion (with bounded eddy viscosities) and the production term. This model also contains a nonlinear friction term to model the interactions between the two fluids across the interface: \begin{eqnarray}\label{P1}\left\{ \begin{array}{rcl}-\nabla \cdot\left(\alpha_i\left(k_i\right)\nabla {\boldsymbol u}_i\right) + \nabla p_i &=& {\boldsymbol f}_i \; \, \mbox{in} {\it{\Omega}}_i, \\ \nabla \cdot {\boldsymbol u}_i &=& 0 \quad \mbox{in} {\it{\Omega}}_i,\\ -\nabla \cdot\left(\gamma_i\left(k_i\right)\nabla k_i\right) &=&\alpha_i\left(k_i\right) \left|\nabla {\boldsymbol u}_i \right|^2 \mbox{in} {\it{\Omega}}_i,\\ {\boldsymbol u}_i &=& {\boldsymbol 0} \mbox{on} {\it{\Gamma}}_i,\\ k_i &=& 0 \mbox {on} {\it{\Gamma}}_i,\\ \alpha_i\left(k_i\right)\partial_{{\boldsymbol n}_i}{\boldsymbol u}_i - p_i {\boldsymbol n}_i + \kappa\left ({\boldsymbol u}_i -{\boldsymbol u}_j\right)\left| {\boldsymbol u}_i-{\boldsymbol u}_j\right| &=& {\boldsymbol 0} \mbox{on} {\it{\Gamma}}, 1\leq i \neq j \leq 2,\\ k_i &=& \lambda\left|{\boldsymbol u}_1 -{\boldsymbol u}_2\right|^2 \mbox {on} {\it{\Gamma}}, \end{array}\right. \end{eqnarray} (1.1) where both domains $${\it{\Omega}}_i$$, $$i=1,2$$ are bounded in $$\mathbb R^d,~ d=2,3$$, which are either convex or of class $$\mathbb C^{1,1}$$, with boundaries $$\partial {\it{\Omega}}_i= {\it{\Gamma}}_i \cup {\it{\Gamma}}$$ and $${\it{\Gamma}} = \overline {\it{\Omega}}_1 \cap \overline {\it{\Omega}}_2$$ being the interface between the two fluids. In this model, $${\it{\Gamma}}$$ is assumed to be flat. This is the ‘rigid lid hypothesis’ introduced by Bryan (1969), an hypothesis that provides a good modeling for flows at large space scales, although needs adjustments for interfaces governed by more complex dynamics, related to atmospheric pressure and tidal variability effects, among others. The main focus of the article is to analyse a full finite element discretization of this model, which has been studied in several articles (see Bernardi et al., 2003, 2004a; Chacón Rebollo et al., 2010), but this analysis remains open. We are mainly interested in the mathematical treatment of the difficulties due to the nonlinear terms, in particular those coming from RANS turbulence modeling. From the mathematical point of view, the steady model applies to large time scales, although it does not make sense from a physical point of view, as the boundary conditions are largely variable in time. An unsteady RANS model would be more appropriate to approximate a realistic ocean–atmosphere coupled flow. Each of the two turbulent fluids is modeled by a simplified one-equation turbulence model whose unknowns are the velocity $${\boldsymbol u}_i$$, the pressure $$p_i$$ and the TKE $$k_i$$. The generation of eddy viscosity in flow $$i$$ is modeled by the term $$-\nabla \cdot\left(\alpha_i\left(k_i\right)\nabla {\boldsymbol u}_i\right)\!.$$ The (positive) quantity $$\alpha_i(k_i)$$ is the eddy viscosity. This is a simplification of the usual modeling of Reynolds stress tensor by $$R_i \simeq - \alpha_i\left(k_i\right)\left(\nabla {\boldsymbol u}_i + \nabla ^t{\boldsymbol u}_i \right)$$ to simplify the mathematical analysis. The equations for the TKE only contains the eddy dissipation and production terms, respectively, given by $$-\nabla \cdot\left(\gamma_i\left(k_i\right)\nabla k_i\right)\quad \mbox{and}\quad \alpha_i\left(k_i\right) \left|\nabla {\boldsymbol u}_i\right|^2,$$ where $$\gamma_i(k_i)$$ is the eddy diffusion for the TKE $$k_i$$. The interface terms model the friction between the two fluids (fifth equation in (1.1)) and the generation of TKE (sixth equation in (1.1)), both are wall laws that model the dissipation of turbulence in the boundary layers on both sides of the interface. The positive parameters $$\kappa$$ and $$\lambda$$ are the friction and energy production coefficients, respectively. The analysis of model (1.1) reported in Bernardi et al. (2002) is based on the reformulation of the equations for the TKEs by transposition combined with a compactness argument. The global system is proved to admit a solution $$\left({\boldsymbol u}_i,p_i,k_i\right)\in {\boldsymbol H}^1\left({\it{\Omega}}_i\right) \times L^2\left({\it{\Omega}}_i\right)\times H^{s}\left({\it{\Omega}}_i\right)$$ regularity for all $$s<1/2$$. Several subsequent works have dealt with the numerical approximation of this model. The same authors perform in Bernardi et al. (2003) an error analysis for spectral discretizations of (1.1) for smooth solutions. In Bernardi et al. (2004a), a finite element discretization by piecewise affine finite elements of this problem is studied using the compactness method. The convergence of a subsequence of discrete solutions to the continuous solution is proved. In Chacón Rebollo et al. (2010), a linearization procedure to solve the same continuous model for large eddy viscosities is introduced. The procedure is proved to converge for large eddy viscosities, assuming that the discrete velocities and TKEs are uniformly bounded in $$W^{1, 3+\varepsilon}$$ norm, for some $$\varepsilon >0$$. In Chacón Rebollo & Lewandowski (2014, Chapter 7), the right-hand side for the TKE equations is regularized by convolution, for a problem similar to (1.1) with a single flow. It is proved that the regularized problems admit a solution and that a subsequence of these converge to a weak solution of the original problem, in $${\boldsymbol H}^1\left({\it{\Omega}}\right) \times L^2\left({\it{\Omega}}\right)\times W^{1,q}\left({\it{\Omega}}\right)$$ for $$1\le q <3/2$$, where $${\it{\Omega}}$$ is the flow domain. The TKE equation is satisfied in the renormalized sense, the Boccardo–Gallouet estimates play an essential role in the proof, being the origin of the estimates in $$W^{1,q}\left({\it{\Omega}}\right)$$ of the TKE. A different approach was followed by Holst et al. (2010) in the context of the convergence analysis of finite element approximations of the Joule heating problem. The main technical tool in this article is an identity that allows to rewrite the right-hand side for the temperature (with $$L^1$$ regularity, as in our case does the TKE) in terms of the data for the electric potential and a controllable diffusion term. This technique does not seem to be applicable in our case because of the presence of the pressure gradient in the momentum equation, combined with the friction boundary conditions. In this article, we follow the approach of Chacón Rebollo & Lewandowski (2014) and study the numerical approximation of a regularized approximation of (1.1) by convolution. More concretely, we consider the numerical solution of a full finite element discretization of the regularized problem by the iterative procedure introduced in Chacón Rebollo et al. (2010). We prove that the procedure converges under similar conditions, i.e., for large enough eddy viscosities, without any additional boundedness hypothesis on the velocities and TKEs. The hardest technical point is the obtaining of estimates of the interface quadratic terms. We treat them by specific discrete lifting operators that are compatible with the discretizations of velocity and TKE. We present some numerical tests, where, on one side, we study the accuracy of the procedure. On another side, we simulate a realistic flow in which an imposed wind in the upper atmosphere generates an upwelling in the oceanic flow. The article is structured as follows. In Section 2, we introduce the regularized approximation of model (1.1). Section 3 describes the full finite element approximation of the regularized problem and its solution by the iterative procedure. Section 4 is devoted to the analysis of convergence of this discretization. Finally, Section 5 reports the numerical tests. 2. Regularized model To describe the regularized approximation of model (1.1) that we shall study, let us denote by $$\omega_\varepsilon \left({\boldsymbol x}\right) = \frac{1}{\varepsilon^d}\omega \left(\frac{{\boldsymbol x}}{\varepsilon} \right)$$ a smoothing (mollifier) sequence, defined as follows, see, for instance, Brezis (1983): $$\label{mollifiers}\omega \in \mathcal C^\infty_c\left(\mathbb R^d\right)\!,\qquad \int_{\mathbb R^d} \omega= 1, \qquad supp\left (\omega\right) \subset B(0,1), \qquad \omega \ge 0; \qquad \mbox{for} \; \varepsilon >0.$$ (2.1) For a function $$\psi \in L^p\left({\it{\Omega}}\right)\!, p > 1$$, consider the convolution $$\left(\psi * \omega_\varepsilon\right) \left({\boldsymbol x}\right)\; = \int_{R^d} \tilde{\psi}\left({\boldsymbol x} -{\boldsymbol y}\right) \, \omega_\varepsilon\left({\boldsymbol y}\right) \; {\rm{d}}{\boldsymbol y},$$ where $$\tilde{\psi}$$ is the extension by zero of $$\psi$$ outside $${\it{\Omega}}$$. Then, $$\psi * \omega_\varepsilon \in C^\infty\left(R^d\right)$$ and the following properties hold (cf. Brezis, 1983): $$\label{regu}\displaystyle \lim _{\varepsilon \rightarrow 0} \psi * \omega_\varepsilon = \; \psi \; \mbox{in}\, L^p\left({\it{\Omega}}\right)\!, \left\| \psi * \omega_\varepsilon \right\|_{W^{k,q}\left(\mathbb R^d\right)} \le C_{\varepsilon} \, \left\| \psi \right\|_{L^p({\it{\Omega}})}\!,$$ (2.2) for any integer $$k \ge 1$$, and real number $$q \in [p,+\infty]$$, for some constant $$C_{\varepsilon}$$, such that $$C_{\varepsilon} \to +\infty$$ as $$\varepsilon \to 0^+$$. We are now in a position to state the regularized problem that we study in this article: \begin{eqnarray}\label{P1Reg}\left\{ \begin{array}{rcl}-\nabla \cdot(\alpha_i(k^\varepsilon_i)\nabla {\boldsymbol u}_i) + \nabla p_i &=& {\boldsymbol f}_i \ \,\mbox{in} {\it{\Omega}}_i, \\ \nabla \cdot {\boldsymbol u}_i &=& 0 \mbox{in} {\it{\Omega}}_i,\\ -\nabla \cdot(\gamma_i(k^\varepsilon_i)\nabla k_i) &=& \alpha_i(k^\varepsilon_i) \nabla {\boldsymbol u}_i : \nabla {\boldsymbol u}_i^\varepsilon \mbox{in} {\it{\Omega}}_i,\\ {\boldsymbol u}_i &=& {\boldsymbol 0} \mbox{on} {\it{\Gamma}}_i,\\ k^\varepsilon_i &=& 0 \mbox {on} {\it{\Gamma}}_i,\\ \sigma_i&=& {\boldsymbol 0} \mbox{on} {\it{\Gamma}},\\ k_i &=& \lambda |{\boldsymbol u}^\varepsilon_1 -{\boldsymbol u}^\varepsilon_2|^2 \mbox{on} {\it{\Gamma}}, \end{array}\right. \end{eqnarray} (2.3) where $${\boldsymbol u}^\varepsilon_i = {\boldsymbol u}_i * \omega_\varepsilon$$ (componentwise), $$k^\varepsilon_i = k_i * \omega_\varepsilon$$ and $$\sigma_i = \alpha_i(k^\varepsilon_i) \partial_{{\boldsymbol n}_i}{\boldsymbol u}_i - p_i {\boldsymbol n}_i + \kappa ({\boldsymbol u}_i - {\boldsymbol u}_j)| {\boldsymbol u}_i-{\boldsymbol u}_j|, \;1\leq i \neq j \leq 2.$$ The analysis performed in Chacón Rebollo & Lewandowski (2014, Section 7.4) readily extends to prove that problem (2.3) admits a solution that converges to a solution of (1.1) in $${\boldsymbol H}^1({\it{\Omega}}_i) \times L^2({\it{\Omega}}_i)\times W^{1,q}({\it{\Omega}}_i)$$, as $$\varepsilon$$ tends to $$0$$, whenever the TKE satisfy a homogeneous Dirichlet boundary condition on $${\it{\Gamma}}$$ (seventh equation of (2.3)). However, here, we prefer to include a more realistic boundary condition, modeling the generation of TKEs by friction at the interfaces. Note that the more singular terms in model (1.1) are those that model the generation of TKEs, either by eddy diffusion (right-hand side of third equation) or by friction at the interface (seventh equation). These terms are regularized in model (2.3), in addition to the eddy viscosities. Let us recall some standard notation that we use throughout the article. We denote by $$W^{s,p}({\it{\Omega}}_i)$$ the real Sobolev space, $$0 \leq s < \infty, 0 \leq p \leq \infty$$, equipped with the norm $$\| \cdot \|_{W^{s,p}({\it{\Omega}}_i)}$$. The space $$W^{s,p}_0({\it{\Omega}}_i)$$ is the completion of the space of the smooth functions compactly supported in $${\it{\Omega}}_i$$ with respect to the $$\| \cdot\|_{W^{s,p}({\it{\Omega}}_i)}$$ norm. For $$s=1$$ and $$p=2$$, we denote the Hilbert spaces $$W^{1,2}({\it{\Omega}}_i)$$ (resp. $$W^{1,2}_0(\varOmega_i)$$) by $$H^1({\it{\Omega}}_i)$$ (resp. $$H^1_0({\it{\Omega}}_i)$$). The related norm is denoted by $$\| \cdot\|_{1,{\it{\Omega}}_i}$$. The case of $$s=0$$ corresponds to the space $$L^2({\it{\Omega}}_i)$$ equipped with its standard norm $$\|\cdot\|_{0,{\it{\Omega}}_i}$$. We finally denote by $$|\cdot|_{1,{\it{\Omega}}_i}$$ the seminorm in $$H^1({\it{\Omega}}_i)$$ given by $$|v|_{1,{\it{\Omega}}_i}=\|\nabla v\|_{0,{\it{\Omega}}_i}$$. To formulate the coupled problem (2.3) in variational form, we introduce the velocity and TKE spaces defined as follows: \begin{align}{\boldsymbol X}_i &= \{{\boldsymbol v}_i \in {\boldsymbol H}^1({\it{\Omega}}_i),\quad {\boldsymbol v}_i = {\boldsymbol 0} \; \mbox{on} \, {\it{\Gamma}}_i \}. \label{EspX} \\ \end{align} (2.4) \begin{align}K_i &= \{\ell_i \in H^1({\it{\Omega}}_i),\quad \ell_i = 0 \; \mbox{on} \, {\it{\Gamma}}_i \}. \label{EspK}\end{align} (2.5) The trace operator is continuous from $${\boldsymbol X}_i$$ (resp. $$K_i$$) onto the space $${\boldsymbol H}^{\frac{1}{2}}_{00}({\it{\Gamma}})$$ (resp. $$H^{\frac{1}{2}}_{00}({\it{\Gamma}})$$), [this is the subspace of $$H^{\frac{1}{2}}({\it{\Gamma}})$$ formed by functions whose prolongation by zero to either $$\partial {\it{\Omega}}_1$$—or $$\partial {\it{\Omega}}_2$$—belongs to $$H^{\frac{1}{2}}(\partial {\it{\Omega}}_1)$$—or $$H^{\frac{1}{2}}(\partial {\it{\Omega}}_2)$$—, see Lions & Magenes, 1970 for its definition]. We next state a relevant hypothesis concerning the eddy viscosities and diffusions: Hypothesis 2.1 The functions $$\alpha_i$$ and $$\gamma_i$$ belong to $$W^{1,\infty}(\mathbb R_+)$$. Moreover, there exist positive constants $$\delta$$ and $$\nu$$, such that \begin{equation*}\label{H1}\forall \ell \in \mathbb R_+,\qquad \nu \leq \alpha^{(m)}_i(\ell) \leq \delta, \quad \nu \leq \gamma^{(m)}_i(\ell) \leq \delta, \quad m=0 \; \mbox{or}\; 1. \end{equation*} This hypothesis holds in practical applications, as some kind of numerical smoothing or truncation is applied to avoid too small or too large eddy viscosities that may lead to instabilities. From now on, the following spaces are introduced to simplify notations: $$\mathcal F_i = {\boldsymbol X}_i \times L^2({\it{\Omega}}_i) \times H^1_0({\it{\Omega}}_i), \qquad \mathcal G_i = {\boldsymbol X}_i \times L^2({\it{\Omega}}_i) \times K_i.$$ We are now in a position to write the weak formulation of problem (2.3): Given $${\boldsymbol f}_i \in {\boldsymbol L}^2({\it{\Omega}}_i)$$, find the triplet $$({\boldsymbol u}_i,p_i,k_i) \in \mathcal G_i$$ such that for all test functions $$({\boldsymbol v}_i,q_i,\phi_i) \in \mathcal F_i$$ it holds \begin{align}& a_i(k_i;{\boldsymbol u}_i,{\boldsymbol v}_i) + b_i({\boldsymbol v}_i,p_i) + \kappa \int_ {{\it{\Gamma}}} |{\boldsymbol u}_i-{\boldsymbol u}_j|({\boldsymbol u}_i -{\boldsymbol u}_j) \cdot {\boldsymbol v}_i \, {\rm{d}}\tau = \int_{{\it{\Omega}}_i}{\boldsymbol f}_i\cdot {\boldsymbol v}_i \,{\rm{d}} {\boldsymbol x}, \label{Var1} \\ \end{align} (2.6) \begin{align}& b_i({\boldsymbol u}_i,q_i) =0, \label{Var1_ps} \\ \end{align} (2.7) \begin{align}& k_i= \,0 \; {\rm{on}} \,{\it{\Gamma}}_i,\quadk_i= \lambda |{\boldsymbol u}^{\varepsilon}_i -{\boldsymbol u}^{\varepsilon}_j|^2 \quad\mbox{on}~ {\it{\Gamma}}, \quad{\rm{and}} \label{Var2_CL} \\ \end{align} (2.8) \begin{align}& {\mathcal N_i}(k_i;k_i,\phi_i) = \displaystyle\int_{{\it{\Omega}}_i}\alpha_i(k^{\varepsilon}_i)\, \nabla{\boldsymbol u}_i : \nabla{\boldsymbol u}^{\varepsilon}_i \; \phi_i \, {\rm{d}}{\boldsymbol x}; \label{Var2}\end{align} (2.9) where the forms $$a_i(\cdot;\cdot,\cdot),~b_i(\cdot,\cdot)$$ and $${\mathcal N_i}(\cdot;\cdot,\cdot)$$ are defined by \begin{align}a_i(\ell_i;{\boldsymbol u}_i,{\boldsymbol v}_i) &= \int_{{\it{\Omega}}_i} \alpha(\ell^{\varepsilon}_i) \nabla {\boldsymbol u}_i : \nabla {\boldsymbol v}_i\, {\rm{d}}{\boldsymbol x}, \quad b_i({\boldsymbol v}_i,q_i) = -\displaystyle \int_{{\it{\Omega}}_i} q_i\,\nabla\cdot {\boldsymbol v}_i\,{\rm~d} {\boldsymbol x}, \notag \\ & {\rm{and}} \quad {\mathcal N_i}(\ell_i; k_i, \phi_i) = \displaystyle \int_{{\it{\Omega}}_i}{\gamma_i(\ell^{\varepsilon}_i)}\nabla k_i \cdot \nabla \phi_i \,{\rm{d}}{\boldsymbol x}.\notag \end{align} Note that since $${\boldsymbol u}_i \in {\boldsymbol H}^1({\it{\Omega}}_i)$$ and according to Proposition IV.20 in Brezis (1983), $$\nabla {\boldsymbol u}_i^\varepsilon$$ belongs to $$C^\infty(\mathbb R^d)$$, then the term $$\displaystyle\int_{{\it{\Omega}}_i}\alpha_i(k^\varepsilon_i) \, \nabla{\boldsymbol u}_i : \nabla{\boldsymbol u}^\varepsilon_i \;\phi_i$$ in (2.9) is well defined for all $$\phi_i \in H^1_0({\it{\Omega}}_i)$$. Also, due to the continuous Sobolev embedding from $${\boldsymbol H}^{\frac{1}{2}}(\partial {\it{\Omega}}_i)$$ into $$L^3(\partial {\it{\Omega}}_i)^d$$, the term $$\displaystyle \int_ {{\it{\Gamma}}} \left|{\boldsymbol u}_i-{\boldsymbol u}_j \right|\left({\boldsymbol u}_i -{\boldsymbol u}_j\right) \cdot{\boldsymbol v}_i \, {\rm{d}}\tau$$ in (2.6) is well defined for all $${\boldsymbol v}_i \in{\boldsymbol H}^1({\it{\Omega}}_i)$$. The source functions $${\boldsymbol f}_i$$ can be taken in $${\boldsymbol H}^{-1}({\it{\Omega}}_i)$$ and then the scalar product in $$L^2$$ must be replaced by the duality product between $$<\cdot,\cdot>_{{\boldsymbol H}^1,{\boldsymbol H}^{-1}}$$. Nevertheless, we prefer to work with $${\boldsymbol f}_i \in L^2({\it{\Omega}}_i)^d$$ for simplicity of notation. 3. Discrete iterative procedure To introduce the finite element approximation of problem (2.3), we assume that the domains $${\it{\Omega}}_1$$ and $${\it{\Omega}}_2$$ are polygonal (when $$d=2$$) or polyhedric (when $$d=3$$). Consider a family of triangular grids $$(\mathcal T_{i,h})_h$$ of $$\overline {\it{\Omega}}_i$$ that we assume to be regular, in the usual sense of the finite element method (FEM) (see Ciarlet, 1978; Girault & Raviart, 1986; Ern & Guermond, 2004). For each non-negative integer $$m$$ and any element $$K$$ in $$\mathcal T_{i,h}$$, let $$\mathcal P_{m}(K)$$ denote the space of restriction to $$K$$ of polynomials with $$d$$ variables and total degree $$\leq m$$. Thus, we choose the following spaces: \begin{eqnarray}\label{eq:espx}&&\quad{\boldsymbol X}_{i,h} =\left \{{\boldsymbol v}_{i,h} \in C^0\left(\overline {\it{\Omega}}_i \right)^d \, \forall K \in \mathcal T_{i,h}, \, \restriction{{\boldsymbol v}_{i,h}}{K} \in \mathcal P_{2}(K)^d \right\} \cap {\boldsymbol X}_i, \\ \end{eqnarray} (3.1) \begin{eqnarray}&&\quad M_{i,h} =\left \{q_{i,h} \in C^0\left(\overline {\it{\Omega}}_i \right) \, \forall K \in \mathcal T_{i,h}, \, \restriction{q_{i,h}}{K} \in \mathcal P_{1}(K) \right \}\!, \; \label{eq:espm} \\ \end{eqnarray} (3.2) \begin{eqnarray}&&\quad K_{i,h}=\left \{\ell_{i,h} \in C^0\left(\overline {\it{\Omega}}_i \right) \, \forall K \in \mathcal T_{i,h}, \, \restriction{\ell_{i,h}}{K} \in \mathcal P_{2}(K), \, \restriction{\ell_{i,h}}{{\it{\Gamma}}_i} =0 \right\}\!,\label{eq:espk}\\ \end{eqnarray} (3.3) \begin{eqnarray}&&\quad \label{W}W_{i,h} = \left \{\phi_{i,h} \in \mathcal C^0 \left({\it{\Gamma}} \right)\; \forall e \in \mathcal E_{i,h}, \, \restriction{\phi_{i,h}}{e} \in \mathcal P_{2}(e),\, \restriction{\phi_{i,h}}{\partial{\it{\Gamma}}} =0 \right \}\!, \end{eqnarray} (3.4) where $$\left(\mathcal E_{i,h}\right)_{i,h}$$ denotes all faces ($$d =3$$) or edges ($$d=2$$) of triangulation $$\mathcal T_{i,h}$$, which are contained in $${\it{\Gamma}}$$. Consider the standard Lagrange interpolation operators \begin{align}\mathcal{Q}_{i,h} &:\quad {\boldsymbol X}_i \cap C^0(\overline{{\it{\Omega}}_i})^d\mapsto {\boldsymbol X}_{i,h} \label{Op_Pih1}, \\ \end{align} (3.5) \begin{align}\mathcal{P}_{i,h} &:\quad M_i \cap C^0(\overline{{\it{\Omega}}_i})\mapsto M_{i,h}, \quad \rm{} \label{Op_Ph1} \\ \end{align} (3.6) \begin{align}\mathcal S_{i,h} &: \quad H^{1}({\it{\Omega}}_i) \cap C^0(\overline{{\it{\Omega}}_i}) \mapsto K_{i,h} \label{Op_Sih}.\end{align} (3.7) For instance $$\mathcal{Q}_{i,h}$$ is defined by: $$\forall K \in \mathcal T_{i,h},\;$$ $$\mathcal{Q}_{i,h} \,{\boldsymbol v}|_K = \mathcal I_K \, {\boldsymbol v} \; {\boldsymbol v} \in C^0\left(\overline {\it{\Omega}}_i \right)^d\!, \;$$ where $$\mathcal I_K \, {\boldsymbol v}$$ is the only polynomial of $$\mathcal P_{2}(K)^d$$ that takes the same values as the function $${\boldsymbol v}$$ at degrees of freedom of the local Lagrange interpolation on $$\mathcal P_{2}(K)^d$$. These interpolation operators satisfy the following approximation and stability properties (for more details, see, for instance, Ciarlet, 1978; Bernardi et al., 2004b; Ern & Guermond, 2004; Chacón Rebollo & Lewandowski, 2014): If $${\boldsymbol v} \in H^{\ell_1}({\it{\Omega}}_i)^d\cap X_i$$, $$\ell_1=0,1, 2$$; $$p \in H^{\ell_2}({\it{\Omega}}_i) \cap M_i \;$$, $$\ell_2=0,1$$; $$k \in W^{\ell_3}({\it{\Omega}}_i) \cap K_i$$, $$\ell_3=0,1, 2$$, \begin{align}\| {\boldsymbol v} - \mathcal{Q}_{i,h}\,{\boldsymbol v} \|_{H^{1}({\it{\Omega}}_i)^d} &\leq c h^{\ell_1 - 1} \, |{\boldsymbol v}|_{H^{\ell_1}({\it{\Omega}}_i)^d}, \label{Err_Interp_Pih} \\ \end{align} (3.8) \begin{align}\| p - \mathcal{P}_{i,h}\,p \|_{H^{1}({\it{\Omega}}_i)} &\leq c h^{\ell_2 - 1} \, |p|_{H^{\ell_1}({\it{\Omega}}_i)}, \label{Err_Interp_Ph} \\ \end{align} (3.9) \begin{align}\| k - \mathcal S_{i,h}\,k \|_{H^{1}({\it{\Omega}}_i)} &\leq c h^{\ell_3 - 1} \, |k|_{W^{\ell_2}({\it{\Omega}}_i)}.\label{Err_Interp_Sih}\end{align} (3.10) Also, the family of spaces $$\left({\boldsymbol X}_{i,h}, M_{i,h}\right)_{h>0}$$ satisfy the discrete Babuska–Brezzi inf–sup condition on $${\it{\Omega}}_i$$ (see, for instance, Brezzi & Fortin, 1991). There exists a constant $$\beta_{i} > 0$$, such that: $$\label{inf-sup}q_{i,h} \in M_{i,h}, \qquad \displaystyle \sup_{{\boldsymbol v}_{i,h} \in {\boldsymbol X}_{i,h}}\displaystyle \frac{b_i\left({\boldsymbol v}_{i,h},q_{i,h} \right)} {\left| {\boldsymbol v}_{i,h} \right|} \geq \beta_{i} \|q_{i,h}\|_{0,{\it{\Omega}}_i}.$$ (3.11) In addition, consider the standard Lagrange interpolation operator $$\mathcal L_{i,h}:\quad H^{\frac{1}{2}}_{00}\left({\it{\Gamma}} \right)\cap C^0(\overline{{\it{\Gamma}}}) \longrightarrow W_{i,h} \label{Op_Lih1}$$ (3.12) given by, for all $$e \in \mathcal E_{i,h}$$, $$\mathcal L_{i,h} \,\omega|_e = \mathcal J_e \, \omega, \; \forall \omega \in C^0\left(\overline {\it{\Gamma}} \right)^d,$$ where $$\mathcal J_e \, \omega$$ is the only polynomial of $$\mathcal P_{2}(e)$$ that takes the same values as the function $$\omega$$ at the degrees of freedom of the local Lagrange interpolation on $$\mathcal P_{2}(e)$$. The operators $$L_{i,h}$$ satisfies the stability property $$\label{contlih}\|\mathcal L_{i,h} w_i\|_{H^{1/2}_{00}({\it{\Gamma}})} \le C \,\|w_i\|_{H^{1/2}_{00}({\it{\Gamma}})}\, \mbox{for any} w_i \in H^{1/2}_{00}({\it{\Gamma}})\cap C^0(\overline{{\it{\Gamma}}}),$$ (3.13) for some constant $$C>0$$. Note also that, as $$W_{i,h}$$ is the ‘trace space’ on $${\it{\Gamma}}$$ of $$K_{i,h}$$, the operators $${\cal L}_{i,h}$$ and $${\cal S}_{i,h}$$ satisfy the following compatibility condition. Hypothesis 3.1 For all $$w_i \in H^{1}({\it{\Omega}}_i)$$, the trace on $${\it{\Gamma}}$$ of the interpolate $$\mathcal S_{i,h} \left(w_i\right)$$ coincides with the interpolate of the trace of $$w_i$$: $$\label{Op_Lih2}\mathcal L_{i,h} \left(w_{i_{|_{\it{\Gamma}}}} \right) = \restriction {\left(\mathcal S_{i,h} \,w_i \right)} {{\it{\Gamma}}}.$$ (3.14) We finally assume that there exists a lifting operator $$\mathcal R_{i,h}:W_{i,h} \mapsto K_{i,h}$$ such that, $$\label{lift}\mathcal R_{i,h}(\phi_{i,h})_{|_{\partial {\it{\Omega}}_i}}=\phi_{i,h}\,\mbox{for any}\phi_{i,h} \in W_{i,h}.$$ (3.15) Furthermore, this operator verifies the stability property: $$\label{Rih}\|\mathcal R_{i,h}(\phi_{i,h})\|_{W^{1,p}({\it{\Omega}}_i)} \, \leq \,C_p\, \| \phi_{i,h}\|_{W^{1-1/p,p}(\partial {\it{\Omega}}_i)}\qquad \,\forall p \in]1,+\infty[,$$ (3.16) for some constant $$C_p>0$$. The existence of a lifting operator verifying (3.15)–(3.16) is proved in Bernardi et al. (2004b, Theorem 4.1). Let us introduce the discrete spaces: $$\mathcal F_{i,h} = {\boldsymbol X}_{i,h} \times M_{i,h} \times K_{i,h}, \quadK^0_{i,h} =K_{i,h}\cap H_{0}^{1}({\it{\Omega}}_i).$$ We are in a position to build the discrete problem from (4.1) to (4.5): Assume known $$\left({\boldsymbol u}^{n}_{i,h},p^{n}_{i,h},k^{n}_{i,h}\right) \in \mathcal F_{i,h},\; n \geq 0$$ 1. Compute $$\left({\boldsymbol u}^{n+1}_{i,h},p^{n+1}_{i,h}\right) \in {\boldsymbol X}_{i,h} \times M_{i,h}$$, such that $$\forall({\boldsymbol v}_{i,h},q_{i,h}) \in {\boldsymbol X}_{i,h} \times M_{i,h}$$, \begin{align}\label{iter1h}&a_i\left(k^{n}_{i,h};{\boldsymbol u}_{i,h}^{n+1},\nabla {\boldsymbol v}_{i,h}\right) \,+\, b_i\left({\boldsymbol v}_{i,h},p_{i,h}^{n+1}\right) \notag\\ &\quad+\kappa \displaystyle \int_{{\it{\Gamma}}}\left | {\boldsymbol u}_{i,h}^{n+1} -{\boldsymbol u}_{j,h}^{n+1}\right| \, \left ({\boldsymbol u}_{i,h}^{n+1} -{\boldsymbol u}_{j,h}^{n+1}\right)\, \cdot \,{\boldsymbol v}_{i,h} \, {\rm{d}}\tau =\displaystyle \int_{{\it{\Omega}}_i} {\boldsymbol f}_i \, \cdot \, {\boldsymbol v}_{i,h},\end{align} (3.17) $$\label{iter2h}b_i\left({\boldsymbol u}_{i,h}^{n+1},q_{i,h}\right) =0.$$ (3.18) 2. Compute $$k^{n+1}_{i,h}\in K_{i,h}$$, such that $$\forall\phi_{i,h} \in K^0_{i,h}$$, \begin{align}k^{n+1}_{i,h} & = 0 \mbox{on}{\it{\Gamma}}_i, \label{iter3h} \\ \end{align} (3.19) \begin{align}k^{n+1}_{i,h} &= \lambda \mathcal L_{i,h} \left(\left|{\boldsymbol u}^{n+1,\varepsilon}_{i,h} -{\boldsymbol u}^{n+1,\varepsilon}_{2,h}\right|^2\right) \mbox{on} {\it{\Gamma}} \quad \mbox{and} \label{iter3hbis}\end{align} (3.20) \begin{align}\label{iter4h}{\mathcal N_i}(k^{n}_{i,h};k^{n+1}_{i,h},\phi_{i,h}) &= \displaystyle \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \nabla {\boldsymbol u}^{n+1}_{i,h} : \nabla {\boldsymbol u}^{n+1,\varepsilon}_{i,h}\phi_{i,h} \, {\rm{d}}{\boldsymbol x}.\end{align} (3.21) This system is readily proved to admit a unique solution, similarly to system (4.1)–(4.2)–(4.3)–(4.5). We have considered here a fully coupled discretization of the friction boundary condition for the TKEs on the interface $${\it{\Gamma}}$$, as our purpose is to study a complete discretization of the fixed point algorithm introduced in Chacón Rebollo et al. (2010). It would also be possible to study the decoupled discretization introduced in Connors et al. (2009), which allows to separately solve the problems in the two subdomains. 4. Numerical analysis of the discrete scheme In this section, we prove the convergence of the solution of the algorithm (3.17)–(3.21) to a solution of the regularized problem (2.3). The proof is based on a recursive estimate of the errors between the finite element sequence $$\left({\boldsymbol u}_{i,h}^n,k_{i,h}^n \right)$$ and the continuous sequence $$\left({\boldsymbol u}_{i}^n,k_{i}^n \right)$$ solution of the iterative algorithm: For $$n \in \mathbb N$$ and for a given $$\left({\boldsymbol u}^{n}_i,p^{n}_i,k^{n}_i\right) \in \mathcal G_i$$, find $$\left({\boldsymbol u}^{n+1}_i,p^{n+1}_i,k^{n+1}_i\right) \in \mathcal G_i$$ by: 1. Find $$\left({\boldsymbol u}^{n+1}_i,p^{n+1}_i\right) \in {\boldsymbol X}_i \times L^2({\it{\Omega}}_i),$$ such that for all $$\left({\boldsymbol v}_i,q_i\right) \in {\boldsymbol X}_i \times L^2({\it{\Omega}}_i)$$, \begin{align}\label{iter1}&a_i\left(k^n_{i};{\boldsymbol u}_{i}^{n+1},\nabla {\boldsymbol v}_{i}\right) + b_i\left({\boldsymbol v}_{i},p_{i}^{n+1}\right) \notag\\ &\quad+ \kappa \displaystyle \int_{{\it{\Gamma}}}\left | {\boldsymbol u}_{i}^{n+1} -{\boldsymbol u}_{j}^{n+1}\right| \left ({\boldsymbol u}_{i}^{n+1} -{\boldsymbol u}_{j}^{n+1}\right)\, \cdot \,{\boldsymbol v}_{i} {\rm{d}}\tau =\displaystyle \int_{{\it{\Omega}}_i} {\boldsymbol f}_i \, \cdot \, {\boldsymbol v}_{i} \quad \forall{\boldsymbol v}_i \in {\boldsymbol X}_i.\end{align} (4.1) $$\label{iter2}\qquad \forall q_i \in L^2({\it{\Omega}}_i), \quad b_i({\boldsymbol u}_i^{n+1},q_i) =0.$$ (4.2) 2. Find $$k^{n+1}_i \in K_i$$ such that $$\forall\phi_i \in H^{1}_0({\it{\Omega}}_i)$$, \begin{align}\label{iter3}k^{n+1}_i & = 0 \mbox{on}{\it{\Gamma}}_i, \\ \end{align} (4.3) \begin{align}k^{n+1}_i & = \lambda \left|{\boldsymbol u}^{n+1,\varepsilon}_1 -{\boldsymbol u}^{n+1,\varepsilon}_2\right|^2 \mbox{on}{\it{\Gamma}},\quad \mbox{} \, \\ \end{align} (4.4) \begin{align}\label{iter4}{\mathcal N_i}(k^n_i;k^{n+1}_i,\phi_i) & = \displaystyle \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_i) \, \nabla {\boldsymbol u}^{n+1}_i : \nabla{\boldsymbol u}^{n+1,\varepsilon}_i \; \phi_i \, {\rm{d}}{\boldsymbol x}.\end{align} (4.5) In Chacón Rebollo et al. (2010), it is proved that problem (4.1)–(4.5), without regularization, admits a unique solution. The existence follows from Brouwer’s fixed-point theorem, based on energy stability estimates (see (4.9) and (4.10) below), and the uniqueness follows because the boundary term in (4.1) is dissipative. This proof readily extends to problems (4.1)–(4.5) and (3.17)–(3.21), and we shall omit it for brevity. In Chacón Rebollo et al. (2010), it is also proved that this iterative procedure is contracting for the nonregularized problem, when the eddy viscosities are large enough: Theorem 4.1 (Convergence of the continuous scheme) Assume that Hypothesis 2.1 holds and that $${\boldsymbol f}_i \in L^2\left({\it{\Omega}}_i\right)^d$$. Then, if $$\nu$$ is large enough then there exists a positive constant $$K < 1$$, depending only on $${\it{\Omega}}_i$$, $$M$$ and on the data $$\kappa$$, $$\nu$$ and $$\lambda$$, such that for all $$n \in \mathbb N^*$$, $$\label{contractivness}\begin{split}\displaystyle \sum _{i=1}^2 \left(\left|{\boldsymbol u}^{n+1}_i -{\boldsymbol u}^{n}_i\right|^2_{1,{\it{\Omega}}_i}+ \left|k^{n+1}_i -k^{n}_i\right|^2_{1,{\it{\Omega}}_i} \right) & \leq K \displaystyle \sum_{i=1}^2 \left|k^{n}_i -k^{n-1}_i\right|^2_{1,{\it{\Omega}}_i}. \end{split}$$ (4.6) In what follows, for simplicity of notation, we consider only the three-dimensional case $$d=3$$. The two-dimensional analysis is similar. We shall denote by $$c$$ a generic positive constant, which may vary from line to line, but are always independent of the $$n$$ and $$h$$ and $$\nu$$. Furthermore, for the sake of simplicity, we take $$\kappa_1 =\kappa_2 = \lambda = 1$$. Let us recall that according to the interpolation error estimates (3.8) and (3.10) we have \begin{align}\displaystyle \sum_{i=1}^2 \left| \mathcal{Q}_{i,h}\,{\boldsymbol u}^{n+1}_{i} - {\boldsymbol u}^{n+1}_{i} \right|_{1,{\it{\Omega}}_i}& \leq c \, h\, \displaystyle \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i}, \label{Est_I} \\ \end{align} (4.7) \begin{align}\displaystyle \sum_{i=1}^2 \left | \mathcal S_{i,h}\,(k ^{n+1}_{i}) - k^{n+1}_{i} \right|_{1,{\it{\Omega}}_i}&\leq c \, h\, \displaystyle \sum_{i=1}^2 | k^{n+1}_{i}|_{2,{\it{\Omega}}_i}. \label{Est_IK}\end{align} (4.8) To go further, we start by proving the boundedness in $$H^1$$ norm of the continuous and discrete sequences $$({\boldsymbol u}^n_{i}, k^n_{i})$$ and $$({\boldsymbol u}^n_{i,h}, k^n_{i,h})$$. To deal with the boundary condition modeling the generation of TKE at the interface, we use the result of continuity of the product of traces on $${\it{\Gamma}}$$ (see, for instance, Grisvard, 1985; Girault & Raviart, 1986). Lemma 4.2 Assume that $${\it{\Omega}}$$ is a bounded Lipschitz-continuous open subset of $$\mathbb R^d$$. Let $$s,\,s_1$$ and $$s_2$$ be three non-negative reals and $$p,\,p_1, \,p_2$$ be three real numbers in $$[1, +\infty)$$, such that $$s_1 \geq s$$, $$s_2 \geq s$$ and either $$s_1+s_2-s \geq d\left(\frac{1}{p_1} + \frac{1}{p_2} - \frac{1}{p}\right)\geq 0,\quad s_i-s >d\left(\frac{1}{p_i}-\frac{1}{p}\right)\quadi=1,2$$ or $$s_1+s_2-s > d\left(\frac{1}{p_1} + \frac{1}{p_2} - \frac{1}{p}\right)\geq 0,\quad s_i-s \geq d\left(\frac{1}{p_i}-\frac{1}{p}\right)\quadi=1,2.$$ Then, the mapping $$(u,\,v) \to uv$$ is a continuous bilinear map from $$W^{s_1,p_1}({\it{\Omega}}) \times W^{s_2,p_2}({\it{\Omega}})$$ to $$W^{s,p}({\it{\Omega}})$$, and there exists a constant $$C$$ depending on $$s_1,\, s_2,\, s,\,p_1,\,p_2$$ and $$p$$ such that for all $$f \in W^{s_1,p_1}({\it{\Omega}})$$, $$g \in W^{s_,p_2}({\it{\Omega}})$$ it holds $$\| f g\|_{W^{s,p}({\it{\Omega}})} \le C \, \|f\|_{W^{s_1,p_1}({\it{\Omega}})}\, \|g\|_{W^{s_,p_2}({\it{\Omega}})}.$$ Using the results of Hebey (1999), this lemma also holds for Sobolev spaces defined on compact Riemannian manifolds, this is the case of $${\it{\Gamma}}$$. We may now state the following result: Proposition 4.3 Assume that Hypothesis (2.1) holds and that $${\boldsymbol f}_i \in L^2({\it{\Omega}}_i)^d$$. Then, there exists a non-negative constant $$c$$ depending only on the domain $${\it{\Omega}}_i$$ and the coefficients $$\kappa$$, such that $$n \in \mathbb N$$: \begin{gather} \label{Velo_bound}\sum_{i=1}^2 \left | {\boldsymbol u}_{i}^{n} \right |_{1,{\it{\Omega}}_i}\leq \frac{c}{\nu} \left(\sum_{i=1}^2 \left \| {\boldsymbol f}_i \right \|_{0,{\it{\Omega}}_i}^2\right)^{\frac{1}{2}}, \qquad \sum_{i=1}^2 \left | {\boldsymbol u}_{i,h}^{n} \right |_{1,{\it{\Omega}}_i}\leq \frac{c}{\nu} \left(\sum_{i=1}^2 \left \| {\boldsymbol f}_i \right \|_{0,{\it{\Omega}}_i}^2\right)^{\frac{1}{2}}.\\ \end{gather} (4.9) \begin{gather}\label{TKE_bound}\sum_{i=1}^2 \left | k_{i}^{n} \right |_{1,{\it{\Omega}}_i}\leq \frac{c\, C_\varepsilon}{\nu^2} \left(\sum_{i=1}^2 \left \| {\boldsymbol f}_i \right \|_{0,{\it{\Omega}}_i}^2\right)\!, \qquad \sum_{i=1}^2 \left | k_{i,h}^{n} \right |_{1,{\it{\Omega}}_i}\leq \frac{c\, C_\varepsilon}{\nu^2} \left(\sum_{i=1}^2 \left \| {\boldsymbol f}_i \right \|_{0,{\it{\Omega}}_i}^2\right)\!. \end{gather} (4.10) Proof. To obtain estimates (4.9), let us respectively take $$({\boldsymbol u}_{i}^{n}, p_i^n)$$ and $$({\boldsymbol u}_{i,h}^{n},p_{i,h}^{n})$$ as test functions in equations (2.6)–(2.7) and (3.17)–(3.18). Summing up for $$i=1,2$$ it follows $$\sum_{i=1}^2 a_i(k_i;{\boldsymbol u}_{i}^{n},{\boldsymbol u}_{i}^{n}) + \kappa \int_ {{\it{\Gamma}}} |{\boldsymbol u}_1^n-{\boldsymbol u}_2^n|^3 \, {\rm{d}}\tau = \sum_{i=1}^2\int_{{\it{\Omega}}_i}{\boldsymbol f}_i\cdot {\boldsymbol u}_i^n \, {\rm{d}} {\boldsymbol x},$$ and the first estimate in (4.9) follows. The second one is similar. The estimates (4.10) are more involved because of the nonlinear boundary equation on $${\it{\Gamma}}$$. Let us consider the lifting operators (see Bernardi et al., 2004b, for details) $$\mathcal R_{i}: H^{\frac{1}{2}}_{00}({\it{\Gamma}}) \mapsto H^1({\it{\Omega}}_i)$$, $$i=1,2$$ such that, $$\label{liftK}\mathcal R_{i}(\phi_{i}) = \phi_{i}, \; \mbox{on} \; {\it{\Gamma}}, \quad\mathcal R_{i}(\phi_{i}) = 0 \; \mbox{on} \; {\it{\Gamma}}_i,\quad\|\mathcal R_{i} (\phi_{i}) \|_{1,{\it{\Omega}}_i} \leq c \, \|\phi_{i} \|_{H^{\frac{1}{2}}_{00}({\it{\Gamma}})}.$$ (4.11) Note that by Proposition 4.3 (with $$s_1=s=1/2, \, s_2=1-1/5, \,p_1=p=2, \,p_2=5$$), and the properties of mollifiers (2.2), \begin{align}& \left\| \mathcal R_{i}(k^{n+1}_i) \right\|_{1,{\it{\Omega}}_i} \le c\,\left\|\, \left|{\boldsymbol u}_1^{n+1,\varepsilon}-{\boldsymbol u}_2^{n+1,\varepsilon}\right|^2\,\right\|_{H^{\frac{1}{2}}_{00}({\it{\Gamma}})} \nonumber\\ &\quad \le c\, \left\|{\boldsymbol u}_1^{n+1,\varepsilon}-{\boldsymbol u}_2^{n+1,\varepsilon}\,\right\|_{H^{\frac{1}{2}}_{00}({\it{\Gamma}})}\, \left\|{\boldsymbol u}_1^{n+1,\varepsilon}-{\boldsymbol u}_2^{n+1,\varepsilon}\,\right\|_{W^{1-\frac{1}{5},5}({\it{\Gamma}})}\nonumber\\ &\quad \le c\, \left (\left|{\boldsymbol u}_1^{n+1,\varepsilon}\right|_{1,{\it{\Omega}}_1}+ \left|{\boldsymbol u}_2^{n+1,\varepsilon}\right|_{1,{\it{\Omega}}_2}\right)\, \left (\left\|{\boldsymbol u}_1^{n+1,\varepsilon}\right\|_{W^{1,5}({\it{\Omega}}_1)}+\left\|{\boldsymbol u}_2^{n+1,\varepsilon}\right\|_{W^{1,5}({\it{\Omega}}_2)}\right)\nonumber\\ &\quad \le c\, C_\varepsilon \left(\left|{\boldsymbol u}_1^{n+1}\right|_{1,{\it{\Omega}}_1}^2+ \left|{\boldsymbol u}_2^{n+1}\right|_{1,{\it{\Omega}}_2}^2\right)\! \label{estri12}.\end{align} (4.12) Next, taking $$\phi_i= k^{n+1}_i - \mathcal R_i(k^{n+1}_i) \in H^1_0({\it{\Omega}}_i)$$ in (4.5), using Holder inequality, (4.11) and (4.12) leads to \begin{align}{\mathcal N_i}(k^n_i;k^{n+1}_i,k^{n+1}_i) &= \displaystyle \int_{{\it{\Omega}}_i}\alpha_i\left(k^{n,\varepsilon}_i\right) \, \nabla {\boldsymbol u}^{n+1}_i : \nabla{\boldsymbol u}^{n+1,\varepsilon}_i \phi_i \, {\rm{d}}{\boldsymbol x}- {\mathcal N_i}\left(k^n_i;k^{n+1}_i,\mathcal R_i\left(k^{n+1}_i\right)\right)\nonumber \\ &\leq c \, \biggl(\left|{\boldsymbol u}^{n+1}_i\right|_{1,{\it{\Omega}}_i} \, \,\left \|\nabla {\boldsymbol u}^{n+1,\varepsilon}_i\right\|_{L^\infty({\it{\Omega}}_i)} \, \left(\left|k^{n+1}_i\right|_{1,{\it{\Omega}}_i}+\left|\mathcal R_{i}\left(k^{n+1}_i\right)\right|_{1,{\it{\Omega}}_i} \right) \nonumber \\ &\quad\qquad+ \left|k^{n+1}_i\right|_{1,{\it{\Omega}}_i} \left|\mathcal R_i\left(k^{n+1}_i\right)\right|_{1,{\it{\Omega}}_i}\biggl)\nonumber \\ & \leq c \, C_\varepsilon \biggl(|{\boldsymbol u}^{n+1}_1|_{1,{\it{\Omega}}_1}^2 + |{\boldsymbol u}^{n+1}_2|_{1,{\it{\Omega}}_2}^2 \biggl)\,|k^{n+1}_i|_{1,{\it{\Omega}}_i}\label{procd}.\end{align} (4.13) Then, $$\displaystyle |k^{n+1}_i|_{1,{\it{\Omega}}_i} \le \frac{c \, C_\varepsilon}{\nu}\, \biggl(|{\boldsymbol u}^{n+1}_1|_{1,{\it{\Omega}}_1}^2 + |{\boldsymbol u}^{n+1}_2|_{1,{\it{\Omega}}_2}^2 \biggl)$$ and by Hypothesis 2.1, the first estimate in (4.10) follows. To obtain the second estimate, observe that \begin{align}\left \| \mathcal R_{i,h}(k^{n+1}_{i,h})\right\|_{1,{\it{\Omega}}_i}& \le c\,\left\|\,\mathcal L_{i,h} \left (\left|{\boldsymbol u}_{1,h}^{n+1,\varepsilon}-{\boldsymbol u}_{2,h}^{n+1,\varepsilon}\right|^2\right)\,\right\|_{H^{\frac{1}{2}}_{00}({\it{\Gamma}})}\le c\,\left\|\left|{\boldsymbol u}_{1,h}^{n+1,\varepsilon}-{\boldsymbol u}_{2,h}^{n+1,\varepsilon}\right|^2\,\right\|_{H^{\frac{1}{2}}_{00}({\it{\Gamma}})} \nonumber\\ & \le c\, C_\varepsilon \left (\left|{\boldsymbol u}_{1,h}^{n+1}\right|_{1,{\it{\Omega}}_1}^2+ \left|{\boldsymbol u}_{2,h}^{n+1}\right|_{1,{\it{\Omega}}_2}^2\right)\!, \nonumber \end{align} where the last inequality is obtained proceeding as in (4.12), Now, taking $$\varphi_{i,h}=k_{i,h}^{n+1}- \mathcal R_{i,h}(k_{i,h}^{n+1}) \in K_{i,h}^0$$ and proceeding as to obtain (4.13) leads to the second estimate in (4.10).   □ 4.1 Analysis of the velocity sequence $${\boldsymbol u}^n_{i,h}$$ To prove the convergence of the discrete scheme, the idea is to estimate the difference between the continuous and discrete sequences, $$\left | {\boldsymbol u}^{n}_{i,h} - {\boldsymbol u}^{n}_i \right |_{1,{\it{\Omega}}_i}$$ and $$\left | k^{n}_{i,h} - k^{n}_i \right |_{1,{\it{\Omega}}_i}$$. We prove at first that the error in velocity is driven by that in TKE (plus interpolation errors): Theorem 4.4 Assume that Hypotheses (2.1) and (3.1) hold, and assume that the solution $$({\boldsymbol u}^{n+1}_{i}, p^{n+1}_{i})$$ belongs to $${\boldsymbol H}^{2}({\it{\Omega}}_i)\times H^{1}({\it{\Omega}}_i)$$. Then, the following estimate holds \begin{align} \label{Est_U_K}\displaystyle \sum_{i=1}^2 \left| {\boldsymbol u}^{n+1}_{i,h} - {\boldsymbol u}^{n+1}_{i}\right|^2_{1,{\it{\Omega}}_i}&\leq \displaystylec\, \sum_{i=1}^2 \biggl (\left(1+\frac{1}{\nu}\right)\, h^{2} \, | {\boldsymbol u}^{n+1}_{i}|^2_{2,{\it{\Omega}}_i}+ \frac{1}{\nu}\, h\, | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i} \nonumber \\ &\quad+ \left(1+\frac{1}{\nu}\right)\, h^{2} \, | p^{n+1}_{i}|^2_{1,{\it{\Omega}}_i} + \; \displaystyle \frac{C^2_\varepsilon}{\nu^4} \, | k^{n}_{i,h} - k^{n}_{i}|^2_{1,{\it{\Omega}}_i} \biggl).\end{align} (4.14) Proof. Let us consider the Stokes projection $${\it{\Pi}}_{i,h}:X_i \mapsto X_{i,h}$$ defined, for $${\boldsymbol z}_i \in X_i$$, as the solution—together with some discrete pressure $$r_{i,h} \in M_{i,h}$$—of the problem $$\label{presdrop} \left \{ \begin{array}{c}\displaystyle\int_{{\it{\Omega}}_i} \nabla ({\it{\Pi}}_{i,h} {\boldsymbol z}_i) \cdot \nabla {\boldsymbol v}_{i,h} +\int_{{\it{\Omega}}_i} r_{i,h}\nabla \cdot {\boldsymbol v}_{i,h} =\displaystyle \int_{{\it{\Omega}}_i} \nabla {\boldsymbol z}_i \cdot \nabla {\boldsymbol v}_{i,h} +\int_{{\it{\Omega}}_i} p_{i}\nabla \cdot {\boldsymbol v}_{i,h},\\ \displaystyle \int_{{\it{\Omega}}_i} \nabla \cdot ({\it{\Pi}}_{i,h} {\boldsymbol z}_i) \, q_{i,h}= \displaystyle\int_{{\it{\Omega}}_i} \nabla \cdot {\boldsymbol z}_i \, q_{i,h}, \end{array}\right.$$ (4.15)$$\forall{\boldsymbol v}_{i,h}\in X_{i,h}$$, $$\forall q_{i,h}\in M_{i,h}$$. This problem admits a unique solution, thanks to the discrete inf–sup condition (3.11). Moreover, it follows (see Girault & Raviart, 1986) \begin{align}& |{\it{\Pi}}_{i,h} {\boldsymbol z}_i-\mathcal Q_{i,h}{\boldsymbol z}_i|_{1,{\it{\Omega}}_i}+\|r_{i,h}-\mathcal P_{i,h} p_i\|_{L^2({\it{\Omega}}_i)}\notag \\ \label{estproy}&\quad\le C_i\, (| {\boldsymbol z}_i-\mathcal Q_{i,h}{\boldsymbol z}_i|_{1,{\it{\Omega}}_i}+\|p_i-\mathcal P_{i,h} p_i\|_{L^2({\it{\Omega}}_i)}),\end{align} (4.16) for some constant $$C_i>0$$, $$i=1,2$$. We multiply (4.1) and (3.17) by $${\boldsymbol v}_{i,h} = {\boldsymbol e}_{i,h}^{n+1}= {\boldsymbol u}_{i,h}^{n+1}- {\it{\Pi}}_{i,h} {\boldsymbol u}_i^{n+1}$$, and we compute the difference between the obtained equations. Summing upon $$i=1,2$$, thanks to (4.15) the pressure terms cancel, yielding \begin{align*}& \underbrace{\sum_{i=1}^2 \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \nabla \left({\boldsymbol u}^{n+1}_{i,h} - {\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i} \right): \nabla {\boldsymbol e}_{i,h}^{n+1}}_{I_1} \\ & \quad+ \underbrace{\sum_{i=1}^2 \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \nabla \left({\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i} - {\boldsymbol u}^{n+1}_{i} \right): \nabla {\boldsymbol e}_{i,h}^{n+1}}_{I_2}\\ &\quad+\underbrace{\sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left(\alpha_i(k^{n,\varepsilon}_{i,h}) - \alpha_i(k^{n,\varepsilon}_{i}) \right) \nabla {\boldsymbol u}^{n+1}_{i} : \nabla {\boldsymbol e}_{i,h}^{n+1}}_{I_3}\\ &\quad+ \int_{\it{\Gamma}} \left[\left | {\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right| \, \left ({\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right) - \left | {\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right| \, \left ({\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right) \right] \\ &\quad \times\underbrace{ \cdot \left[\left ({\boldsymbol u}^{n+1}_{1} - {\it{\Pi}}_{1,h}\,{\boldsymbol u}^{n+1}_{1} \right) - \left ({\boldsymbol u}^{n+1}_{2} - {\it{\Pi}}_{2,h} \,{\boldsymbol u}^{n+1}_{2} \right) \right]}_{I_4} \\ & \quad + \int_{\it{\Gamma}} \left[\left | {\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right| \, \left ({\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right) - \left | {\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right| \, \left ({\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right) \right]\\ &\quad \times\underbrace{ \cdot \left[\left ({\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right) - \left ({\boldsymbol u}^{n+1}_{1} -{\boldsymbol u}^{n+1}_{2} \right) \right]}_{I_5} \\ &= 0 \, \left(I_1 +I_2 +I_3 +I_4 +I_5 \right)\!. \end{align*} Thanks to the convexity inequality $$\left (|{\boldsymbol b}| {\boldsymbol b} -|{\boldsymbol a}|{\boldsymbol a} \right) \cdot \left({\boldsymbol b} -{\boldsymbol a}\right) \geq 0 \, \forall \left({\boldsymbol a},{\boldsymbol b}\right) \in \mathbb R^d$$, we deduce that $$I_5 \geq 0$$. Consequently, $$\label{I1}|I_1| \leq |I_2| + |I_3| + |I_4|.$$ (4.17) Estimation of $$I_1$$: It comes from Hypothesis (2.1) that $$\label{Est_I1}|I_1| =\; \displaystyle \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \nabla \left({\boldsymbol u}^{n+1}_{i,h} - {\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i} \right) : \nabla {\boldsymbol e}_{i,h}^{n+1}\; \geq \; \displaystyle \nu\sum_{i=1}^2 |{\boldsymbol e}_{i,h}^{n+1} |^2_{1,{\it{\Omega}}_i}.$$ (4.18) Estimation of $$I_2$$: Using Young’s inequality, estimates (4.7) and (4.16), we deduce \begin{align}I_2 = & \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \nabla \left({\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i} - {\boldsymbol u}^{n+1}_{i} \right): \nabla {\boldsymbol e}_{i,h}^{n+1} \notag \\ & \leq c \sum_{i=1}^2 \left |{\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i} - {\boldsymbol u}^{n+1}_{i} \right |_{1,{\it{\Omega}}_i} \; \left |{\boldsymbol e}_{i,h}^{n+1} \right|_{1,{\it{\Omega}}_i}\notag \\ & \leq \displaystyle\frac{c}{\nu} \sum_{i=1}^2 \left |{\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i} - {\boldsymbol u}^{n+1}_{i} \right |^2_{1,{\it{\Omega}}_i} \; + \; \frac{\nu}{4} \sum_{i=1}^2 \left |{\boldsymbol e}_{i,h}^{n+1} \right|^2_{1,{\it{\Omega}}_i} \notag \\ & \leq \frac{c}{\nu}\, h^{2} \, \sum_{i=1}^2 \, |{\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \frac{c}{\nu}\, h^{2} \, \sum_{i=1}^2 \, |p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}\; + \; \frac{\nu}{4} \sum_{i=1}^2 \left |{\boldsymbol e}_{i,h}^{n+1} \right|^2_{1,{\it{\Omega}}_i}\!.\label{Est_I2}\end{align} (4.19) Estimation of $$I_3$$: Using successively the mean value theorem, Hölder inequality, Hypothesis 2.1 and relations (2.2) and (4.9), we obtain \begin{align}I_3 &=\sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left(\alpha_i(k^{n,\varepsilon}_{i,h}) - \alpha_i(k^{n,\varepsilon}_{i}) \right) \nabla {\boldsymbol u}^{n+1}_{i} : \nabla {\boldsymbol e}_{i,h}^{n+1} \notag \\ & \leq c \sum_{i=1}^2 \| k^{n,\varepsilon}_{i,h} - k^{n,\varepsilon}_{i}\|_{L^\infty({\it{\Omega}}_i)} \; |\nabla {\boldsymbol u}^{n+1}_{i}|_{1,{\it{\Omega}}_i} \; \left| {\boldsymbol e}^{n+1}_{i,h}\right|_{1,{\it{\Omega}}_i} \notag \\ & \leq c\, C_\varepsilon \sum_{i=1}^2 | k^{n}_{i,h} - k^{n}_{i}|_{1,{\it{\Omega}}_i} \; |\nabla {\boldsymbol u}^{n+1}_{i}|_{1,{\it{\Omega}}_i} \; \left| {\boldsymbol e}^{n+1}_{i,h}\right|_{1,{\it{\Omega}}_i} \notag \\ & \leq \frac{c \, C^2_\varepsilon}{\nu^3} \; \sum_{i=1}^2 | k^{n}_{i,h} - k^{n}_{i}|^2_{1,{\it{\Omega}}_i} + \frac{\nu}{4} \sum_{i=1}^2 \left| {\boldsymbol e}^{n+1}_{i,h}\right|^2_{1,{\it{\Omega}}_i}\!.\label{Est_I3}\end{align} (4.20) Estimation of $$I_4$$: Using Hölder inequality, the continuity of the injection from $${\boldsymbol H}^{1/2}({\it{\Gamma}})$$ into $$L^3({\it{\Gamma}})^d$$ and the trace operator from $${\boldsymbol H}^{1}({\it{\Omega}}_i)$$ into $${\boldsymbol H}^{1/2}({\it{\Gamma}})$$, it holds \begin{align}I_4 &= \int_{\it{\Gamma}} \left({\left | {\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right|}\, \left ({\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right) - \left | {\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right| \, {\left ({\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right)} \right) \notag \\ & \quad\cdot \left[\left ({\boldsymbol u}^{n+1}_{1} - {\it{\Pi}}_{1,h}\,{\boldsymbol u}^{n+1}_{1} \right) - \left ({\boldsymbol u}^{n+1}_{2} - {\it{\Pi}}_{2,h} \,{\boldsymbol u}^{n+1}_{2} \right) \right] \notag\\ & \leq \left ( \left \| {\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right\|^2_{L^3({\it{\Gamma}})}+ \left \| {\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right \|^2_{L^3({\it{\Gamma}})}\right) \notag \\ & \quad\times \left ( \left \| {\boldsymbol u}^{n+1}_{1} - {\it{\Pi}}_{1,h}\,{\boldsymbol u}^{n+1}_{1} \right\|_{L^3({\it{\Gamma}})} + \left \| {\boldsymbol u}^{n+1}_{2} - {\it{\Pi}}_{2,h} \,{\boldsymbol u}^{n+1}_{2} \right\|_{L^3({\it{\Gamma}})}\right) \notag \\ & \leq c \left ( \left \| {\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right\|^2_{{\boldsymbol H}^{1/2}({\it{\Gamma}})}+ \left \| {\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right \|^2_{{\boldsymbol H}^{1/2}({\it{\Gamma}})}\right) \notag \\ & \quad\times \left ( \left \| {\boldsymbol u}^{n+1}_{1} - {\it{\Pi}}_{1,h}\,{\boldsymbol u}^{n+1}_{1} \right\|_{{\boldsymbol H}^{1/2}({\it{\Gamma}})}+ \left \| {\boldsymbol u}^{n+1}_{2} - {\it{\Pi}}_{2,h} \,{\boldsymbol u}^{n+1}_{2} \right\|_{{\boldsymbol H}^{1/2}({\it{\Gamma}})}\right) \notag \\ & \leq c \left(| {\boldsymbol u}^{n+1}_{1,h}|^2_{1,{\it{\Omega}}_i} + | {\boldsymbol u}^{n+1}_{2,h}|^2_{1,{\it{\Omega}}_i} + | {\boldsymbol u}^{n+1}_{1}|^2_{1,{\it{\Omega}}_i} + | {\boldsymbol u}^{n+1}_{2}|^2_{1,{\it{\Omega}}_i}\right) \notag \\ & \quad\times \left(\left | {\boldsymbol u}^{n+1}_{1} - {\it{\Pi}}_{1,h}\,{\boldsymbol u}^{n+1}_{1} \right|_{1,{\it{\Omega}}_i} + \left | {\boldsymbol u}^{n+1}_{2} - {\it{\Pi}}_{1,h}\,{\boldsymbol u}^{n+1}_{2} \right|_{1,{\it{\Omega}}_i} \right)\!. \notag \end{align} Thanks to relations (4.7), (4.9) and (4.16), we obtain $$\label{Est_I4}I_4 \leq \; \displaystyle \frac{c}{\nu} \; h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |_{2,{\it{\Omega}}_i}+\frac{c}{\nu} \; h \; \sum_{i=1}^2 | p^{n+1}_{i} |_{1,{\it{\Omega}}_i}.$$ (4.21) Then, estimate (4.14) is obtained easily by combining estimates (4.17)–(4.21) with the interpolation error estimate (4.7). □ 4.2 Analysis of the TKE sequence The purpose of this section is to prove that the errors in TKE $$k^n_i - k^n_{i,h}$$ converge to 0 as $$h$$ tends to 0 and $$n$$ tends to $$\infty$$, for large enough eddy viscosities. To obtain an estimate for the error in the TKEs, the standard choice for the test function in (4.5) and in (3.21) would be $$\ell^{n+1}_{i,h} =k^{n+1}_{i,h} - \mathcal S_{i,h} (k^{n+1}_i)$$. However, in general, this function does not vanish on the boundary $$\partial {\it{\Omega}}_i$$. Hence, it is necessary to introduce the lifting $$\mathcal R_{i,h}$$ and to use the function test $$\phi_{i,h} = \ell^{n+1}_{i,h} - \mathcal R_{i,h}(\ell^{n+1}_{i,h})$$. This choice requires to estimate the norm of the correction term $$\mathcal R_{i,h}(\ell^{n+1}_{i,h})$$. This is done in the following result: Lemma 4.5 Assume that Hypotheses (2.1) and (3.1) hold, and assume that the solution $$({\boldsymbol u}^{n+1}_{i}, p^{n+1}_{i})$$ belongs to $${\boldsymbol H}^{2}({\it{\Omega}}_i)\times H^{1}({\it{\Omega}}_i)$$. Then, the following error estimate holds \begin{align}\label{Est_Rih_k}\left |\mathcal R_{i,h} \left(\ell_{i,h}^{n+1}\right)\right|^2_{1,{\it{\Omega}}_i} &\leq \displaystyle \frac{c\,C^2_\varepsilon}{\nu^3}\left(\displaystyle \left(\nu +1\right)\, h^{2} \sum_{i=1}^2 \left| {\boldsymbol u}^{n+1}_{i} \right|^2_{2,{\it{\Omega}}_i} +h\sum_{i=1}^2 \left| {\boldsymbol u}^{n+1}_{i}\right|_{2,{\it{\Omega}}_i} \right.\notag\\ &\qquad\qquad \left.+ \, \displaystyle h\sum_{i=1}^2 \left| p^{n+1}_{i}\right|_{1,{\it{\Omega}}_i} + \displaystyle \frac{C^2_\varepsilon}{\nu^2} \; \sum_{i=1}^2 \left| k^{n}_{i,h} - k^{n}_{i}\right|^2_{1,{\it{\Omega}}_i}\right)\!.\end{align} (4.22) Proof. Since for all $$n$$, the sequences $$k^n_i$$ and $$k^n_{i,h}$$ are included in $$H^{1}({\it{\Omega}}_i)$$, their trace on $${\it{\Gamma}}$$ belong to $$H^{1/2}({\it{\Gamma}})$$. As both traces on $${\it{\Gamma}}_i$$ vanish, i.e., $$\restriction{k^n_i}{{\it{\Gamma}}_i} \,=\, \restriction{k^n_{i,h}}{{\it{\Gamma}}_i}\,=\,0$$, then, they belong to $$H^{1/2}_{00}({\it{\Gamma}})$$. Furthermore, by (3.16), there exists a positive constant $$c$$ such that: $$\label{Rih_Ell}\left|\mathcal R_{i,h}\left(\ell^{n+1}_{i,h}\right)\right|_{1,{\it{\Omega}}_i} \, \leq \, c\, \left\|\ell^{n+1}_{i,h}\right \|_{H^{1/2}_{00}({\it{\Gamma}})}\!.$$ (4.23) On the other hand, due to Hypothesis 3.1, the trace on $${\it{\Gamma}}$$ of the function $$\ell^{n+1}_{i,h}$$ can be rewritten as \begin{align*}\restriction{\ell^{n+1}_{i,h}}{{\it{\Gamma}}} &= \restriction{k^{n+1}_{i,h}}{{\it{\Gamma}}} - \restriction{\left(\mathcal S_{i,h} \left(k^{n+1}_i \right) \right)}{{\it{\Gamma}}} \\ &= \, \mathcal L_{i,h} \left(\left|{\boldsymbol u}^{n+1,\varepsilon}_{1,h} - {\boldsymbol u}^{n+1,\varepsilon}_{2,h}\right|^2 \right) \, - \, \mathcal L_{i,h} \left(\left|{\boldsymbol u}^{n+1,\varepsilon}_{1} - {\boldsymbol u}^{n+1,\varepsilon}_{2}\right|^2 \right) \\ &= \mathcal L_{i,h}\left(\left|{\boldsymbol u}^{n+1,\varepsilon}_{1,h} - {\boldsymbol u}^{n+1,\varepsilon}_{2,h}\right|^2 - \left|{\boldsymbol u}^{n+1,\varepsilon}_{1} - {\boldsymbol u}^{n+1,\varepsilon}_{2}\right|^2 \right)\!. \end{align*} Next, using (2.2), yielding \begin{align}{3}\|\ell_{i,h}^{n+1}\|_{H^{1/2}_{00}({\it{\Gamma}})} & \leq c \left\| \left|{\boldsymbol u}^{n+1,\varepsilon}_{1,h} - {\boldsymbol u}^{n+1,\varepsilon}_{2,h}\right|^2 - \left|{\boldsymbol u}^{n+1,\varepsilon}_{1} - {\boldsymbol u}^{n+1,\varepsilon}_{2}\right|^2 \right\|_{H^{1/2}_{00}({\it{\Gamma}})} \notag \\ & = c \, \left\|\left [\left({\boldsymbol u}^{n+1,\varepsilon}_{1,h} - {\boldsymbol u}^{n+1,\varepsilon}_{1}\right) -\left({\boldsymbol u}^{n+1,\varepsilon}_{2,h} - {\boldsymbol u}^{n+1,\varepsilon}_{2} \right)\right] \right. \notag \\ & \quad\times \left. \left[\left({\boldsymbol u}^{n+1,\varepsilon}_{1,h} +{\boldsymbol u}^{n+1,\varepsilon}_{1} \right) -\left({\boldsymbol u}^{n+1,\varepsilon}_{2,h} +{\boldsymbol u}^{n+1,\varepsilon}_{2} \right)\right]\right\|_{H^{1/2}_{00}({\it{\Gamma}})} \notag \\ & \leq \, c \, \sum_{i=1}^2 c_i \, \left \|{\boldsymbol u}^{n+1,\varepsilon}_{i,h} -{\boldsymbol u}^{n+1,\varepsilon}_{i}\right\|_{{L^\infty}({\it{\Gamma}})} \; \sum_{i=1}^2 \left \|{\boldsymbol u}^{n+1,\varepsilon}_{i,h} +{\boldsymbol u}^{n+1,\varepsilon}_{i}\right\|_{{L^\infty}({\it{\Gamma}})} \notag \\ & \leq \, c \, \sum_{i=1}^2 \left \|{\boldsymbol u}^{n+1,\varepsilon}_{i,h} -{\boldsymbol u}^{n+1,\varepsilon}_{i}\right\|_{{L^\infty}({\it{\Omega}}_i)} \; \sum_{i=1}^2 \left \|{\boldsymbol u}^{n+1,\varepsilon}_{i,h} +{\boldsymbol u}^{n+1,\varepsilon}_{i}\right\|_{{L^\infty}({\it{\Omega}}_i)}\!. \notag \end{align} According to (4.9) and Poincaré–Friedrichs inequality, we obtain $$\label{LU}\|\ell_{i,h}^{n+1}\|_{H^{1/2}_{00}({\it{\Gamma}})} \leq \, \displaystyle \frac{c \, C_\varepsilon}{\nu} \; \sum_{i=1}^2 \left |{\boldsymbol u}^{n+1}_{i,h} -{\boldsymbol u}^{n+1}_{i}\right|_{1,{\it{\Omega}}_i}\!.$$ (4.24) When adding and subtracting the quantity $${\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i,h}$$, and using (4.23), we obtain \begin{equation*}\left |\mathcal R_{i,h} (\ell_{i,h}^{n+1})\right|^2_{1,{\it{\Omega}}_i} \, \leq \displaystyle \frac{c\, C^2_\varepsilon}{\nu^2} \; \left(\sum_{i=1}^2 \left| {\boldsymbol e}^{n+1}_{i,h} \right|^2_{1,{\it{\Omega}}_i} + \sum_{i=1}^2 \left | {\it{\Pi}}_{i,h} \, {\boldsymbol u}^{n+1}_{i} - {\boldsymbol u}^{n+1}_{i} \right|^2_{1,{\it{\Omega}}_i}\right)\!. \end{equation*} Finally, estimate (4.22) follows, thanks to (4.7), (4.14) and (4.16). □ We may now estimate the test function $$\ell_{i,h}^{n+1}$$: Theorem 4.6 Assume that Hypotheses (2.1) and (3.1) hold, and assume that the solution $$\left({\boldsymbol u}^{n}_{i},p^{n}_{i},k^{n}_{i}\right)$$ belongs to $${\boldsymbol H}^{2}({\it{\Omega}}_i) \times H^{1}({\it{\Omega}}_i) \times H^{2}({\it{\Omega}}_i)$$. Then, the following error estimate holds \begin{align} \nonumber \displaystyle \nu \, \sum_{i=1}^2 \left|\ell_{i,h}^{n+1}\right|^2_{1,{\it{\Omega}}_i}& \leq \; \displaystyle \frac{c \, C_\varepsilon^2}{\nu^6} \; \left(\nu \, C_\varepsilon^2 \, +\, \nu^4 \, + \, C_\varepsilon^2 \, + \, \frac{C_\varepsilon^4}{\nu^2}+ \, \frac{C_\varepsilon^4}{\nu^3}\right) \; \sum_{i=1}^2 \left|k_{i,h}^{n}-k_i^n\right|^2_{1,{\it{\Omega}}_i} \\ & \quad\displaystyle +c\, h^{2} \; \left[\frac{C_\varepsilon^2}{\nu^6} \; \left(\nu \, C_\varepsilon^2 \, +\, \nu^4 \, + \, C_\varepsilon^2 \, + \, \frac{C_\varepsilon^4}{\nu^2}\right) \; \sum_{i=1}^2 \left| k^{n}_{i}\right|^2_{2,{\it{\Omega}}_i}\, + \, \displaystyle \frac{\nu+1}{\nu} \; \sum_{i=1}^2 \left| k^{n+1}_{i}\right|^2_{2,{\it{\Omega}}_i}\right] \; \nonumber \\ &\quad +\, \displaystyle \displaystyle \frac{c\, C^2_\varepsilon}{\nu^5} \, \left(\nu \,+ \, C^2_\varepsilon +\frac{C^2_\varepsilon}{\nu}\right) \biggl[\, \displaystyle \left(\nu +1\right)\, h^{2} \; \sum_{i=1}^2 \left| {\boldsymbol u}^{n+1}_{i} \right|^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 \left| {\boldsymbol u}^{n+1}_{i}\right|_{2,{\it{\Omega}}_i} \biggl]\nonumber \\ &\quad +\, \displaystyle \displaystyle \frac{c\, C^2_\varepsilon}{\nu^5} \, \left(\nu \,+ \, C^2_\varepsilon +\frac{C^2_\varepsilon}{\nu}\right) \biggl[\, \displaystyle \left(\nu +1 \right)\, h^{2} \; \sum_{i=1}^2 \left| p^{n+1}_{i} \right|^2_{1,{\it{\Omega}}_i}\biggl].\label{Est_Ell_h}\end{align} (4.25) Proof. To estimate $$\ell_{i,h}^{n+1}= k_{i,h}^{n+1}-\mathcal S_{i,h} \left(k_{i}^{n+1}\right)$$, we use the difference between equations (4.5) and (3.21), taking as test function $$\phi_{i,h} = \ell_{i,h}^{n+1} \, -\, \mathcal R_{i,h} \left(\ell_{i,h}^{n+1}\right)$$: \begin{align}A &:= \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left[\gamma_i(k^{n,\varepsilon}_{i,h}) \, \nabla \,k^{n+1}_{i,h} \,- \, \gamma_i(k^{n,\varepsilon}_{i}) \, \nabla \,k^{n+1}_{i} \right] \, \cdot \, \nabla \,\phi_{i,h} \notag\\ & = B := \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left[\alpha_i(k^{n,\varepsilon}_{i,h}) \, \nabla {\boldsymbol u}^{n+1}_{i,h} \,: \, \nabla \, {\boldsymbol u}^{n+1,\varepsilon}_{i,h} \,- \, \alpha_i(k^{n,\varepsilon}_{i}) \, \nabla {\boldsymbol u}^{n+1}_{i} \,: \, \nabla {\boldsymbol u}^{n+1,\varepsilon}_{i}\right] \, \phi_{i,h}.\notag \end{align} Adding and subtracting $$\ell_{i,h}^{n+1}$$ in the first factor, then $$A$$ can be rewritten as \begin{align}A &= \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left[\gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \, \nabla \,k^{n+1}_{i,h} \,- \, \gamma_i\left(k^{n,\varepsilon}_{i}\right) \, \nabla \,k^{n+1}_{i} \right] \, \cdot \, \nabla\left (\ell_{i,h}^{n+1} \, -\, \mathcal R_{i,h} \left(\ell_{i,h}^{n+1}\right) \right) \notag \\ &= \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left(\gamma_i\left(k^{n,\varepsilon}_{i,h}\right)- \gamma_i\left(k^{n,\varepsilon}_{i}\right) \right) \nabla k^{n+1}_{i} \cdot \nabla \ell_{i,h}^{n+1} \notag \\ & \quad + \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \nabla\left (\mathcal S_{i,h}\,\left(k^{n+1}_{i}\right) - k^{n+1}_{i} \right) \cdot \nabla \ell_{i,h}^{n+1} \notag \\ &\quad + \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \nabla | \ell_{i,h}^{n+1}|^2 \notag \\ &\quad - \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\biggl( \left(\gamma_i\left(k^{n,\varepsilon}_{i,h}\right) - \gamma_i\left(k^{n,\varepsilon}_{i}\right)\right)\nabla \,k^{n+1}_{i} + \gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \nabla \ell_{i,h}^{n+1}\notag \\ & \qquad\qquad\qquad +\gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \nabla (\mathcal S_{i,h}\,\left(k^{n+1}_{i} - k^{n+1}_{i} \right) \biggr) \cdot \, \nabla \mathcal R_{i,h} \left(\ell_{i,h}^{n+1}\right)\!.\notag \end{align} The relation $$A= B$$ yields \begin{align}\nu \sum_{i=1}^2 |\ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i}&\leq |B| \notag \\ (A_1&:=) + \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left|\left(\gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \, -\, \gamma_i\left(k^{n,\varepsilon}_{i}\right) \right)\, \nabla \,k^{n+1}_{i} \, \cdot \, \nabla \, \ell_{i,h}^{n+1} \right| \notag \\ (A_2&:=) + \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left| \gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \,\nabla \, \left(\mathcal S_{i,h}\,\left(k^{n+1}_{i}\right) \,- \, k^{n+1}_{i} \right) \, \cdot \, \nabla \, \ell_{i,h}^{n+1} \right| \notag \\ (A_3 & :=) + \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left|\left(\gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \, -\, \gamma_i\left(k^{n,\varepsilon}_{i}\right) \right)\, \nabla \,k^{n+1}_{i} \,\cdot \nabla \, \mathcal R_{i,h}\left(\ell_{i,h}^{n+1}\right) \right| \notag \\ (A_4 & :=) +\sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left| \gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \, \nabla \, \ell_{i,h}^{n+1} \, \cdot \, \nabla \, \mathcal R_{i,h}\left(\ell_{i,h}^{n+1}\right) \right| \notag \\ (A_5 & :=) + \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left| \gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \,\nabla \, \left(\mathcal S_{i,h}\,\left(k^{n+1}_{i}\right) \,- \, k^{n+1}_{i} \right) \, \cdot \, \nabla \, \mathcal R_{i,h} \left(\ell_{i,h}^{n+1}\right) \right|\!. \label{Ell_AB}\end{align} (4.26) The next step is to estimate $$(A_j)_{1 \leq j \leq 5}$$ and $$B$$. Estimation of $$A_1$$:   Using successively the mean value theorem, Hölder and Poincaré-Friedrichs inequalities and relations (2.2) and (4.10), there exists positive a constant $$c$$ depending only on $${\it{\Omega}}_i$$ such that \begin{align}A_1 & \leq c \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left|(k^{n,\varepsilon}_{i,h}- k^{n,\varepsilon}_{i}) \nabla \,k^{n+1}_{i} \cdot \nabla \ell_{i,h}^{n+1} \right| \notag \\ & \leq c \sum_{i=1}^2 \| k^{n,\varepsilon}_{i,h} - k^{n,\varepsilon}_{i}\|_{L^\infty({\it{\Omega}}_i)}\| \nabla \,k^{n+1}_{i}\|_{L^2({\it{\Omega}}_i)}| \ell_{i,h}^{n+1}|_{1,{\it{\Omega}}_i} \notag \\ & \leq \, \frac{c \, C_\varepsilon^3}{\nu^2}\; \sum_{i=1}^2 | k^n_{i,h} \, -\, k^n_{i}|_{1,{\it{\Omega}}_i} \; | \ell_{i,h}^{n+1}|_{1,{\it{\Omega}}_i}.\notag \end{align} To simplify the calculations, we introduce a positive number $$\beta$$ which we shall fix later. According to Young’s inequality, this yields for $$\beta>0$$, which we shall fix later \begin{align}A_1 \; \leq \; \frac{\nu}{\beta} \, \sum_{i=1}^2 | \ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i} \, + \, \beta \; \frac{c \, C_\varepsilon^6}{\nu^5}\; \sum_{i=1}^2 | k^n_{i,h} \, -\, k^n_{i}|^2_{1,{\it{\Omega}}_i}.\label{A1}\end{align} (4.27) Estimation of $$A_2$$:   Using (4.8), (4.10) and Young and Cauchy–Schwarz inequalities, \begin{align}A_2 &= \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left| \gamma_i(k^{n,\varepsilon}_{i,h}) \,\nabla \, \left(\mathcal S_{i,h}\,(k^{n+1}_{i}) \,- \, k^{n+1}_{i} \right) \, \cdot \, \nabla \, \ell_{i,h}^{n+1} \right| \notag \\ & \leq \, \frac{\nu}{\beta} \, \sum_{i=1}^2 | \ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i} \, + \, \frac{c\, \beta}{\nu} \, \sum_{i=1}^2 \left |\mathcal S_{i,h}\,(k^{n+1}_{i}) \,- \, k^{n+1}_{i} \right|^2_{1,{\it{\Omega}}_i} \notag \\ & \leq \, \frac{\nu}{\beta} \, \sum_{i=1}^2 | \ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i} \, + \, \beta \, \frac{c}{\nu} \, h^{2} \, \sum_{i=1}^2 | k^{n+1}_{i}|^2_{2,{\it{\Omega}}_i}. \label{A2}\end{align} (4.28) Estimation of $$A_3$$:   The same arguments used to estimate $$A_1$$ and due to (4.22), the following inequalities hold \begin{align}A_3 & \leq \, c \sum_{i=1}^2 \| k^{n,\varepsilon}_{i,h} \, -\, k^{n,\varepsilon}_{i}\|_{L^\infty({\it{\Omega}}_i)} \, | k^{n+1}_{i}|_{1,{\it{\Omega}}_i} \, | \mathcal R_{i,h} (\ell_{i,h}^{n+1})|_{1,{\it{\Omega}}_i} \notag \\ & \leq \, \frac{c \, C^2_\varepsilon}{\nu^2}\; \sum_{i=1}^2 | k^n_{i,h} \, -\, k^n_{i}|_{1,{\it{\Omega}}_i} \; | \mathcal R_{i,h} (\ell_{i,h}^{n+1})|_{1,{\it{\Omega}}_i} \notag \\ &\leq \, \frac{c \, C^2_\varepsilon}{\nu^2}\; \left(\sum_{i=1}^2 | k^n_{i,h} \, -\, k^n_{i}|^2_{1,{\it{\Omega}}_i} \; +\; | \mathcal R_{i,h} (\ell_{i,h}^{n+1})|^2_{1,{\it{\Omega}}_i} \right) \notag \\ & \leq \, \frac{c \, C^4_\varepsilon}{\nu^5}\; \left(\displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i}\right) \notag \\ &\quad + \frac{c \, C^4_\varepsilon}{\nu^5}\; \displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}+ \displaystyle \frac{c\, C^2_\varepsilon}{\nu^5} \; \left(\nu^3 \, + \, C^2_\varepsilon \right) \sum_{i=1}^2 | k^{n}_{i,h} - k^{n}_{i}|^2_{1,{\it{\Omega}}_i}.\label{A3}\end{align} (4.29) Estimation of $$A_4$$:   Using the same procedure, we deduce \begin{align}A_4 & \leq \, c \sum_{i=1}^2 |\ell_{i,h}^{n+1} |_{1,{\it{\Omega}}_i} \, |\mathcal R_{i,h} (\ell_{i,h}^{n+1})|_{1,{\it{\Omega}}_i} \notag \\ &\leq \, \frac{\nu}{\beta} \, \sum_{i=1}^2 | \ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i} \, + \, \frac{c\, \beta}{\nu} \, \sum_{i=1}^2 | \mathcal R_{i,h} (\ell_{i,h}^{n+1})|^2_{1,{\it{\Omega}}_i} \notag \\ & \leq \, \frac{\nu}{\beta} \, \sum_{i=1}^2 | \ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i} \, + \, \displaystyle \frac{c\, C^2_\varepsilon\, \beta}{\nu^4} \, \biggl(\, + \displaystyle \frac{C^2_\varepsilon}{\nu^2} \; \sum_{i=1}^2 | k^{n}_{i,h} - k^{n}_{i}|^2_{1,{\it{\Omega}}_i}\notag \\ &\quad\displaystyle + (\nu +1)\, h^{2} \; \sum_{i=1}^2 | p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}+ (\nu +1)\, h^{2} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i}\biggr).\label{A4}\end{align} (4.30) Estimation of $$A_5$$:   Using the same techniques as previously and the error estimates (4.8) for operator $$\mathcal S_{i,h}$$, we obtain \begin{align}A_5 & \leq \, c \sum_{i=1}^2 |\mathcal S_{i,h}\,(k^{n+1}_{i}) \,- \, k^{n+1}_{i}|_{1,{\it{\Omega}}_i} \, | \mathcal R_{i,h} (\ell_{i,h}^{n+1})|_{1,{\it{\Omega}}_i} \notag \\ &\leq \, \frac{\nu}{\beta} \, \sum_{i=1}^2 |\mathcal S_{i,h}\,(k^{n+1}_{i}) \,- \, k^{n+1}_{i}|^2_{1,{\it{\Omega}}_i} \,+\, \frac{c \beta}{\nu} \,\sum_{i=1}^2| \mathcal R_{i,h} (\ell_{i,h}^{n+1})|^2_{1,{\it{\Omega}}_i}\notag \\ &\leq \, \frac{\nu}{\beta} \,h^{2}\, \sum_{i=1}^2 | k^{n+1}_{i} |^2_{2,{\it{\Omega}}_i} \,+\, \displaystyle \frac{c\, C^2_\varepsilon\, \beta}{\nu^4} \, \biggl(\, \displaystyle \frac{C^2_\varepsilon}{\nu^2} \; \sum_{i=1}^2 | k^{n}_{i,h} - k^{n}_{i}|^2_{1,{\it{\Omega}}_i}\notag \\ &\quad \displaystyle + (\nu +1)\, h^{2} \; \sum_{i=1}^2 | p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}+(\nu +1)\, h^{2} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i}\biggr).\label{A5}\end{align} (4.31) Summing up estimates for the $$(A_j)_{1 \leq j \leq 5}$$ and choosing $$\beta=3$$, for instance, we can write \begin{align}\label{ell_Aj}\displaystyle \nu \, \sum_{i=1}^2 |\ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i}& \leq |B| \;+\; c\, \displaystyle \frac{\nu+1}{\nu} \, h^{2}\, \; \sum_{i=1}^2 | k^{n+1}_{i}|^2_{2,{\it{\Omega}}_i} \notag\\ & \quad+ \displaystyle \frac{c \, C_\varepsilon^2}{\nu^6} \; \left(\nu \, C_\varepsilon^2 \, +\, \nu^4 \, + \, C_\varepsilon^2 \right) \; \sum_{i=1}^2 | k^n_{i,h} \, -\, k^n_{i}|^2_{1,{\it{\Omega}}_i} \notag\\ & \quad+ \displaystyle \displaystyle \frac{c\, C^2_\varepsilon}{\nu^5} \, \left(\nu \,+ \, C^2_\varepsilon \right) \left(\, \displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i} \right)\notag\\ & \quad+ \displaystyle \displaystyle \frac{c\, C^2_\varepsilon}{\nu^5} \, \left(\nu \,+ \, C^2_\varepsilon \right) \, \displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}.\end{align} (4.32) Estimation of $$B$$: Let us write the term $$B$$ as \begin{align}B & = \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left[\alpha_i(k^{n,\varepsilon}_{i,h}) \, \nabla {\boldsymbol u}^{n+1}_{i,h} \,: \, \nabla \, {\boldsymbol u}^{n+1,\varepsilon}_{i,h} \,- \, \alpha_i(k^{n,\varepsilon}_{i}) \, \nabla {\boldsymbol u}^{n+1}_{i} \,: \, \nabla {\boldsymbol u}^{n+1,\varepsilon}_{i}\right] \, \phi_{i,h}\notag \\ & = \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \, \nabla({\boldsymbol u}^{n+1}_{i,h} - {\boldsymbol u}^{n+1}_{i}) \,:\, \nabla {\boldsymbol u}^{n+1,\varepsilon}_{i,h} \; \phi_{i,h} \notag \\ & \quad + \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \, \nabla \, {\boldsymbol u}^{n+1}_{i} \,:\, \nabla ({\boldsymbol u}^{n+1,\varepsilon}_{i,h} \, - \, {\boldsymbol u}^{n+1,\varepsilon}_{i}) \; \phi_{i,h} \notag \\ & \quad+ \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left(\alpha_i(k^{n,\varepsilon}_{i,h}) \, - \, \alpha_i(k^{n,\varepsilon}_{i}) \right) \, \, \nabla \, {\boldsymbol u}^{n+1}_{i} \,:\, \nabla {\boldsymbol u}^{n+1,\varepsilon}_{i} \; \phi_{i,h} \notag \\ & = \; J_1 + J_2 +J_3. \notag \end{align} The terms $$J_1, J_2$$ and $$J_3$$ can be bounded using $$\phi_{i,h} = \ell_{i,h}^{n+1} - \mathcal R_{i,h} (\ell_{i,h}^{n+1})$$, Young inequality, (2.2), (4.9) and (4.10), as follows: \begin{align}|J_1| + |J_2| & \leq \; \displaystyle \frac{c \, C^2_\varepsilon}{\nu^2} \; \sum_{i=1}^2 |{\boldsymbol u}^{n+1}_{i,h} - {\boldsymbol u}^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}\,+\, |\mathcal R_{i,h} (\ell_{i,h}^{n+1})|^2_{1,{\it{\Omega}}_i}\,+\, \frac{\nu}{4} |\ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i},\notag \\ |J_3| & \leq \; \displaystyle \frac{c \, C^6_\varepsilon}{\nu^8} \; \sum_{i=1}^2 |k^{n}_{i,h} - k^{n}_{i} |^2_{1,{\it{\Omega}}_i}\,+\, |\mathcal R_{i,h} (\ell_{i,h}^{n+1})|^2_{1,{\it{\Omega}}_i}\,+\, \frac{\nu}{4} |\ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i}.\notag \end{align} Then, the following inequality holds, thanks to (4.22) \begin{align}\label{B}|B| &\leq \displaystyle \frac{\nu}{2} |\ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i}\notag\\ &\quad +\, \displaystyle\left(\displaystyle \frac{c \, C^6_\varepsilon}{\nu^8} \,+\, \displaystyle \frac{c \, C^4_\varepsilon}{\nu^5} \right)\; \sum_{i=1}^2 |k^{n}_{i,h} - k^{n}_{i} |^2_{1,{\it{\Omega}}_i}+ \; \displaystyle \frac{C^4_\varepsilon}{\nu^5} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i,h} - {\boldsymbol u}^{n+1}_{i}|^2_{1,{\it{\Omega}}_i} \notag \\ &\quad +\, \displaystyle \frac{c\, C^2_\varepsilon}{\nu^3} \, \left(\, \displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i} \right) \notag\\ & \quad+\, \displaystyle \frac{c\, C^2_\varepsilon}{\nu^3} \, (\nu +1)\, h^{2} \; \sum_{i=1}^2 | p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}.\end{align} (4.33) Combining (4.33) with (4.32) leads to \begin{equation*}\begin{array}{rcl}\displaystyle \nu \, \sum_{i=1}^2 |\ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i}&\leq& \displaystyle \frac{c \, C_\varepsilon^2}{\nu^6}\left(\nu \, C_\varepsilon^2 \, +\, \nu^4 \, + \, C_\varepsilon^2 \, + \, \frac{C_\varepsilon^4}{\nu^2}\right) \sum_{i=1}^2 | k^n_{i,h} \, -\, k^n_{i}|^2_{1,{\it{\Omega}}_i}\, +\, \displaystyle \frac{C^4_\varepsilon}{\nu^5} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i,h} - {\boldsymbol u}^{n+1}_{i}|^2_{1,{\it{\Omega}}_i} \\ && +\, \displaystyle \displaystyle \frac{c\, C^2_\varepsilon}{\nu^5} \, \left(\nu \,+ \, C^2_\varepsilon \right) \left(\, \displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i} \right) \\ && \displaystyle+ \frac{c\, C^2_\varepsilon}{\nu^5} \, \left(\nu \,+ \, C^2_\varepsilon \right) \, \displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}+ c\, \displaystyle \frac{\nu+1}{\nu} \, h^{2}\, \; \sum_{i=1}^2 | k^{n+1}_{i}|^2_{2,{\it{\Omega}}_i}. \end{array}\end{equation*} Finally, using (4.14), estimate (4.25) follows.   □ We may now state the following: Theorem 4.7 Assume that Hypotheses 2.1 and 3.1 hold, and assume that the solution $$\left({\boldsymbol u}^{n}_{i},p^{n}_{i},k^{n}_{i}\right)$$ belongs to $${\boldsymbol H}^{2}({\it{\Omega}}_i) \times H^{1}({\it{\Omega}}_i) \times H^{2}({\it{\Omega}}_i)$$. Then, there exists a $$\nu_0>0$$ such that id $$0<\nu_0 <\nu$$, it holds for all $$n=1,2,\cdots$$ \begin{align} \label{Est_Cont_k}&\displaystyle \sum_{i=1}^2 \left | k^{n}_i - k^{n}_{i,h} \right|^2_{1,{\it{\Omega}}_i}\leq \; \displaystyle A_{\varepsilon,\nu}^{n}\, \sum_{i=1}^2 \left | k^{0}_i - k^{0}_{i,h} \right|^2_{1,{\it{\Omega}}_i} +h\, B_{\varepsilon,\nu},\end{align} (4.34) for some constants $$A_{\varepsilon,\nu}\, \in (0,1)$$, $$B_{\varepsilon,\nu}\, >0$$. Proof. By estimates (3.10) and (4.25), \begin{align}\sum_{i=1}^2 \left|k_{i}^{n+1} - k_{i,h}^{n+1} \right|^2_ {1,{\it{\Omega}}_i}\notag & \leq \displaystyle 2 \sum_{i=1}^2 \biggl (\left|k_{i}^{n+1} \, - \, \mathcal S_{i,h}(k_{i}^{n+1}) \right|^2_ {1,{\it{\Omega}}_i} \, + \, \left| \mathcal S_{i,h}(k_{i}^{n+1}) \,-\, k_{i,h}^{n+1} \right|^2_ {1,{\it{\Omega}}_i} \biggl) \notag \\ & \leq \displaystyle 2 \sum_{i=1}^2 |\ell_{i,h}^{n+1} |^2_ {1,{\it{\Omega}}_i} \,+\, c\, h^{2} \; \sum_{i=1}^2 |k_{i}^{n+1}|^2_{1,{\it{\Omega}}_i} \notag \\ & \leq \; \displaystyle A_{\varepsilon,\nu}\, \sum_{i=1}^2 \left | k^{n}_i - k^{n}_{i,h} \right|^2_{1,{\it{\Omega}}_i} +h\,D_{\varepsilon,\nu}\,,\mbox{with}\notag \end{align} \begin{gather*}\displaystyle A_{\varepsilon,\nu}=\frac{c \, C_\varepsilon^2}{\nu^7} \; \left(\nu \, C_\varepsilon^2 \, +\, \nu^4 \, + \, C_\varepsilon^2 \, + \, \frac{C_\varepsilon^4}{\nu^2}+ \, \frac{C_\varepsilon^4}{\nu^3}\right)\!,\\ \displaystyle D_{\varepsilon,\nu}= c\, \max\biggl\{\frac{C_\varepsilon^2}{\nu^6} \; \left(\nu \, C_\varepsilon^2 \, +\frac{C_\varepsilon^2}{\nu}\right)\!, \left (1+\frac{1}{\nu} \right) \biggl\}. \end{gather*} For large enough $$\nu$$, $$A_{\varepsilon,\nu}^n<1$$. Let $$\displaystyle\sigma_n=\sum_{i=1}^2 |k_{i}^{n+1} - k_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i}$$. Arguing recursively, we deduce $$\sigma_{n} \le A_{\varepsilon,\nu}^n \, \sigma_{0} + h\, D_{\varepsilon,\nu}\, \left (\sum_{k=0}^{n-1} A_{\varepsilon,\nu}^k \right) \le A_{\varepsilon,\nu}^n\, \sigma_{0} + h\, D_{\varepsilon,\nu}\, \frac{A_{\varepsilon,\nu}}{1- A_{\varepsilon,\nu}}.$$ This yields (4.34). □ From this result, it follows: Corollary 4.8 Assume that the hypotheses of Theorem 4.7 hold. Then, the following error estimates hold, \begin{align} \label{Est_up}\displaystyle \sum_{i=1}^2 \left (\left | {\boldsymbol u}^{n}_i - {\boldsymbol u}^{n}_{i,h} \right|^2_{1,{\it{\Omega}}_i}+ \left\| p^{n}_i - p^{n}_{i,h} \right\|^2_{0,{\it{\Omega}}_i} \right) & \leq \; E_{\varepsilon,\nu}\,\displaystyle A_{\varepsilon,\nu}^{n}\, \sum_{i=1}^2 \left | k^{0}_i - k^{0}_{i,h} \right|^2_{1,{\it{\Omega}}_i} \notag \\ &\quad + h\, E_{\varepsilon,\nu}\, B_{\varepsilon,\nu},\end{align} (4.35) for some constant $$E_{\varepsilon,\nu}>0$$, where $$A_{\varepsilon,\nu}$$ and $$B_{\varepsilon,\nu}$$ are the constants appearing in estimate (4.34). Proof. The estimate for the error in velocity in (4.35) follows from estimates (4.4) and (4.34). The estimates for the error in pressures follows from the discrete inf–sup condition and a treatment for the nonlinear boundary terms similar to that of the term $$I_4$$ in the proof of Theorem 4.4, which we omit for brevity.  □ Remark 4.9 The hypothesis that $$\nu$$ (which is a lower bound for the eddy viscosities) is large enough, which seems to be unavoidable by our analysis, however, is not verified by oceanic and atmospheric flows. It would be much closer to reality to weaken this hypothesis to a condition that some average value of the eddy viscosities be large enough. In the numerical experiments reported in Section 5, it is rather this last condition that appears to be sufficient to ensure the convergence of the fixed-point algorithm (3.17)–(3.21). Remark 4.10 The previous error analysis may be extended to higher-order finite elements. Indeed, if spaces $${\boldsymbol X}_{i,h},\,M_{i,h}$$ and $$K_{i,h}$$ are respectively built with finite elements of orders $$s+1, \, s$$ and $$r+1$$ for $$s, r \ge 1$$ then under the hypotheses of Theorem 4.7, if the solution $$\left({\boldsymbol u}^{n}_{i},p^{n}_{i},k^{n}_{i}\right)$$ belongs to $${\boldsymbol H}^{2}({\it{\Omega}}_i) \times H^{1}({\it{\Omega}}_i) \times H^{2}({\it{\Omega}}_i)$$, for $$\nu$$ small enough a similar estimate to (4.35) holds: \begin{align} \label{Est_upsr}\displaystyle \sum_{i=1}^2 \left (\left | {\boldsymbol u}^{n}_i - {\boldsymbol u}^{n}_{i,h} \right|^2_{1,{\it{\Omega}}_i}+ \left \| p^{n}_i - p^{n}_{i,h} \right\|^2_{0,{\it{\Omega}}_i} \right) & \leq \; E_{\varepsilon,\nu}\,\displaystyle A_{\varepsilon,\nu}^{n}\, \sum_{i=1}^2 \left | k^{0}_i - k^{0}_{i,h} \right|^2_{1,{\it{\Omega}}_i} \notag \\ &\quad+ (h^s+h^{2s}+h^{2r})\, E_{\varepsilon,\nu}\, B_{\varepsilon,\nu},\end{align} (4.36) so the overall order of convergence of discretization (3.17)–(3.21) is $$h^{\min\{s/2,r\}}$$ for smooth solutions. 5. Numerical experiments The aim of this section is twofold: we intend to test the convergence order (in $$h$$) of the algorithm (3.17)–(3.21) and also the ability of the discretization to reproduce overall qualitative features of a realistic flow. Actually, we simulate an upwelling oceanic flow generated by an atmospheric cavity flow. Although for analysis purposes we have considered a regularized problem, we perform our numerical tests with the direct discretization of problem (1.1), without any smoothing; that is, we replace $$u_{i,h}^{n+1,\varepsilon}$$ by $$u_{i,h}^{n+1}$$ and $$k_{i,h}^{n,\varepsilon}$$ by $$k_{i,h}^{n}$$ in (3.20) and (3.21). In both cases, we use the with FreeFEM++ code to perform our tests (cf. Hecht, 2012). The solver uses a Taylor–Hood $$\mathcal P_2-\mathcal P_1$$ FEM for the space discretization of velocity–pressure and $$\mathcal P_2$$ FEM for the TKEs equation. At each step, linear systems are obtained and solved using a preconditioned generalized minimal residual method iterative routine (see, for instance, Saad, 2003). 5.1 Accuracy test The first test is aimed at estimating the convergence order (in $$h$$) of the fixed point provided by algorithm (3.17)–(3.21). To do this, we have compared an exact (or very accurate) solution of problem (1.1) with this discrete fixed point, computed as the limit as $$n \to \infty$$ of the solution $$u_{i,h}^{n+1}$$, $$p_{i,h}^{n+1}$$, $$k_{i,h}^{n,\varepsilon}$$. We have performed two tests, one corresponding to a smooth solution and another corresponding to a low-regularity solution. Test 1: Convergence order for smooth solutions. In this test, we have constructed a problem similar to (1.1) with nonhomogeneous data verified by an analytic solution: The domains occupied by the flows are $${\it{\Omega}}_1 =]0,0.5[^3 \qquad \mbox{and} \qquad {\it{\Omega}}_2 =]-0.5,0[^3,$$ and then the interface $${\it{\Gamma}}$$ corresponds to the plane $$z=0$$. The exact solutions $${\boldsymbol u}_{e,i} \,=\, \left(u_{e,ix},u_{e,iy},u_{e,iz} \right)$$, $$p_{e,i}$$ and $$k_{e,i}$$ for $$i=1,2$$ are given by: \begin{eqnarray*} \label{SolAnal1}\left\{ \begin{array}{rcl}u_{e,1x} \,&=&\, \sin(\pi \,x)\, \sin(\pi\,y)\, \sin(\pi\,z)\, - 4xz^3 +y^2,\\ u_{e,1y} &=& \cos(\pi \,x)\, \cos(\pi\,y)\, \sin(\pi\,z)\, + \pi\cos(\pi \,z)y + x^4,\\ u_{e,1z} &=& z^4 - \sin(\pi\,z),\\ p_{e,1} &=& \sin(\pi\,(x+y+z)),\\ k_{e,1} &=& \lambda |{\boldsymbol u}_1 - {\boldsymbol u}_2|^2, \end{array}\right. \end{eqnarray*} \begin{eqnarray*} \label{SolAnal2}\left\{ \begin{array}{rcl}u_{e,2x} &=& \sin(\pi\,y)\, \sin(\pi\,z) \,+yz, \\ u_{e,2y} &=& \cos(\pi\,x)\, + \sin(\pi\,z)\, -x + z^2, \\ u_{e,2z} &=& x^2+y^2, \\ p_{e,2} &=& x+y+z, \\ k_{e,2} &= & \lambda (z^2 +1)\, |{\boldsymbol u}_1 - {\boldsymbol u}_2|^2. \end{array}\right. \end{eqnarray*} Then, it holds $$k_1 \,=\, k_2 \qquad \mbox{on the interface}\; {\it{\Gamma}}.$$ The eddy viscosities and diffusions are: \begin{eqnarray*}\left\{ \begin{array}{rcl}\alpha_i(\ell) &=& \nu \, + \nu_i \, \sqrt{\ell} \\ \gamma_i(\ell) &=& \nu \, + a_i \, \ell. \end{array}\right. \end{eqnarray*} The preceding functions are solution of a modified system (1.1) with nonzero source terms, as follows: Energy equations: $$-\nabla \cdot(\gamma_i(k_i)\nabla k_i) \,-\, \alpha_i(k_i) |\nabla {\boldsymbol u}_i|^2 \;=\; g_i \; \mbox{in}\; {\it{\Omega}}_i.$$ Manning friction law on the interface $${\it{\Gamma}}$$: $$\alpha_i(k_i)\partial_{{\boldsymbol n}_i}{\boldsymbol u}_i - p_i {\boldsymbol n}_i + \kappa ({\boldsymbol u}_i - {\boldsymbol u}_j) | {\boldsymbol u}_i-{\boldsymbol u}_j| = {\boldsymbol h}_i \; \mbox{sur}\; {\it{\Gamma}}.$$ Dirichlet conditions on $$\partial {\it{\Omega}}_i \setminus {\it{\Gamma}}$$: $${\boldsymbol u}_i = {\boldsymbol u}_{e,i}$$ and $$k_i = k_{e,i}$$. The values of the parameters are: $$\lambda =1, \; \nu = 1, \; \nu_1 = 0.1, \; \nu_2 = 0.2, \; a_1 = 0.01, \; a_2 = 0.02\; \mbox{and}\; \kappa = 10^{-3}.$$ We estimate the convergence rate using several grid sizes by \begin{align}\mathcal {\rm{O}}_h &= \displaystyle \frac {{\rm{log}}\left(\frac {\mathcal E_{h_1}} {\mathcal E_{h_2}} \right)} {{\rm{log}}\left(\frac{h_1}{h_2}\right)} \approx p, \qquad \mathcal E_h & := \, \left(\displaystyle \sum_{i=1}^2 \left | k_{i,h} - k_i \right|^2_{1,{\it{\Omega}}_i}\, + \, \left| {\boldsymbol u}_{i,h}- {\boldsymbol u}_{i}\right|^2_{1,{\it{\Omega}}_i} \right)^{\frac{1}{2}}\!.\label{Est_Converg_h}\end{align} (5.1) We present in Tables 1 and 2 the computed errors and estimates of convergence orders for this test. We recover a second-order accuracy in $$H^1$$ norms, better than the one predicted by Theorem 4.7, but also better than the one predicted by the standard finite element estimates (3.8)–(3.10). We thus encounter a superconvergence effect, probably due to the symmetries of the grid. Table 1 Estimated convergence order for smooth solutions: velocities $$h$$ $$| {\boldsymbol u}_{e,1} - {\boldsymbol u}_1|_{1,{\it{\Omega}}_1}$$ $$\| {\boldsymbol u}_{e,2} - {\boldsymbol u}_2\|_{1,{\it{\Omega}}_2}$$ $$\mathcal{\rm{O}}({\boldsymbol u}_{1})-\mathcal{\rm{O}}({\boldsymbol u}_{2})$$ 0.176777 0.0264 0.0112 — 0.117852 0.0118 0.00504 1.979–1.987 0.0883883 0.00668 0.00285 1.986–1.995 0.0707107 0.00428 0.00183 1.99–1.998 0.0589261 0.00298 0.00127 1.992–2.00 0.0505087 0.00219 0.000939 1.994–2.001 0.0441942 0.00167 0.000721 1.994–2.001 0.0392841 0.00132 0.000573 1.995–2.001 $$h$$ $$| {\boldsymbol u}_{e,1} - {\boldsymbol u}_1|_{1,{\it{\Omega}}_1}$$ $$\| {\boldsymbol u}_{e,2} - {\boldsymbol u}_2\|_{1,{\it{\Omega}}_2}$$ $$\mathcal{\rm{O}}({\boldsymbol u}_{1})-\mathcal{\rm{O}}({\boldsymbol u}_{2})$$ 0.176777 0.0264 0.0112 — 0.117852 0.0118 0.00504 1.979–1.987 0.0883883 0.00668 0.00285 1.986–1.995 0.0707107 0.00428 0.00183 1.99–1.998 0.0589261 0.00298 0.00127 1.992–2.00 0.0505087 0.00219 0.000939 1.994–2.001 0.0441942 0.00167 0.000721 1.994–2.001 0.0392841 0.00132 0.000573 1.995–2.001 Table 1 Estimated convergence order for smooth solutions: velocities $$h$$ $$| {\boldsymbol u}_{e,1} - {\boldsymbol u}_1|_{1,{\it{\Omega}}_1}$$ $$\| {\boldsymbol u}_{e,2} - {\boldsymbol u}_2\|_{1,{\it{\Omega}}_2}$$ $$\mathcal{\rm{O}}({\boldsymbol u}_{1})-\mathcal{\rm{O}}({\boldsymbol u}_{2})$$ 0.176777 0.0264 0.0112 — 0.117852 0.0118 0.00504 1.979–1.987 0.0883883 0.00668 0.00285 1.986–1.995 0.0707107 0.00428 0.00183 1.99–1.998 0.0589261 0.00298 0.00127 1.992–2.00 0.0505087 0.00219 0.000939 1.994–2.001 0.0441942 0.00167 0.000721 1.994–2.001 0.0392841 0.00132 0.000573 1.995–2.001 $$h$$ $$| {\boldsymbol u}_{e,1} - {\boldsymbol u}_1|_{1,{\it{\Omega}}_1}$$ $$\| {\boldsymbol u}_{e,2} - {\boldsymbol u}_2\|_{1,{\it{\Omega}}_2}$$ $$\mathcal{\rm{O}}({\boldsymbol u}_{1})-\mathcal{\rm{O}}({\boldsymbol u}_{2})$$ 0.176777 0.0264 0.0112 — 0.117852 0.0118 0.00504 1.979–1.987 0.0883883 0.00668 0.00285 1.986–1.995 0.0707107 0.00428 0.00183 1.99–1.998 0.0589261 0.00298 0.00127 1.992–2.00 0.0505087 0.00219 0.000939 1.994–2.001 0.0441942 0.00167 0.000721 1.994–2.001 0.0392841 0.00132 0.000573 1.995–2.001 Table 2 Estimated convergence order for smooth solutions: energies $$h$$ $$\| k_{1,e} - k_1 \|_{1,{\it{\Omega}}_1}$$ $$\| k_{2,e} - k_2 \|_{1,{\it{\Omega}}_2}$$ $$\mathcal {\rm{O}}(k_{1})-\mathcal{\rm{O}}(k_{2})$$ 0.176777 0.0796 0.126 — 0.117852 0.0355 0.0574 1.987–1.956 0.0883883 0.0200 0.0325 1.995–1.975 0.0707107 0.0128 0.0208 1.998–1.984 0.0589261 0.00890 0.0145 2.000–1.989 0.0505087 0.00654 0.0106 2.001–1.993 0.0441942 0.00500 0.00818 2.001–1.995 0.0392841 0.00395 0.00647 2.001–1.998 $$h$$ $$\| k_{1,e} - k_1 \|_{1,{\it{\Omega}}_1}$$ $$\| k_{2,e} - k_2 \|_{1,{\it{\Omega}}_2}$$ $$\mathcal {\rm{O}}(k_{1})-\mathcal{\rm{O}}(k_{2})$$ 0.176777 0.0796 0.126 — 0.117852 0.0355 0.0574 1.987–1.956 0.0883883 0.0200 0.0325 1.995–1.975 0.0707107 0.0128 0.0208 1.998–1.984 0.0589261 0.00890 0.0145 2.000–1.989 0.0505087 0.00654 0.0106 2.001–1.993 0.0441942 0.00500 0.00818 2.001–1.995 0.0392841 0.00395 0.00647 2.001–1.998 Table 2 Estimated convergence order for smooth solutions: energies $$h$$ $$\| k_{1,e} - k_1 \|_{1,{\it{\Omega}}_1}$$ $$\| k_{2,e} - k_2 \|_{1,{\it{\Omega}}_2}$$ $$\mathcal {\rm{O}}(k_{1})-\mathcal{\rm{O}}(k_{2})$$ 0.176777 0.0796 0.126 — 0.117852 0.0355 0.0574 1.987–1.956 0.0883883 0.0200 0.0325 1.995–1.975 0.0707107 0.0128 0.0208 1.998–1.984 0.0589261 0.00890 0.0145 2.000–1.989 0.0505087 0.00654 0.0106 2.001–1.993 0.0441942 0.00500 0.00818 2.001–1.995 0.0392841 0.00395 0.00647 2.001–1.998 $$h$$ $$\| k_{1,e} - k_1 \|_{1,{\it{\Omega}}_1}$$ $$\| k_{2,e} - k_2 \|_{1,{\it{\Omega}}_2}$$ $$\mathcal {\rm{O}}(k_{1})-\mathcal{\rm{O}}(k_{2})$$ 0.176777 0.0796 0.126 — 0.117852 0.0355 0.0574 1.987–1.956 0.0883883 0.0200 0.0325 1.995–1.975 0.0707107 0.0128 0.0208 1.998–1.984 0.0589261 0.00890 0.0145 2.000–1.989 0.0505087 0.00654 0.0106 2.001–1.993 0.0441942 0.00500 0.00818 2.001–1.995 0.0392841 0.00395 0.00647 2.001–1.998 Test 2: Convergence order for low-regularity solution. We have solved in this test a dimensional flow, with realistic data, used in Chacón Rebollo et al. (2010): \begin{align}\gamma_1(k_1) = 3 \times 10^{-3} + 0.277 \times 10^{-4} \sqrt{k_1}; & \quad \gamma_1(k_2) = 3 \times 10^{-2} + 0.185 \times 10^{-5} \sqrt{k_2}. \notag \\ \alpha_i(\cdot)= \gamma_i(\cdot), \qquad \kappa = 10^{-3}\;{\rm{and}}\;& \lambda = 5 \times 10^{-2}. \notag \end{align} The computational domains we consider here are $${\it{\Omega}}_1 =]0, 2[\times]0,1[\times]0,1[$$ for the ‘atmosphere’ and $${\it{\Omega}}_2 =]0, 2[\times]0,1[\times]-1,0[$$ for the ‘ocean’. We have imposed rigid lid velocity boundary conditions on the top of the atmosphere $$\tilde {\it{\Gamma}}_1$$ (that corresponds to $$z=1$$): $${\boldsymbol u}_1 = {\boldsymbol 0}$$ on $${\it{\Gamma}}_1 / \tilde {\it{\Gamma}}_1$$, $${\boldsymbol u}_1 = (1,0,0)$$ on $$\tilde {\it{\Gamma}}_1$$ and $${\boldsymbol u}_2 = {\boldsymbol 0}$$ on $${\it{\Gamma}}_2$$. We consider homogenous Dirichlet boundary conditions for the TKE on all border $$\partial {\it{\Omega}}_1 \cup \partial {\it{\Omega}}_2$$ and equal to $$\lambda |{\boldsymbol u}_1 -{\boldsymbol u}_2|^2$$ on the interface $${\it{\Gamma}}$$. In this way, a cavity flow is induced in the atmosphere, which drives the oceanic flow through the nonlinear boundary conditions at the interface. The ‘exact’ $${\boldsymbol u}_i$$ and $$k_i$$ are computed with a very fine grid. Table 3 displays the computed convergence orders. These seem to approximate the value $$p\approx 0.25$$. Note that the Dirichlet boundary condition for the velocity $${\boldsymbol u}_1$$ is a piecewise constant function, which belongs to $$H^{1/2-\epsilon}({\it{\Gamma}}_1)$$ for all $$\epsilon >0$$, but not to $$H^{1/2}({\it{\Gamma}}_1)$$ (see Chen, 2016). We may then expect that the velocities have a reduced regularity. In fact, from the error estimates (4.34), the value $$p=0.25$$ would correspond to $${\boldsymbol u}_i \in H^{3/2}({\it{\Omega}}_i)^d$$. Table 3 Estimated convergence order for low-regularity solution Mesh size Order $$\mathcal {\rm{O}}_h$$ $$h$$ — $$h/2$$ 0.12 $$h/4$$ 0.16 $$h/8$$ 0.22 $$h/16$$ 0.23 Mesh size Order $$\mathcal {\rm{O}}_h$$ $$h$$ — $$h/2$$ 0.12 $$h/4$$ 0.16 $$h/8$$ 0.22 $$h/16$$ 0.23 Table 3 Estimated convergence order for low-regularity solution Mesh size Order $$\mathcal {\rm{O}}_h$$ $$h$$ — $$h/2$$ 0.12 $$h/4$$ 0.16 $$h/8$$ 0.22 $$h/16$$ 0.23 Mesh size Order $$\mathcal {\rm{O}}_h$$ $$h$$ — $$h/2$$ 0.12 $$h/4$$ 0.16 $$h/8$$ 0.22 $$h/16$$ 0.23 Using high-order finite element spaces to solve problem (1.1) for low-regularity solutions needs an un-useful computational effort, as the convergence order of the discretization is limited by the low regularity of the solution. Table 4 Number of iterations vs. $$\nu$$ $$\nu$$ 1 0.5 0.1 0.05 $$1e^{-2}$$ $$1e^{-3}$$ $$1e^{-4}$$ $$1e^{-5}$$ $$1e^{-8}$$ Number of iterations 4 5 9 14 18 24 20 20 20 $$\nu$$ 1 0.5 0.1 0.05 $$1e^{-2}$$ $$1e^{-3}$$ $$1e^{-4}$$ $$1e^{-5}$$ $$1e^{-8}$$ Number of iterations 4 5 9 14 18 24 20 20 20 Table 4 Number of iterations vs. $$\nu$$ $$\nu$$ 1 0.5 0.1 0.05 $$1e^{-2}$$ $$1e^{-3}$$ $$1e^{-4}$$ $$1e^{-5}$$ $$1e^{-8}$$ Number of iterations 4 5 9 14 18 24 20 20 20 $$\nu$$ 1 0.5 0.1 0.05 $$1e^{-2}$$ $$1e^{-3}$$ $$1e^{-4}$$ $$1e^{-5}$$ $$1e^{-8}$$ Number of iterations 4 5 9 14 18 24 20 20 20 5.2 Test 3: Dependency of convergence rate upon $$\nu$$ This test aims at determining the convergence and the practical convergence rate of algorithm (3.17)–(3.21) with respect to the laminar viscosity $$\nu$$. To illustrate the effects of the viscosity we have used the analytic solution described in Test 2 computed in tetrahedral mesh, where $$h \approx 0.07$$ with the viscosity $$\nu$$ ranging in the interval $$[10^{-8},1]$$, and we determine the convergence rate $$K$$ of the fixed-point algorithm (3.17)–(3.21) as follows: $$K \,=\, \left(\frac{ \displaystyle \sum_{i=1}^2 \| {{\boldsymbol u}}_{i}^{j+1} - {{\boldsymbol u}}_{i}^{j}\|^2 \,+\, \| k_{i}^{j+1} - k_{i}^{j}\|^2}{\displaystyle\sum_{i=1}^2 \| {{\boldsymbol u}}_{i}^{j} - {{\boldsymbol u}}_{i}^{j-1}\|^2 \,+\, \| k_{i}^{j} - k_{i}^{j-1}\|^2}\right)^{\frac{1}{2}},$$ where the index $$j$$ is the first iteration in the fixed-point algorithm for which $$\left(\displaystyle \sum_{i=1}^2\| {{\boldsymbol u}}_{i}^{j+1} - {{\boldsymbol u}}_{i}^{j}\|^2 \,+\, \| k_{i}^{j+1} - k_{i}^{j}\|^2 \right)^{\frac{1}{2}} \leq \varepsilon = 1e^{-10}.$$ Figure 1 shows that algorithm (3.17)–(3.21) converges with a rather constant rate $$K \simeq 0.55$$ when $$\nu \in [1e^{-8}, 1e^{-3}]$$. This seems to indicate that the eddy viscosities and diffusion remain uniformly bounded from below, thus ensuring a good convergence rate for this fixed-point algorithm. Note that the convergence rate improves as $$\nu$$ increases in the range $$[1e^{-3}, 1]$$. Fig. 1. View largeDownload slide Constant rate $$K$$ with respect to $$\nu$$. Fig. 1. View largeDownload slide Constant rate $$K$$ with respect to $$\nu$$. Also, the errors between the exact solution and the fixed point provided by algorithm (3.17)–(3.21) decrease as $$\nu$$ increases in $$[1e^{-3},1]$$, as predicted by Theorem 4.7, but remains constant for $$\nu \le 1e^{-3}$$, see Figs 2 and 3. This is still consistent with the existence of a uniform lower bound for the eddy viscosities and diffusions as $$\nu$$ decreases. Fig. 2. View largeDownload slide $${\boldsymbol H}^1$$-error between the exact and approximate solutions of velocities with respect to $$\nu$$. Fig. 2. View largeDownload slide $${\boldsymbol H}^1$$-error between the exact and approximate solutions of velocities with respect to $$\nu$$. Fig. 3. View largeDownload slide $${\boldsymbol H}^1$$-error between the exact and approximate solutions of energies with respect to $$\nu$$. Fig. 3. View largeDownload slide $${\boldsymbol H}^1$$-error between the exact and approximate solutions of energies with respect to $$\nu$$. 5.3 Upwelling effects near the shores This test is designed to determine the ability of our iterative algorithm presented above to simulate genuine three-dimensional effects that arise in geophysical flows. Concretely, we test the formation of the upwelling effect due to the interaction between surface wind tension and Coriolis forces. We have considered the following problem that includes transport and Coriolis terms, instead of system (1.1), $$\label{P1-b}\left \{ \begin{array}{rcl}\left({\boldsymbol u}_i \cdot \nabla \right) {\boldsymbol u}_i + \tau \left(-u_{i,y},u_{i,x},0\right) \qquad \qquad&& \\ - \nabla \cdot(\alpha_i(k_i)\nabla {\boldsymbol u}_i) + \nabla p_i &=& {\boldsymbol f}_i \\ \nabla \cdot {\boldsymbol u}_i &=& 0 \\ {\boldsymbol u}_i \nabla k_i -\nabla \cdot(\gamma_i(k_i)\nabla k_i) &=&\alpha_i(k_i) |\nabla {\boldsymbol u}_i|^2, \end{array}\right.$$ (5.2) where the component of the velocity fields are denoted by $${\boldsymbol u}_i = (u_{i,x},u_{i,y},u_{i,z})$$. The term $$\tau \left(-u_{i,y},u_{i,x},0\right)$$ models the Coriolis forces, where the parameter $$\tau$$ depends on the angular velocity of the earth and the latitude. Furthermore, to take into account the inertial effects we have added the convection term $$\left({\boldsymbol u}_i \cdot \nabla \right) {\boldsymbol u}_i$$ in the first equation and the transport term $${\boldsymbol u}_i \nabla k_i$$ in the TKEs equation. The boundary conditions are the same as for system (1.1). We have considered the computational domains $${\it{\Omega}}_1 =]0,\,10^4[\times]0, 5 \cdot10^3[\times]0,\,500[\, (m)$$ for atmosphere and a swimming pool-like domain to model the geometry of the ocean (see Fig. 4), $${\it{\Omega}}_2 = \omega \times \{z = D(x,y); \quad (x,y) \in \omega \}$$, such that Fig. 4. View largeDownload slide Computational domain of ocean. Fig. 4. View largeDownload slide Computational domain of ocean. Horizontal dimensions $$(m)$$:   $$\omega \,= \,]0,\,10^4[\times]0,\, 5 \cdot10^3[$$ Bathimetry $$(m)$$: \begin{equation*}\label{pool}D(x,y) = \left \{ \begin{array}{rclr}-50 \quad &\mbox{if}& 0 \leq x \leq 4 \cdot 10^3 \\ \displaystyle -50 \cdot \frac{5\cdot 10^3 -x}{10^3} -100 \cdot \frac{4\cdot 10^3 -x}{10^3}\quad &\mbox{if}& 4\cdot 10^3 \leq x \leq 5 \cdot 10^3 \\ -100 \quad &\mbox{if}& 5 \cdot 10^3 \leq x \leq 10^4. \end{array}\right. \end{equation*} We use the following set of data: $$\alpha_1(k_1)= \gamma_1(k_1)= 3\cdot 10^-2 + 0,277\cdot 10^{-4} \sqrt{k_1} \,m^2/s$$ $$\alpha_2(k_2)= \gamma_2(k_2)=\nu+ 0,185\cdot 10^{-5} \sqrt{k_2}$$, where $$\nu = (10^{-1},10^{-2},10^{-4}) \,m^2/s$$ $$\lambda = 5\cdot 10^{-2}$$, $$\kappa = 10^{-3}$$, $${\boldsymbol f}_i = {\boldsymbol 0},$$ $$\; \tau = 2 \theta \, \sin(\phi)$$, where $$\theta = 7,3\cdot 10^{-5}\,s^{-1}$$, $$\phi = 45^0\,N$$. The viscosity $$\nu$$ is scaled by the different sizes of $$\omega$$ in the directions $$OX$$, $$OY$$ and $$OZ$$ to take into account the anisotropy of the domain. Also, we have again imposed lid-driven cavity boundary conditions, $${\boldsymbol u}_1 = {\boldsymbol 0}$$ on $${\it{\Gamma}}_1 / \tilde {\it{\Gamma}}_1$$, $${\boldsymbol u}_1 = (-1,0,0)$$ on $$\tilde {\it{\Gamma}}_1$$ and $${\boldsymbol u}_2 = {\boldsymbol 0}$$ on $${\it{\Gamma}}_2$$, where $$\tilde {\it{\Gamma}}_1$$ is the upper face $$z=500$$ of $${\it{\Omega}}_1$$. We consider homogenous Dirichlet boundary conditions for the TKE on all the border $$\partial {\it{\Omega}}_1 \cup \partial {\it{\Omega}}_2$$ and equal to $$\lambda |{\boldsymbol u}_1 -{\boldsymbol u}_2|^2$$ on the interface $${\it{\Gamma}}$$. These settings are chosen to create a driven cavity-like flow in atmosphere domain $${\it{\Omega}}_1$$. The atmospheric flow generates a wind flow at the top of the pool, i.e., the interface air–water and, subsequently, the formation of the upwelling flow besides a lateral wall of the pool. The relatively short dimension of the domain in the cross-wind direction also originates a downwelling flow in the vertical face of the pool opposite to the upwelling. On the other hand, the relatively short dimension of the domain in the wind direction originates a longitudinal recirculation that accelerates due to the bottom ramp, in a direction opposite to the wind. Note that in the representation of the numerical results, the depth and the vertical velocity have been increased by a factor 10 to provide a good visualization. Figure 5 shows the vertical velocity, where Coriolis acceleration effects are apparent. In the Northern Hemisphere, the Earth rotation deviates the flow to its right. Figure 6 represents the velocity profile along a vertical cut of the domain (the plane $$y = 2500$$). We observe a global recirculation of the flow that produces an acceleration along the ramp, in the direction opposite to the wind. The flow presents a quasi-parabolic vertical profile in the less depth part of the domain. Figures 7 and 8 show the projection of the three-dimensional velocity on two planes orthogonal to the wind direction ($$x = 5000,\,x = 8000$$, respectively). We observe not only the upwelling effect on the left of the domain, but also a downwelling effect on the right of the domain. The overall flow is a recirculation transversal to the wind direction. The trajectory of a flow particle describes spirals around an axis parallel to the wind direction. Fig. 5. View largeDownload slide Surface velocity in the ocean. Fig. 5. View largeDownload slide Surface velocity in the ocean. Fig. 6. View largeDownload slide Projection of three-dimensional velocity in the ocean onto plane y = 2500. Fig. 6. View largeDownload slide Projection of three-dimensional velocity in the ocean onto plane y = 2500. Fig. 7. View largeDownload slide Projection of three-dimensional velocity in the ocean onto plane x = 8000. Fig. 7. View largeDownload slide Projection of three-dimensional velocity in the ocean onto plane x = 8000. Fig. 8. View largeDownload slide Projection of three-dimensional velocity in the ocean onto plane x = 5000. Fig. 8. View largeDownload slide Projection of three-dimensional velocity in the ocean onto plane x = 5000. 6. Conclusion In this article, we have analysed a numerical model for the coupling two steady turbulent fluids. The coupling is modeled by the Manning law and the generation of TKE due to shear stress at the interface. Both the viscosity and diffusion depend on the TKE. We have considered an iterative scheme by linearization besides a full finite element discretization of a regularized problem. We have proved that the iterative scheme converges for large enough eddy viscosities to the solution of the continuous problem. Also, numerical tests for smooth solutions are in good agreement with our theoretical results. Furthermore, in these tests, the convergence rate of the fixed-point iteration procedure remains uniformly bounded as the laminar viscosity decreases. This is consistent with the existence of a uniform lower bound for eddy viscosities and diffusions, what opens an interesting research subject from both the theoretical and the numerical viewpoints. Furthermore, some tests for realistic flows show the ability of the discretization introduced to correctly simulate the three-dimensional features associated with the interaction between wind stress induced by atmosphere and Coriolis forces. In future works, we will study the application of the fixed-point method introduced here to evaluate large eddy simulation models similar to the one studied and also to de-couple discretizations of the friction interface boundary condition for the TKE, introduced in Connors et al. (2009). Funding Proyecto de Excelencia (P12-FQM-454), in part; Junta de Andalucía Regional Gand FEDER EU Fund. References Bénilan P. , Boccardo L. , Gallouët T. , Gariepy R. , Pierre M. & Vázquez J. L. ( 1995 ) An $$L^1$$-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) , 22 , 241 – 273 . Bernardi C. , Chacén Rebollo T. , Gémez Mármol M. , Lewandowski R. & Murat F. ( 2004a ) A model for two coupled turbulent fluids. III. Numerical approximation by finite elements. Numer. Math. , 98 , 33 – 66 . Google Scholar Crossref Search ADS Bernardi C. , Chacén Rebollo T. , Lewandowski R. & Murat F. ( 2002 ) A model for two coupled turbulent fluids. I. Analysis of the system. Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar (Paris, 1997/1998), vol. 14. Studies in Applied Mathematics, vol, 31. Amsterdam : North-Holland, pp. 69 – 102 . Bernardi C. , Chacén Rebollo T. , Lewandowski R. & Murat F. ( 2003 ) A model for two coupled turbulent fluids. II. Numerical analysis of a spectral discretization. SIAM J. Numer. Anal. , 40 , 2368 – 2394 . Google Scholar Crossref Search ADS Bernardi C. , Maday Y. & Rapetti F. ( 2004b ) Discrétisation variationnelles de problèmes aux limites elliptiques , vol. 45 Mathématiques & Applications (Berlin) [Mathematics & Applications]. Paris : Springer. Betta M. F. , Mercaldo A. , Murat F. & Porzio M. M. ( 2002 ) Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $$L^1(\Omega)$$. ESAIM Control Optim. Calc. Var. , 8 , 239 – 272 . A tribute to J. L. Lions. Google Scholar Crossref Search ADS Betta M. 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M2AN Math. Model. Numer. Anal. , 44 , 693 – 713 . Google Scholar Crossref Search ADS Chacén Rebollo T. & Lewandowski R. ( 2014 ) Mathematical and Numerical Foundations of Turbulence Models and Applications . Modeling and Simulation in Science, Engineering and Technology. New York : Birkhäuser/Springer. Chen L. ( 2016 ) Sobolev spaces and elliptic equations. Course Notes . Ciarlet P. G. ( 1978 ) The Finite Element Method for Elliptic Problems , vol. 4. Studies in Mathematics and its Applications. Amsterdam : North-Holland Publishing Co. Codina R. ( 1993 ) A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation. Comput. Methods Appl. Mech. Engrg. , 110 , 325 – 342 . Google Scholar Crossref Search ADS Connors J. F. , Howell J. S. & Layton W. J. ( 2009 ) Partitioned time stepping for a parabolic two domain problem. SIAM J. Numer. Anal. , 47 , 3526 – 3549 . Google Scholar Crossref Search ADS Davidson L. ( 2013 ) Fluid Mechanics, Turbulent Flow and Turbulence Modeling . Lecture Notes in Chalmers University. Available at http://www.tfd.chalmers.se/?lada/mof/lecture notes.htm. Ern A. & Guermond J.-L. ( 2004 ) Theory and Practice of Finite Elements , vol. 159. Applied Mathematical Sciences. New York : Springer. Girault V. & Raviart P.-A. ( 1986 ) Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms . Springer. Grisvard P. ( 1985 ) Elliptic Problems in Nonsmooth Domains , vol. 24 . Monographs and Studies in Mathematics. Boston : Pitman ( Advanced Publishing Program ). Hebey E. ( 1999 ) Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities , vol. 5. CIMS Lecture Notes, Courant Institute of Mathematical Sciences. Hecht F. ( 2012 ) New development in FreeFem++. J. Numer. Math. , 20 , 251 – 266 . Google Scholar Crossref Search ADS Holst M. J. , Larson M. G. , Målqvist A. & Söderlund R. 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# A three-dimensional model for two coupled turbulent fluids: numerical analysis of a finite element approximation

IMA Journal of Numerical Analysis, Volume 38 (4) – Oct 16, 2018
32 pages

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Oxford University Press
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0272-4979
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1464-3642
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10.1093/imanum/drx049
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### Abstract

Abstract This article deals with the numerical analysis of a coupled two-fluid Reynolds-averaged Navier–Stokes (RANS) turbulence model, such as atmosphere–ocean flow. Each fluid is modeled by the coupled steady Stokes equations with the equation for the turbulent kinetic energy (TKE). In this model, the eddy viscosities for velocity and TKE depend on the TKE, the production (source) term for the TKEs is only in $$L^1$$ and the boundary condition for the TKEs on the interface between the two flows depends quadratically on the difference of velocities. To overcome the lack of regularity, we approximate the initial system by a regularized system, in which the eddy viscosities and source terms for the TKEs are regularized by convolution. We perform a full finite element discretization of the regularized model, combined with a decoupled iterative linearization procedure. We prove that the discrete scheme converges to the continuous scheme for large enough eddy viscosities in natural norms. Finally, we present some numerical tests where we study the accuracy of the procedure, and simulate a realistic flow in which an imposed wind in the upper atmosphere generates an upwelling in the oceanic flow. 1. Introduction This article deals with the numerical analysis of the approximation of Reynolds-averaged Navier–Stokes (RANS) turbulence models and, more specifically, on coupled two-fluid RANS models. Typically, this model addresses the coupled atmosphere–ocean system, whose accurate numerical simulation is crucial to analyse the main issues related to climate change. From the practical point of view, RANS models are rather diffusive and provide overall predictions of many flows of engineering interest (cf. Davidson, 2013). However, from the mathematical point of view, RANS models are more singular problems than the Navier–Stokes equations. The main mathematical difficulties in the analysis of RANS models are that the source term (the production term) for the turbulent kinetic energy (TKE) has just $$L^1({\it{\Omega}})$$ regularity, and that the equation for the TKE does not make sense in $$H^{-1}({\it{\Omega}})$$. Its solution must be understood in the renormalized—or entropy—sense (cf. Boccardo & Gallou, 1992; Bénilan et al., 1995; Boccardo et al., 1996a,b; Betta et al., 2002, 2003). The numerical analysis of the finite element approximation of elliptic equations with right-hand side in $$L^1$$ has been performed for linear diffusion equations (cf. Casado-Díaz et al., 2007). The main reason is that the extension of the basic estimates of renormalized solutions (the Boccardo–Gallouet estimates) only hold if the numerical scheme satisfies a discrete maximum principle. For convection–diffusion equations, there exist a few finite element schemes that satisfy this principle, based on the multidimensional unwinding techniques (cf. Tabata, 1978; Codina, 1993; Burman & Ern, 2005). However, the extension of the numerical analysis of these schemes to the solution of equations with $$L^1$$ right-hand side has not been performed yet. For finite volume discretizations, there exist more schemes satisfying the maximum principle, whose analysis has been performed in some cases. Because of these difficulties, the mathematical and numerical analyses of RANS models performed up to date has been applied to simplified equations. In Bernardi et al. (2002) study, a model for two-coupled turbulent fluids and its numerical approximation, where the turbulent kinetic energy (TKE) equation only contains eddy diffusion (with bounded eddy viscosities) and the production term. This model also contains a nonlinear friction term to model the interactions between the two fluids across the interface: \begin{eqnarray}\label{P1}\left\{ \begin{array}{rcl}-\nabla \cdot\left(\alpha_i\left(k_i\right)\nabla {\boldsymbol u}_i\right) + \nabla p_i &=& {\boldsymbol f}_i \; \, \mbox{in} {\it{\Omega}}_i, \\ \nabla \cdot {\boldsymbol u}_i &=& 0 \quad \mbox{in} {\it{\Omega}}_i,\\ -\nabla \cdot\left(\gamma_i\left(k_i\right)\nabla k_i\right) &=&\alpha_i\left(k_i\right) \left|\nabla {\boldsymbol u}_i \right|^2 \mbox{in} {\it{\Omega}}_i,\\ {\boldsymbol u}_i &=& {\boldsymbol 0} \mbox{on} {\it{\Gamma}}_i,\\ k_i &=& 0 \mbox {on} {\it{\Gamma}}_i,\\ \alpha_i\left(k_i\right)\partial_{{\boldsymbol n}_i}{\boldsymbol u}_i - p_i {\boldsymbol n}_i + \kappa\left ({\boldsymbol u}_i -{\boldsymbol u}_j\right)\left| {\boldsymbol u}_i-{\boldsymbol u}_j\right| &=& {\boldsymbol 0} \mbox{on} {\it{\Gamma}}, 1\leq i \neq j \leq 2,\\ k_i &=& \lambda\left|{\boldsymbol u}_1 -{\boldsymbol u}_2\right|^2 \mbox {on} {\it{\Gamma}}, \end{array}\right. \end{eqnarray} (1.1) where both domains $${\it{\Omega}}_i$$, $$i=1,2$$ are bounded in $$\mathbb R^d,~ d=2,3$$, which are either convex or of class $$\mathbb C^{1,1}$$, with boundaries $$\partial {\it{\Omega}}_i= {\it{\Gamma}}_i \cup {\it{\Gamma}}$$ and $${\it{\Gamma}} = \overline {\it{\Omega}}_1 \cap \overline {\it{\Omega}}_2$$ being the interface between the two fluids. In this model, $${\it{\Gamma}}$$ is assumed to be flat. This is the ‘rigid lid hypothesis’ introduced by Bryan (1969), an hypothesis that provides a good modeling for flows at large space scales, although needs adjustments for interfaces governed by more complex dynamics, related to atmospheric pressure and tidal variability effects, among others. The main focus of the article is to analyse a full finite element discretization of this model, which has been studied in several articles (see Bernardi et al., 2003, 2004a; Chacón Rebollo et al., 2010), but this analysis remains open. We are mainly interested in the mathematical treatment of the difficulties due to the nonlinear terms, in particular those coming from RANS turbulence modeling. From the mathematical point of view, the steady model applies to large time scales, although it does not make sense from a physical point of view, as the boundary conditions are largely variable in time. An unsteady RANS model would be more appropriate to approximate a realistic ocean–atmosphere coupled flow. Each of the two turbulent fluids is modeled by a simplified one-equation turbulence model whose unknowns are the velocity $${\boldsymbol u}_i$$, the pressure $$p_i$$ and the TKE $$k_i$$. The generation of eddy viscosity in flow $$i$$ is modeled by the term $$-\nabla \cdot\left(\alpha_i\left(k_i\right)\nabla {\boldsymbol u}_i\right)\!.$$ The (positive) quantity $$\alpha_i(k_i)$$ is the eddy viscosity. This is a simplification of the usual modeling of Reynolds stress tensor by $$R_i \simeq - \alpha_i\left(k_i\right)\left(\nabla {\boldsymbol u}_i + \nabla ^t{\boldsymbol u}_i \right)$$ to simplify the mathematical analysis. The equations for the TKE only contains the eddy dissipation and production terms, respectively, given by $$-\nabla \cdot\left(\gamma_i\left(k_i\right)\nabla k_i\right)\quad \mbox{and}\quad \alpha_i\left(k_i\right) \left|\nabla {\boldsymbol u}_i\right|^2,$$ where $$\gamma_i(k_i)$$ is the eddy diffusion for the TKE $$k_i$$. The interface terms model the friction between the two fluids (fifth equation in (1.1)) and the generation of TKE (sixth equation in (1.1)), both are wall laws that model the dissipation of turbulence in the boundary layers on both sides of the interface. The positive parameters $$\kappa$$ and $$\lambda$$ are the friction and energy production coefficients, respectively. The analysis of model (1.1) reported in Bernardi et al. (2002) is based on the reformulation of the equations for the TKEs by transposition combined with a compactness argument. The global system is proved to admit a solution $$\left({\boldsymbol u}_i,p_i,k_i\right)\in {\boldsymbol H}^1\left({\it{\Omega}}_i\right) \times L^2\left({\it{\Omega}}_i\right)\times H^{s}\left({\it{\Omega}}_i\right)$$ regularity for all $$s<1/2$$. Several subsequent works have dealt with the numerical approximation of this model. The same authors perform in Bernardi et al. (2003) an error analysis for spectral discretizations of (1.1) for smooth solutions. In Bernardi et al. (2004a), a finite element discretization by piecewise affine finite elements of this problem is studied using the compactness method. The convergence of a subsequence of discrete solutions to the continuous solution is proved. In Chacón Rebollo et al. (2010), a linearization procedure to solve the same continuous model for large eddy viscosities is introduced. The procedure is proved to converge for large eddy viscosities, assuming that the discrete velocities and TKEs are uniformly bounded in $$W^{1, 3+\varepsilon}$$ norm, for some $$\varepsilon >0$$. In Chacón Rebollo & Lewandowski (2014, Chapter 7), the right-hand side for the TKE equations is regularized by convolution, for a problem similar to (1.1) with a single flow. It is proved that the regularized problems admit a solution and that a subsequence of these converge to a weak solution of the original problem, in $${\boldsymbol H}^1\left({\it{\Omega}}\right) \times L^2\left({\it{\Omega}}\right)\times W^{1,q}\left({\it{\Omega}}\right)$$ for $$1\le q <3/2$$, where $${\it{\Omega}}$$ is the flow domain. The TKE equation is satisfied in the renormalized sense, the Boccardo–Gallouet estimates play an essential role in the proof, being the origin of the estimates in $$W^{1,q}\left({\it{\Omega}}\right)$$ of the TKE. A different approach was followed by Holst et al. (2010) in the context of the convergence analysis of finite element approximations of the Joule heating problem. The main technical tool in this article is an identity that allows to rewrite the right-hand side for the temperature (with $$L^1$$ regularity, as in our case does the TKE) in terms of the data for the electric potential and a controllable diffusion term. This technique does not seem to be applicable in our case because of the presence of the pressure gradient in the momentum equation, combined with the friction boundary conditions. In this article, we follow the approach of Chacón Rebollo & Lewandowski (2014) and study the numerical approximation of a regularized approximation of (1.1) by convolution. More concretely, we consider the numerical solution of a full finite element discretization of the regularized problem by the iterative procedure introduced in Chacón Rebollo et al. (2010). We prove that the procedure converges under similar conditions, i.e., for large enough eddy viscosities, without any additional boundedness hypothesis on the velocities and TKEs. The hardest technical point is the obtaining of estimates of the interface quadratic terms. We treat them by specific discrete lifting operators that are compatible with the discretizations of velocity and TKE. We present some numerical tests, where, on one side, we study the accuracy of the procedure. On another side, we simulate a realistic flow in which an imposed wind in the upper atmosphere generates an upwelling in the oceanic flow. The article is structured as follows. In Section 2, we introduce the regularized approximation of model (1.1). Section 3 describes the full finite element approximation of the regularized problem and its solution by the iterative procedure. Section 4 is devoted to the analysis of convergence of this discretization. Finally, Section 5 reports the numerical tests. 2. Regularized model To describe the regularized approximation of model (1.1) that we shall study, let us denote by $$\omega_\varepsilon \left({\boldsymbol x}\right) = \frac{1}{\varepsilon^d}\omega \left(\frac{{\boldsymbol x}}{\varepsilon} \right)$$ a smoothing (mollifier) sequence, defined as follows, see, for instance, Brezis (1983): $$\label{mollifiers}\omega \in \mathcal C^\infty_c\left(\mathbb R^d\right)\!,\qquad \int_{\mathbb R^d} \omega= 1, \qquad supp\left (\omega\right) \subset B(0,1), \qquad \omega \ge 0; \qquad \mbox{for} \; \varepsilon >0.$$ (2.1) For a function $$\psi \in L^p\left({\it{\Omega}}\right)\!, p > 1$$, consider the convolution $$\left(\psi * \omega_\varepsilon\right) \left({\boldsymbol x}\right)\; = \int_{R^d} \tilde{\psi}\left({\boldsymbol x} -{\boldsymbol y}\right) \, \omega_\varepsilon\left({\boldsymbol y}\right) \; {\rm{d}}{\boldsymbol y},$$ where $$\tilde{\psi}$$ is the extension by zero of $$\psi$$ outside $${\it{\Omega}}$$. Then, $$\psi * \omega_\varepsilon \in C^\infty\left(R^d\right)$$ and the following properties hold (cf. Brezis, 1983): $$\label{regu}\displaystyle \lim _{\varepsilon \rightarrow 0} \psi * \omega_\varepsilon = \; \psi \; \mbox{in}\, L^p\left({\it{\Omega}}\right)\!, \left\| \psi * \omega_\varepsilon \right\|_{W^{k,q}\left(\mathbb R^d\right)} \le C_{\varepsilon} \, \left\| \psi \right\|_{L^p({\it{\Omega}})}\!,$$ (2.2) for any integer $$k \ge 1$$, and real number $$q \in [p,+\infty]$$, for some constant $$C_{\varepsilon}$$, such that $$C_{\varepsilon} \to +\infty$$ as $$\varepsilon \to 0^+$$. We are now in a position to state the regularized problem that we study in this article: \begin{eqnarray}\label{P1Reg}\left\{ \begin{array}{rcl}-\nabla \cdot(\alpha_i(k^\varepsilon_i)\nabla {\boldsymbol u}_i) + \nabla p_i &=& {\boldsymbol f}_i \ \,\mbox{in} {\it{\Omega}}_i, \\ \nabla \cdot {\boldsymbol u}_i &=& 0 \mbox{in} {\it{\Omega}}_i,\\ -\nabla \cdot(\gamma_i(k^\varepsilon_i)\nabla k_i) &=& \alpha_i(k^\varepsilon_i) \nabla {\boldsymbol u}_i : \nabla {\boldsymbol u}_i^\varepsilon \mbox{in} {\it{\Omega}}_i,\\ {\boldsymbol u}_i &=& {\boldsymbol 0} \mbox{on} {\it{\Gamma}}_i,\\ k^\varepsilon_i &=& 0 \mbox {on} {\it{\Gamma}}_i,\\ \sigma_i&=& {\boldsymbol 0} \mbox{on} {\it{\Gamma}},\\ k_i &=& \lambda |{\boldsymbol u}^\varepsilon_1 -{\boldsymbol u}^\varepsilon_2|^2 \mbox{on} {\it{\Gamma}}, \end{array}\right. \end{eqnarray} (2.3) where $${\boldsymbol u}^\varepsilon_i = {\boldsymbol u}_i * \omega_\varepsilon$$ (componentwise), $$k^\varepsilon_i = k_i * \omega_\varepsilon$$ and $$\sigma_i = \alpha_i(k^\varepsilon_i) \partial_{{\boldsymbol n}_i}{\boldsymbol u}_i - p_i {\boldsymbol n}_i + \kappa ({\boldsymbol u}_i - {\boldsymbol u}_j)| {\boldsymbol u}_i-{\boldsymbol u}_j|, \;1\leq i \neq j \leq 2.$$ The analysis performed in Chacón Rebollo & Lewandowski (2014, Section 7.4) readily extends to prove that problem (2.3) admits a solution that converges to a solution of (1.1) in $${\boldsymbol H}^1({\it{\Omega}}_i) \times L^2({\it{\Omega}}_i)\times W^{1,q}({\it{\Omega}}_i)$$, as $$\varepsilon$$ tends to $$0$$, whenever the TKE satisfy a homogeneous Dirichlet boundary condition on $${\it{\Gamma}}$$ (seventh equation of (2.3)). However, here, we prefer to include a more realistic boundary condition, modeling the generation of TKEs by friction at the interfaces. Note that the more singular terms in model (1.1) are those that model the generation of TKEs, either by eddy diffusion (right-hand side of third equation) or by friction at the interface (seventh equation). These terms are regularized in model (2.3), in addition to the eddy viscosities. Let us recall some standard notation that we use throughout the article. We denote by $$W^{s,p}({\it{\Omega}}_i)$$ the real Sobolev space, $$0 \leq s < \infty, 0 \leq p \leq \infty$$, equipped with the norm $$\| \cdot \|_{W^{s,p}({\it{\Omega}}_i)}$$. The space $$W^{s,p}_0({\it{\Omega}}_i)$$ is the completion of the space of the smooth functions compactly supported in $${\it{\Omega}}_i$$ with respect to the $$\| \cdot\|_{W^{s,p}({\it{\Omega}}_i)}$$ norm. For $$s=1$$ and $$p=2$$, we denote the Hilbert spaces $$W^{1,2}({\it{\Omega}}_i)$$ (resp. $$W^{1,2}_0(\varOmega_i)$$) by $$H^1({\it{\Omega}}_i)$$ (resp. $$H^1_0({\it{\Omega}}_i)$$). The related norm is denoted by $$\| \cdot\|_{1,{\it{\Omega}}_i}$$. The case of $$s=0$$ corresponds to the space $$L^2({\it{\Omega}}_i)$$ equipped with its standard norm $$\|\cdot\|_{0,{\it{\Omega}}_i}$$. We finally denote by $$|\cdot|_{1,{\it{\Omega}}_i}$$ the seminorm in $$H^1({\it{\Omega}}_i)$$ given by $$|v|_{1,{\it{\Omega}}_i}=\|\nabla v\|_{0,{\it{\Omega}}_i}$$. To formulate the coupled problem (2.3) in variational form, we introduce the velocity and TKE spaces defined as follows: \begin{align}{\boldsymbol X}_i &= \{{\boldsymbol v}_i \in {\boldsymbol H}^1({\it{\Omega}}_i),\quad {\boldsymbol v}_i = {\boldsymbol 0} \; \mbox{on} \, {\it{\Gamma}}_i \}. \label{EspX} \\ \end{align} (2.4) \begin{align}K_i &= \{\ell_i \in H^1({\it{\Omega}}_i),\quad \ell_i = 0 \; \mbox{on} \, {\it{\Gamma}}_i \}. \label{EspK}\end{align} (2.5) The trace operator is continuous from $${\boldsymbol X}_i$$ (resp. $$K_i$$) onto the space $${\boldsymbol H}^{\frac{1}{2}}_{00}({\it{\Gamma}})$$ (resp. $$H^{\frac{1}{2}}_{00}({\it{\Gamma}})$$), [this is the subspace of $$H^{\frac{1}{2}}({\it{\Gamma}})$$ formed by functions whose prolongation by zero to either $$\partial {\it{\Omega}}_1$$—or $$\partial {\it{\Omega}}_2$$—belongs to $$H^{\frac{1}{2}}(\partial {\it{\Omega}}_1)$$—or $$H^{\frac{1}{2}}(\partial {\it{\Omega}}_2)$$—, see Lions & Magenes, 1970 for its definition]. We next state a relevant hypothesis concerning the eddy viscosities and diffusions: Hypothesis 2.1 The functions $$\alpha_i$$ and $$\gamma_i$$ belong to $$W^{1,\infty}(\mathbb R_+)$$. Moreover, there exist positive constants $$\delta$$ and $$\nu$$, such that \begin{equation*}\label{H1}\forall \ell \in \mathbb R_+,\qquad \nu \leq \alpha^{(m)}_i(\ell) \leq \delta, \quad \nu \leq \gamma^{(m)}_i(\ell) \leq \delta, \quad m=0 \; \mbox{or}\; 1. \end{equation*} This hypothesis holds in practical applications, as some kind of numerical smoothing or truncation is applied to avoid too small or too large eddy viscosities that may lead to instabilities. From now on, the following spaces are introduced to simplify notations: $$\mathcal F_i = {\boldsymbol X}_i \times L^2({\it{\Omega}}_i) \times H^1_0({\it{\Omega}}_i), \qquad \mathcal G_i = {\boldsymbol X}_i \times L^2({\it{\Omega}}_i) \times K_i.$$ We are now in a position to write the weak formulation of problem (2.3): Given $${\boldsymbol f}_i \in {\boldsymbol L}^2({\it{\Omega}}_i)$$, find the triplet $$({\boldsymbol u}_i,p_i,k_i) \in \mathcal G_i$$ such that for all test functions $$({\boldsymbol v}_i,q_i,\phi_i) \in \mathcal F_i$$ it holds \begin{align}& a_i(k_i;{\boldsymbol u}_i,{\boldsymbol v}_i) + b_i({\boldsymbol v}_i,p_i) + \kappa \int_ {{\it{\Gamma}}} |{\boldsymbol u}_i-{\boldsymbol u}_j|({\boldsymbol u}_i -{\boldsymbol u}_j) \cdot {\boldsymbol v}_i \, {\rm{d}}\tau = \int_{{\it{\Omega}}_i}{\boldsymbol f}_i\cdot {\boldsymbol v}_i \,{\rm{d}} {\boldsymbol x}, \label{Var1} \\ \end{align} (2.6) \begin{align}& b_i({\boldsymbol u}_i,q_i) =0, \label{Var1_ps} \\ \end{align} (2.7) \begin{align}& k_i= \,0 \; {\rm{on}} \,{\it{\Gamma}}_i,\quadk_i= \lambda |{\boldsymbol u}^{\varepsilon}_i -{\boldsymbol u}^{\varepsilon}_j|^2 \quad\mbox{on}~ {\it{\Gamma}}, \quad{\rm{and}} \label{Var2_CL} \\ \end{align} (2.8) \begin{align}& {\mathcal N_i}(k_i;k_i,\phi_i) = \displaystyle\int_{{\it{\Omega}}_i}\alpha_i(k^{\varepsilon}_i)\, \nabla{\boldsymbol u}_i : \nabla{\boldsymbol u}^{\varepsilon}_i \; \phi_i \, {\rm{d}}{\boldsymbol x}; \label{Var2}\end{align} (2.9) where the forms $$a_i(\cdot;\cdot,\cdot),~b_i(\cdot,\cdot)$$ and $${\mathcal N_i}(\cdot;\cdot,\cdot)$$ are defined by \begin{align}a_i(\ell_i;{\boldsymbol u}_i,{\boldsymbol v}_i) &= \int_{{\it{\Omega}}_i} \alpha(\ell^{\varepsilon}_i) \nabla {\boldsymbol u}_i : \nabla {\boldsymbol v}_i\, {\rm{d}}{\boldsymbol x}, \quad b_i({\boldsymbol v}_i,q_i) = -\displaystyle \int_{{\it{\Omega}}_i} q_i\,\nabla\cdot {\boldsymbol v}_i\,{\rm~d} {\boldsymbol x}, \notag \\ & {\rm{and}} \quad {\mathcal N_i}(\ell_i; k_i, \phi_i) = \displaystyle \int_{{\it{\Omega}}_i}{\gamma_i(\ell^{\varepsilon}_i)}\nabla k_i \cdot \nabla \phi_i \,{\rm{d}}{\boldsymbol x}.\notag \end{align} Note that since $${\boldsymbol u}_i \in {\boldsymbol H}^1({\it{\Omega}}_i)$$ and according to Proposition IV.20 in Brezis (1983), $$\nabla {\boldsymbol u}_i^\varepsilon$$ belongs to $$C^\infty(\mathbb R^d)$$, then the term $$\displaystyle\int_{{\it{\Omega}}_i}\alpha_i(k^\varepsilon_i) \, \nabla{\boldsymbol u}_i : \nabla{\boldsymbol u}^\varepsilon_i \;\phi_i$$ in (2.9) is well defined for all $$\phi_i \in H^1_0({\it{\Omega}}_i)$$. Also, due to the continuous Sobolev embedding from $${\boldsymbol H}^{\frac{1}{2}}(\partial {\it{\Omega}}_i)$$ into $$L^3(\partial {\it{\Omega}}_i)^d$$, the term $$\displaystyle \int_ {{\it{\Gamma}}} \left|{\boldsymbol u}_i-{\boldsymbol u}_j \right|\left({\boldsymbol u}_i -{\boldsymbol u}_j\right) \cdot{\boldsymbol v}_i \, {\rm{d}}\tau$$ in (2.6) is well defined for all $${\boldsymbol v}_i \in{\boldsymbol H}^1({\it{\Omega}}_i)$$. The source functions $${\boldsymbol f}_i$$ can be taken in $${\boldsymbol H}^{-1}({\it{\Omega}}_i)$$ and then the scalar product in $$L^2$$ must be replaced by the duality product between $$<\cdot,\cdot>_{{\boldsymbol H}^1,{\boldsymbol H}^{-1}}$$. Nevertheless, we prefer to work with $${\boldsymbol f}_i \in L^2({\it{\Omega}}_i)^d$$ for simplicity of notation. 3. Discrete iterative procedure To introduce the finite element approximation of problem (2.3), we assume that the domains $${\it{\Omega}}_1$$ and $${\it{\Omega}}_2$$ are polygonal (when $$d=2$$) or polyhedric (when $$d=3$$). Consider a family of triangular grids $$(\mathcal T_{i,h})_h$$ of $$\overline {\it{\Omega}}_i$$ that we assume to be regular, in the usual sense of the finite element method (FEM) (see Ciarlet, 1978; Girault & Raviart, 1986; Ern & Guermond, 2004). For each non-negative integer $$m$$ and any element $$K$$ in $$\mathcal T_{i,h}$$, let $$\mathcal P_{m}(K)$$ denote the space of restriction to $$K$$ of polynomials with $$d$$ variables and total degree $$\leq m$$. Thus, we choose the following spaces: \begin{eqnarray}\label{eq:espx}&&\quad{\boldsymbol X}_{i,h} =\left \{{\boldsymbol v}_{i,h} \in C^0\left(\overline {\it{\Omega}}_i \right)^d \, \forall K \in \mathcal T_{i,h}, \, \restriction{{\boldsymbol v}_{i,h}}{K} \in \mathcal P_{2}(K)^d \right\} \cap {\boldsymbol X}_i, \\ \end{eqnarray} (3.1) \begin{eqnarray}&&\quad M_{i,h} =\left \{q_{i,h} \in C^0\left(\overline {\it{\Omega}}_i \right) \, \forall K \in \mathcal T_{i,h}, \, \restriction{q_{i,h}}{K} \in \mathcal P_{1}(K) \right \}\!, \; \label{eq:espm} \\ \end{eqnarray} (3.2) \begin{eqnarray}&&\quad K_{i,h}=\left \{\ell_{i,h} \in C^0\left(\overline {\it{\Omega}}_i \right) \, \forall K \in \mathcal T_{i,h}, \, \restriction{\ell_{i,h}}{K} \in \mathcal P_{2}(K), \, \restriction{\ell_{i,h}}{{\it{\Gamma}}_i} =0 \right\}\!,\label{eq:espk}\\ \end{eqnarray} (3.3) \begin{eqnarray}&&\quad \label{W}W_{i,h} = \left \{\phi_{i,h} \in \mathcal C^0 \left({\it{\Gamma}} \right)\; \forall e \in \mathcal E_{i,h}, \, \restriction{\phi_{i,h}}{e} \in \mathcal P_{2}(e),\, \restriction{\phi_{i,h}}{\partial{\it{\Gamma}}} =0 \right \}\!, \end{eqnarray} (3.4) where $$\left(\mathcal E_{i,h}\right)_{i,h}$$ denotes all faces ($$d =3$$) or edges ($$d=2$$) of triangulation $$\mathcal T_{i,h}$$, which are contained in $${\it{\Gamma}}$$. Consider the standard Lagrange interpolation operators \begin{align}\mathcal{Q}_{i,h} &:\quad {\boldsymbol X}_i \cap C^0(\overline{{\it{\Omega}}_i})^d\mapsto {\boldsymbol X}_{i,h} \label{Op_Pih1}, \\ \end{align} (3.5) \begin{align}\mathcal{P}_{i,h} &:\quad M_i \cap C^0(\overline{{\it{\Omega}}_i})\mapsto M_{i,h}, \quad \rm{} \label{Op_Ph1} \\ \end{align} (3.6) \begin{align}\mathcal S_{i,h} &: \quad H^{1}({\it{\Omega}}_i) \cap C^0(\overline{{\it{\Omega}}_i}) \mapsto K_{i,h} \label{Op_Sih}.\end{align} (3.7) For instance $$\mathcal{Q}_{i,h}$$ is defined by: $$\forall K \in \mathcal T_{i,h},\;$$ $$\mathcal{Q}_{i,h} \,{\boldsymbol v}|_K = \mathcal I_K \, {\boldsymbol v} \; {\boldsymbol v} \in C^0\left(\overline {\it{\Omega}}_i \right)^d\!, \;$$ where $$\mathcal I_K \, {\boldsymbol v}$$ is the only polynomial of $$\mathcal P_{2}(K)^d$$ that takes the same values as the function $${\boldsymbol v}$$ at degrees of freedom of the local Lagrange interpolation on $$\mathcal P_{2}(K)^d$$. These interpolation operators satisfy the following approximation and stability properties (for more details, see, for instance, Ciarlet, 1978; Bernardi et al., 2004b; Ern & Guermond, 2004; Chacón Rebollo & Lewandowski, 2014): If $${\boldsymbol v} \in H^{\ell_1}({\it{\Omega}}_i)^d\cap X_i$$, $$\ell_1=0,1, 2$$; $$p \in H^{\ell_2}({\it{\Omega}}_i) \cap M_i \;$$, $$\ell_2=0,1$$; $$k \in W^{\ell_3}({\it{\Omega}}_i) \cap K_i$$, $$\ell_3=0,1, 2$$, \begin{align}\| {\boldsymbol v} - \mathcal{Q}_{i,h}\,{\boldsymbol v} \|_{H^{1}({\it{\Omega}}_i)^d} &\leq c h^{\ell_1 - 1} \, |{\boldsymbol v}|_{H^{\ell_1}({\it{\Omega}}_i)^d}, \label{Err_Interp_Pih} \\ \end{align} (3.8) \begin{align}\| p - \mathcal{P}_{i,h}\,p \|_{H^{1}({\it{\Omega}}_i)} &\leq c h^{\ell_2 - 1} \, |p|_{H^{\ell_1}({\it{\Omega}}_i)}, \label{Err_Interp_Ph} \\ \end{align} (3.9) \begin{align}\| k - \mathcal S_{i,h}\,k \|_{H^{1}({\it{\Omega}}_i)} &\leq c h^{\ell_3 - 1} \, |k|_{W^{\ell_2}({\it{\Omega}}_i)}.\label{Err_Interp_Sih}\end{align} (3.10) Also, the family of spaces $$\left({\boldsymbol X}_{i,h}, M_{i,h}\right)_{h>0}$$ satisfy the discrete Babuska–Brezzi inf–sup condition on $${\it{\Omega}}_i$$ (see, for instance, Brezzi & Fortin, 1991). There exists a constant $$\beta_{i} > 0$$, such that: $$\label{inf-sup}q_{i,h} \in M_{i,h}, \qquad \displaystyle \sup_{{\boldsymbol v}_{i,h} \in {\boldsymbol X}_{i,h}}\displaystyle \frac{b_i\left({\boldsymbol v}_{i,h},q_{i,h} \right)} {\left| {\boldsymbol v}_{i,h} \right|} \geq \beta_{i} \|q_{i,h}\|_{0,{\it{\Omega}}_i}.$$ (3.11) In addition, consider the standard Lagrange interpolation operator $$\mathcal L_{i,h}:\quad H^{\frac{1}{2}}_{00}\left({\it{\Gamma}} \right)\cap C^0(\overline{{\it{\Gamma}}}) \longrightarrow W_{i,h} \label{Op_Lih1}$$ (3.12) given by, for all $$e \in \mathcal E_{i,h}$$, $$\mathcal L_{i,h} \,\omega|_e = \mathcal J_e \, \omega, \; \forall \omega \in C^0\left(\overline {\it{\Gamma}} \right)^d,$$ where $$\mathcal J_e \, \omega$$ is the only polynomial of $$\mathcal P_{2}(e)$$ that takes the same values as the function $$\omega$$ at the degrees of freedom of the local Lagrange interpolation on $$\mathcal P_{2}(e)$$. The operators $$L_{i,h}$$ satisfies the stability property $$\label{contlih}\|\mathcal L_{i,h} w_i\|_{H^{1/2}_{00}({\it{\Gamma}})} \le C \,\|w_i\|_{H^{1/2}_{00}({\it{\Gamma}})}\, \mbox{for any} w_i \in H^{1/2}_{00}({\it{\Gamma}})\cap C^0(\overline{{\it{\Gamma}}}),$$ (3.13) for some constant $$C>0$$. Note also that, as $$W_{i,h}$$ is the ‘trace space’ on $${\it{\Gamma}}$$ of $$K_{i,h}$$, the operators $${\cal L}_{i,h}$$ and $${\cal S}_{i,h}$$ satisfy the following compatibility condition. Hypothesis 3.1 For all $$w_i \in H^{1}({\it{\Omega}}_i)$$, the trace on $${\it{\Gamma}}$$ of the interpolate $$\mathcal S_{i,h} \left(w_i\right)$$ coincides with the interpolate of the trace of $$w_i$$: $$\label{Op_Lih2}\mathcal L_{i,h} \left(w_{i_{|_{\it{\Gamma}}}} \right) = \restriction {\left(\mathcal S_{i,h} \,w_i \right)} {{\it{\Gamma}}}.$$ (3.14) We finally assume that there exists a lifting operator $$\mathcal R_{i,h}:W_{i,h} \mapsto K_{i,h}$$ such that, $$\label{lift}\mathcal R_{i,h}(\phi_{i,h})_{|_{\partial {\it{\Omega}}_i}}=\phi_{i,h}\,\mbox{for any}\phi_{i,h} \in W_{i,h}.$$ (3.15) Furthermore, this operator verifies the stability property: $$\label{Rih}\|\mathcal R_{i,h}(\phi_{i,h})\|_{W^{1,p}({\it{\Omega}}_i)} \, \leq \,C_p\, \| \phi_{i,h}\|_{W^{1-1/p,p}(\partial {\it{\Omega}}_i)}\qquad \,\forall p \in]1,+\infty[,$$ (3.16) for some constant $$C_p>0$$. The existence of a lifting operator verifying (3.15)–(3.16) is proved in Bernardi et al. (2004b, Theorem 4.1). Let us introduce the discrete spaces: $$\mathcal F_{i,h} = {\boldsymbol X}_{i,h} \times M_{i,h} \times K_{i,h}, \quadK^0_{i,h} =K_{i,h}\cap H_{0}^{1}({\it{\Omega}}_i).$$ We are in a position to build the discrete problem from (4.1) to (4.5): Assume known $$\left({\boldsymbol u}^{n}_{i,h},p^{n}_{i,h},k^{n}_{i,h}\right) \in \mathcal F_{i,h},\; n \geq 0$$ 1. Compute $$\left({\boldsymbol u}^{n+1}_{i,h},p^{n+1}_{i,h}\right) \in {\boldsymbol X}_{i,h} \times M_{i,h}$$, such that $$\forall({\boldsymbol v}_{i,h},q_{i,h}) \in {\boldsymbol X}_{i,h} \times M_{i,h}$$, \begin{align}\label{iter1h}&a_i\left(k^{n}_{i,h};{\boldsymbol u}_{i,h}^{n+1},\nabla {\boldsymbol v}_{i,h}\right) \,+\, b_i\left({\boldsymbol v}_{i,h},p_{i,h}^{n+1}\right) \notag\\ &\quad+\kappa \displaystyle \int_{{\it{\Gamma}}}\left | {\boldsymbol u}_{i,h}^{n+1} -{\boldsymbol u}_{j,h}^{n+1}\right| \, \left ({\boldsymbol u}_{i,h}^{n+1} -{\boldsymbol u}_{j,h}^{n+1}\right)\, \cdot \,{\boldsymbol v}_{i,h} \, {\rm{d}}\tau =\displaystyle \int_{{\it{\Omega}}_i} {\boldsymbol f}_i \, \cdot \, {\boldsymbol v}_{i,h},\end{align} (3.17) $$\label{iter2h}b_i\left({\boldsymbol u}_{i,h}^{n+1},q_{i,h}\right) =0.$$ (3.18) 2. Compute $$k^{n+1}_{i,h}\in K_{i,h}$$, such that $$\forall\phi_{i,h} \in K^0_{i,h}$$, \begin{align}k^{n+1}_{i,h} & = 0 \mbox{on}{\it{\Gamma}}_i, \label{iter3h} \\ \end{align} (3.19) \begin{align}k^{n+1}_{i,h} &= \lambda \mathcal L_{i,h} \left(\left|{\boldsymbol u}^{n+1,\varepsilon}_{i,h} -{\boldsymbol u}^{n+1,\varepsilon}_{2,h}\right|^2\right) \mbox{on} {\it{\Gamma}} \quad \mbox{and} \label{iter3hbis}\end{align} (3.20) \begin{align}\label{iter4h}{\mathcal N_i}(k^{n}_{i,h};k^{n+1}_{i,h},\phi_{i,h}) &= \displaystyle \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \nabla {\boldsymbol u}^{n+1}_{i,h} : \nabla {\boldsymbol u}^{n+1,\varepsilon}_{i,h}\phi_{i,h} \, {\rm{d}}{\boldsymbol x}.\end{align} (3.21) This system is readily proved to admit a unique solution, similarly to system (4.1)–(4.2)–(4.3)–(4.5). We have considered here a fully coupled discretization of the friction boundary condition for the TKEs on the interface $${\it{\Gamma}}$$, as our purpose is to study a complete discretization of the fixed point algorithm introduced in Chacón Rebollo et al. (2010). It would also be possible to study the decoupled discretization introduced in Connors et al. (2009), which allows to separately solve the problems in the two subdomains. 4. Numerical analysis of the discrete scheme In this section, we prove the convergence of the solution of the algorithm (3.17)–(3.21) to a solution of the regularized problem (2.3). The proof is based on a recursive estimate of the errors between the finite element sequence $$\left({\boldsymbol u}_{i,h}^n,k_{i,h}^n \right)$$ and the continuous sequence $$\left({\boldsymbol u}_{i}^n,k_{i}^n \right)$$ solution of the iterative algorithm: For $$n \in \mathbb N$$ and for a given $$\left({\boldsymbol u}^{n}_i,p^{n}_i,k^{n}_i\right) \in \mathcal G_i$$, find $$\left({\boldsymbol u}^{n+1}_i,p^{n+1}_i,k^{n+1}_i\right) \in \mathcal G_i$$ by: 1. Find $$\left({\boldsymbol u}^{n+1}_i,p^{n+1}_i\right) \in {\boldsymbol X}_i \times L^2({\it{\Omega}}_i),$$ such that for all $$\left({\boldsymbol v}_i,q_i\right) \in {\boldsymbol X}_i \times L^2({\it{\Omega}}_i)$$, \begin{align}\label{iter1}&a_i\left(k^n_{i};{\boldsymbol u}_{i}^{n+1},\nabla {\boldsymbol v}_{i}\right) + b_i\left({\boldsymbol v}_{i},p_{i}^{n+1}\right) \notag\\ &\quad+ \kappa \displaystyle \int_{{\it{\Gamma}}}\left | {\boldsymbol u}_{i}^{n+1} -{\boldsymbol u}_{j}^{n+1}\right| \left ({\boldsymbol u}_{i}^{n+1} -{\boldsymbol u}_{j}^{n+1}\right)\, \cdot \,{\boldsymbol v}_{i} {\rm{d}}\tau =\displaystyle \int_{{\it{\Omega}}_i} {\boldsymbol f}_i \, \cdot \, {\boldsymbol v}_{i} \quad \forall{\boldsymbol v}_i \in {\boldsymbol X}_i.\end{align} (4.1) $$\label{iter2}\qquad \forall q_i \in L^2({\it{\Omega}}_i), \quad b_i({\boldsymbol u}_i^{n+1},q_i) =0.$$ (4.2) 2. Find $$k^{n+1}_i \in K_i$$ such that $$\forall\phi_i \in H^{1}_0({\it{\Omega}}_i)$$, \begin{align}\label{iter3}k^{n+1}_i & = 0 \mbox{on}{\it{\Gamma}}_i, \\ \end{align} (4.3) \begin{align}k^{n+1}_i & = \lambda \left|{\boldsymbol u}^{n+1,\varepsilon}_1 -{\boldsymbol u}^{n+1,\varepsilon}_2\right|^2 \mbox{on}{\it{\Gamma}},\quad \mbox{} \, \\ \end{align} (4.4) \begin{align}\label{iter4}{\mathcal N_i}(k^n_i;k^{n+1}_i,\phi_i) & = \displaystyle \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_i) \, \nabla {\boldsymbol u}^{n+1}_i : \nabla{\boldsymbol u}^{n+1,\varepsilon}_i \; \phi_i \, {\rm{d}}{\boldsymbol x}.\end{align} (4.5) In Chacón Rebollo et al. (2010), it is proved that problem (4.1)–(4.5), without regularization, admits a unique solution. The existence follows from Brouwer’s fixed-point theorem, based on energy stability estimates (see (4.9) and (4.10) below), and the uniqueness follows because the boundary term in (4.1) is dissipative. This proof readily extends to problems (4.1)–(4.5) and (3.17)–(3.21), and we shall omit it for brevity. In Chacón Rebollo et al. (2010), it is also proved that this iterative procedure is contracting for the nonregularized problem, when the eddy viscosities are large enough: Theorem 4.1 (Convergence of the continuous scheme) Assume that Hypothesis 2.1 holds and that $${\boldsymbol f}_i \in L^2\left({\it{\Omega}}_i\right)^d$$. Then, if $$\nu$$ is large enough then there exists a positive constant $$K < 1$$, depending only on $${\it{\Omega}}_i$$, $$M$$ and on the data $$\kappa$$, $$\nu$$ and $$\lambda$$, such that for all $$n \in \mathbb N^*$$, $$\label{contractivness}\begin{split}\displaystyle \sum _{i=1}^2 \left(\left|{\boldsymbol u}^{n+1}_i -{\boldsymbol u}^{n}_i\right|^2_{1,{\it{\Omega}}_i}+ \left|k^{n+1}_i -k^{n}_i\right|^2_{1,{\it{\Omega}}_i} \right) & \leq K \displaystyle \sum_{i=1}^2 \left|k^{n}_i -k^{n-1}_i\right|^2_{1,{\it{\Omega}}_i}. \end{split}$$ (4.6) In what follows, for simplicity of notation, we consider only the three-dimensional case $$d=3$$. The two-dimensional analysis is similar. We shall denote by $$c$$ a generic positive constant, which may vary from line to line, but are always independent of the $$n$$ and $$h$$ and $$\nu$$. Furthermore, for the sake of simplicity, we take $$\kappa_1 =\kappa_2 = \lambda = 1$$. Let us recall that according to the interpolation error estimates (3.8) and (3.10) we have \begin{align}\displaystyle \sum_{i=1}^2 \left| \mathcal{Q}_{i,h}\,{\boldsymbol u}^{n+1}_{i} - {\boldsymbol u}^{n+1}_{i} \right|_{1,{\it{\Omega}}_i}& \leq c \, h\, \displaystyle \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i}, \label{Est_I} \\ \end{align} (4.7) \begin{align}\displaystyle \sum_{i=1}^2 \left | \mathcal S_{i,h}\,(k ^{n+1}_{i}) - k^{n+1}_{i} \right|_{1,{\it{\Omega}}_i}&\leq c \, h\, \displaystyle \sum_{i=1}^2 | k^{n+1}_{i}|_{2,{\it{\Omega}}_i}. \label{Est_IK}\end{align} (4.8) To go further, we start by proving the boundedness in $$H^1$$ norm of the continuous and discrete sequences $$({\boldsymbol u}^n_{i}, k^n_{i})$$ and $$({\boldsymbol u}^n_{i,h}, k^n_{i,h})$$. To deal with the boundary condition modeling the generation of TKE at the interface, we use the result of continuity of the product of traces on $${\it{\Gamma}}$$ (see, for instance, Grisvard, 1985; Girault & Raviart, 1986). Lemma 4.2 Assume that $${\it{\Omega}}$$ is a bounded Lipschitz-continuous open subset of $$\mathbb R^d$$. Let $$s,\,s_1$$ and $$s_2$$ be three non-negative reals and $$p,\,p_1, \,p_2$$ be three real numbers in $$[1, +\infty)$$, such that $$s_1 \geq s$$, $$s_2 \geq s$$ and either $$s_1+s_2-s \geq d\left(\frac{1}{p_1} + \frac{1}{p_2} - \frac{1}{p}\right)\geq 0,\quad s_i-s >d\left(\frac{1}{p_i}-\frac{1}{p}\right)\quadi=1,2$$ or $$s_1+s_2-s > d\left(\frac{1}{p_1} + \frac{1}{p_2} - \frac{1}{p}\right)\geq 0,\quad s_i-s \geq d\left(\frac{1}{p_i}-\frac{1}{p}\right)\quadi=1,2.$$ Then, the mapping $$(u,\,v) \to uv$$ is a continuous bilinear map from $$W^{s_1,p_1}({\it{\Omega}}) \times W^{s_2,p_2}({\it{\Omega}})$$ to $$W^{s,p}({\it{\Omega}})$$, and there exists a constant $$C$$ depending on $$s_1,\, s_2,\, s,\,p_1,\,p_2$$ and $$p$$ such that for all $$f \in W^{s_1,p_1}({\it{\Omega}})$$, $$g \in W^{s_,p_2}({\it{\Omega}})$$ it holds $$\| f g\|_{W^{s,p}({\it{\Omega}})} \le C \, \|f\|_{W^{s_1,p_1}({\it{\Omega}})}\, \|g\|_{W^{s_,p_2}({\it{\Omega}})}.$$ Using the results of Hebey (1999), this lemma also holds for Sobolev spaces defined on compact Riemannian manifolds, this is the case of $${\it{\Gamma}}$$. We may now state the following result: Proposition 4.3 Assume that Hypothesis (2.1) holds and that $${\boldsymbol f}_i \in L^2({\it{\Omega}}_i)^d$$. Then, there exists a non-negative constant $$c$$ depending only on the domain $${\it{\Omega}}_i$$ and the coefficients $$\kappa$$, such that $$n \in \mathbb N$$: \begin{gather} \label{Velo_bound}\sum_{i=1}^2 \left | {\boldsymbol u}_{i}^{n} \right |_{1,{\it{\Omega}}_i}\leq \frac{c}{\nu} \left(\sum_{i=1}^2 \left \| {\boldsymbol f}_i \right \|_{0,{\it{\Omega}}_i}^2\right)^{\frac{1}{2}}, \qquad \sum_{i=1}^2 \left | {\boldsymbol u}_{i,h}^{n} \right |_{1,{\it{\Omega}}_i}\leq \frac{c}{\nu} \left(\sum_{i=1}^2 \left \| {\boldsymbol f}_i \right \|_{0,{\it{\Omega}}_i}^2\right)^{\frac{1}{2}}.\\ \end{gather} (4.9) \begin{gather}\label{TKE_bound}\sum_{i=1}^2 \left | k_{i}^{n} \right |_{1,{\it{\Omega}}_i}\leq \frac{c\, C_\varepsilon}{\nu^2} \left(\sum_{i=1}^2 \left \| {\boldsymbol f}_i \right \|_{0,{\it{\Omega}}_i}^2\right)\!, \qquad \sum_{i=1}^2 \left | k_{i,h}^{n} \right |_{1,{\it{\Omega}}_i}\leq \frac{c\, C_\varepsilon}{\nu^2} \left(\sum_{i=1}^2 \left \| {\boldsymbol f}_i \right \|_{0,{\it{\Omega}}_i}^2\right)\!. \end{gather} (4.10) Proof. To obtain estimates (4.9), let us respectively take $$({\boldsymbol u}_{i}^{n}, p_i^n)$$ and $$({\boldsymbol u}_{i,h}^{n},p_{i,h}^{n})$$ as test functions in equations (2.6)–(2.7) and (3.17)–(3.18). Summing up for $$i=1,2$$ it follows $$\sum_{i=1}^2 a_i(k_i;{\boldsymbol u}_{i}^{n},{\boldsymbol u}_{i}^{n}) + \kappa \int_ {{\it{\Gamma}}} |{\boldsymbol u}_1^n-{\boldsymbol u}_2^n|^3 \, {\rm{d}}\tau = \sum_{i=1}^2\int_{{\it{\Omega}}_i}{\boldsymbol f}_i\cdot {\boldsymbol u}_i^n \, {\rm{d}} {\boldsymbol x},$$ and the first estimate in (4.9) follows. The second one is similar. The estimates (4.10) are more involved because of the nonlinear boundary equation on $${\it{\Gamma}}$$. Let us consider the lifting operators (see Bernardi et al., 2004b, for details) $$\mathcal R_{i}: H^{\frac{1}{2}}_{00}({\it{\Gamma}}) \mapsto H^1({\it{\Omega}}_i)$$, $$i=1,2$$ such that, $$\label{liftK}\mathcal R_{i}(\phi_{i}) = \phi_{i}, \; \mbox{on} \; {\it{\Gamma}}, \quad\mathcal R_{i}(\phi_{i}) = 0 \; \mbox{on} \; {\it{\Gamma}}_i,\quad\|\mathcal R_{i} (\phi_{i}) \|_{1,{\it{\Omega}}_i} \leq c \, \|\phi_{i} \|_{H^{\frac{1}{2}}_{00}({\it{\Gamma}})}.$$ (4.11) Note that by Proposition 4.3 (with $$s_1=s=1/2, \, s_2=1-1/5, \,p_1=p=2, \,p_2=5$$), and the properties of mollifiers (2.2), \begin{align}& \left\| \mathcal R_{i}(k^{n+1}_i) \right\|_{1,{\it{\Omega}}_i} \le c\,\left\|\, \left|{\boldsymbol u}_1^{n+1,\varepsilon}-{\boldsymbol u}_2^{n+1,\varepsilon}\right|^2\,\right\|_{H^{\frac{1}{2}}_{00}({\it{\Gamma}})} \nonumber\\ &\quad \le c\, \left\|{\boldsymbol u}_1^{n+1,\varepsilon}-{\boldsymbol u}_2^{n+1,\varepsilon}\,\right\|_{H^{\frac{1}{2}}_{00}({\it{\Gamma}})}\, \left\|{\boldsymbol u}_1^{n+1,\varepsilon}-{\boldsymbol u}_2^{n+1,\varepsilon}\,\right\|_{W^{1-\frac{1}{5},5}({\it{\Gamma}})}\nonumber\\ &\quad \le c\, \left (\left|{\boldsymbol u}_1^{n+1,\varepsilon}\right|_{1,{\it{\Omega}}_1}+ \left|{\boldsymbol u}_2^{n+1,\varepsilon}\right|_{1,{\it{\Omega}}_2}\right)\, \left (\left\|{\boldsymbol u}_1^{n+1,\varepsilon}\right\|_{W^{1,5}({\it{\Omega}}_1)}+\left\|{\boldsymbol u}_2^{n+1,\varepsilon}\right\|_{W^{1,5}({\it{\Omega}}_2)}\right)\nonumber\\ &\quad \le c\, C_\varepsilon \left(\left|{\boldsymbol u}_1^{n+1}\right|_{1,{\it{\Omega}}_1}^2+ \left|{\boldsymbol u}_2^{n+1}\right|_{1,{\it{\Omega}}_2}^2\right)\! \label{estri12}.\end{align} (4.12) Next, taking $$\phi_i= k^{n+1}_i - \mathcal R_i(k^{n+1}_i) \in H^1_0({\it{\Omega}}_i)$$ in (4.5), using Holder inequality, (4.11) and (4.12) leads to \begin{align}{\mathcal N_i}(k^n_i;k^{n+1}_i,k^{n+1}_i) &= \displaystyle \int_{{\it{\Omega}}_i}\alpha_i\left(k^{n,\varepsilon}_i\right) \, \nabla {\boldsymbol u}^{n+1}_i : \nabla{\boldsymbol u}^{n+1,\varepsilon}_i \phi_i \, {\rm{d}}{\boldsymbol x}- {\mathcal N_i}\left(k^n_i;k^{n+1}_i,\mathcal R_i\left(k^{n+1}_i\right)\right)\nonumber \\ &\leq c \, \biggl(\left|{\boldsymbol u}^{n+1}_i\right|_{1,{\it{\Omega}}_i} \, \,\left \|\nabla {\boldsymbol u}^{n+1,\varepsilon}_i\right\|_{L^\infty({\it{\Omega}}_i)} \, \left(\left|k^{n+1}_i\right|_{1,{\it{\Omega}}_i}+\left|\mathcal R_{i}\left(k^{n+1}_i\right)\right|_{1,{\it{\Omega}}_i} \right) \nonumber \\ &\quad\qquad+ \left|k^{n+1}_i\right|_{1,{\it{\Omega}}_i} \left|\mathcal R_i\left(k^{n+1}_i\right)\right|_{1,{\it{\Omega}}_i}\biggl)\nonumber \\ & \leq c \, C_\varepsilon \biggl(|{\boldsymbol u}^{n+1}_1|_{1,{\it{\Omega}}_1}^2 + |{\boldsymbol u}^{n+1}_2|_{1,{\it{\Omega}}_2}^2 \biggl)\,|k^{n+1}_i|_{1,{\it{\Omega}}_i}\label{procd}.\end{align} (4.13) Then, $$\displaystyle |k^{n+1}_i|_{1,{\it{\Omega}}_i} \le \frac{c \, C_\varepsilon}{\nu}\, \biggl(|{\boldsymbol u}^{n+1}_1|_{1,{\it{\Omega}}_1}^2 + |{\boldsymbol u}^{n+1}_2|_{1,{\it{\Omega}}_2}^2 \biggl)$$ and by Hypothesis 2.1, the first estimate in (4.10) follows. To obtain the second estimate, observe that \begin{align}\left \| \mathcal R_{i,h}(k^{n+1}_{i,h})\right\|_{1,{\it{\Omega}}_i}& \le c\,\left\|\,\mathcal L_{i,h} \left (\left|{\boldsymbol u}_{1,h}^{n+1,\varepsilon}-{\boldsymbol u}_{2,h}^{n+1,\varepsilon}\right|^2\right)\,\right\|_{H^{\frac{1}{2}}_{00}({\it{\Gamma}})}\le c\,\left\|\left|{\boldsymbol u}_{1,h}^{n+1,\varepsilon}-{\boldsymbol u}_{2,h}^{n+1,\varepsilon}\right|^2\,\right\|_{H^{\frac{1}{2}}_{00}({\it{\Gamma}})} \nonumber\\ & \le c\, C_\varepsilon \left (\left|{\boldsymbol u}_{1,h}^{n+1}\right|_{1,{\it{\Omega}}_1}^2+ \left|{\boldsymbol u}_{2,h}^{n+1}\right|_{1,{\it{\Omega}}_2}^2\right)\!, \nonumber \end{align} where the last inequality is obtained proceeding as in (4.12), Now, taking $$\varphi_{i,h}=k_{i,h}^{n+1}- \mathcal R_{i,h}(k_{i,h}^{n+1}) \in K_{i,h}^0$$ and proceeding as to obtain (4.13) leads to the second estimate in (4.10).   □ 4.1 Analysis of the velocity sequence $${\boldsymbol u}^n_{i,h}$$ To prove the convergence of the discrete scheme, the idea is to estimate the difference between the continuous and discrete sequences, $$\left | {\boldsymbol u}^{n}_{i,h} - {\boldsymbol u}^{n}_i \right |_{1,{\it{\Omega}}_i}$$ and $$\left | k^{n}_{i,h} - k^{n}_i \right |_{1,{\it{\Omega}}_i}$$. We prove at first that the error in velocity is driven by that in TKE (plus interpolation errors): Theorem 4.4 Assume that Hypotheses (2.1) and (3.1) hold, and assume that the solution $$({\boldsymbol u}^{n+1}_{i}, p^{n+1}_{i})$$ belongs to $${\boldsymbol H}^{2}({\it{\Omega}}_i)\times H^{1}({\it{\Omega}}_i)$$. Then, the following estimate holds \begin{align} \label{Est_U_K}\displaystyle \sum_{i=1}^2 \left| {\boldsymbol u}^{n+1}_{i,h} - {\boldsymbol u}^{n+1}_{i}\right|^2_{1,{\it{\Omega}}_i}&\leq \displaystylec\, \sum_{i=1}^2 \biggl (\left(1+\frac{1}{\nu}\right)\, h^{2} \, | {\boldsymbol u}^{n+1}_{i}|^2_{2,{\it{\Omega}}_i}+ \frac{1}{\nu}\, h\, | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i} \nonumber \\ &\quad+ \left(1+\frac{1}{\nu}\right)\, h^{2} \, | p^{n+1}_{i}|^2_{1,{\it{\Omega}}_i} + \; \displaystyle \frac{C^2_\varepsilon}{\nu^4} \, | k^{n}_{i,h} - k^{n}_{i}|^2_{1,{\it{\Omega}}_i} \biggl).\end{align} (4.14) Proof. Let us consider the Stokes projection $${\it{\Pi}}_{i,h}:X_i \mapsto X_{i,h}$$ defined, for $${\boldsymbol z}_i \in X_i$$, as the solution—together with some discrete pressure $$r_{i,h} \in M_{i,h}$$—of the problem $$\label{presdrop} \left \{ \begin{array}{c}\displaystyle\int_{{\it{\Omega}}_i} \nabla ({\it{\Pi}}_{i,h} {\boldsymbol z}_i) \cdot \nabla {\boldsymbol v}_{i,h} +\int_{{\it{\Omega}}_i} r_{i,h}\nabla \cdot {\boldsymbol v}_{i,h} =\displaystyle \int_{{\it{\Omega}}_i} \nabla {\boldsymbol z}_i \cdot \nabla {\boldsymbol v}_{i,h} +\int_{{\it{\Omega}}_i} p_{i}\nabla \cdot {\boldsymbol v}_{i,h},\\ \displaystyle \int_{{\it{\Omega}}_i} \nabla \cdot ({\it{\Pi}}_{i,h} {\boldsymbol z}_i) \, q_{i,h}= \displaystyle\int_{{\it{\Omega}}_i} \nabla \cdot {\boldsymbol z}_i \, q_{i,h}, \end{array}\right.$$ (4.15)$$\forall{\boldsymbol v}_{i,h}\in X_{i,h}$$, $$\forall q_{i,h}\in M_{i,h}$$. This problem admits a unique solution, thanks to the discrete inf–sup condition (3.11). Moreover, it follows (see Girault & Raviart, 1986) \begin{align}& |{\it{\Pi}}_{i,h} {\boldsymbol z}_i-\mathcal Q_{i,h}{\boldsymbol z}_i|_{1,{\it{\Omega}}_i}+\|r_{i,h}-\mathcal P_{i,h} p_i\|_{L^2({\it{\Omega}}_i)}\notag \\ \label{estproy}&\quad\le C_i\, (| {\boldsymbol z}_i-\mathcal Q_{i,h}{\boldsymbol z}_i|_{1,{\it{\Omega}}_i}+\|p_i-\mathcal P_{i,h} p_i\|_{L^2({\it{\Omega}}_i)}),\end{align} (4.16) for some constant $$C_i>0$$, $$i=1,2$$. We multiply (4.1) and (3.17) by $${\boldsymbol v}_{i,h} = {\boldsymbol e}_{i,h}^{n+1}= {\boldsymbol u}_{i,h}^{n+1}- {\it{\Pi}}_{i,h} {\boldsymbol u}_i^{n+1}$$, and we compute the difference between the obtained equations. Summing upon $$i=1,2$$, thanks to (4.15) the pressure terms cancel, yielding \begin{align*}& \underbrace{\sum_{i=1}^2 \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \nabla \left({\boldsymbol u}^{n+1}_{i,h} - {\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i} \right): \nabla {\boldsymbol e}_{i,h}^{n+1}}_{I_1} \\ & \quad+ \underbrace{\sum_{i=1}^2 \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \nabla \left({\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i} - {\boldsymbol u}^{n+1}_{i} \right): \nabla {\boldsymbol e}_{i,h}^{n+1}}_{I_2}\\ &\quad+\underbrace{\sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left(\alpha_i(k^{n,\varepsilon}_{i,h}) - \alpha_i(k^{n,\varepsilon}_{i}) \right) \nabla {\boldsymbol u}^{n+1}_{i} : \nabla {\boldsymbol e}_{i,h}^{n+1}}_{I_3}\\ &\quad+ \int_{\it{\Gamma}} \left[\left | {\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right| \, \left ({\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right) - \left | {\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right| \, \left ({\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right) \right] \\ &\quad \times\underbrace{ \cdot \left[\left ({\boldsymbol u}^{n+1}_{1} - {\it{\Pi}}_{1,h}\,{\boldsymbol u}^{n+1}_{1} \right) - \left ({\boldsymbol u}^{n+1}_{2} - {\it{\Pi}}_{2,h} \,{\boldsymbol u}^{n+1}_{2} \right) \right]}_{I_4} \\ & \quad + \int_{\it{\Gamma}} \left[\left | {\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right| \, \left ({\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right) - \left | {\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right| \, \left ({\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right) \right]\\ &\quad \times\underbrace{ \cdot \left[\left ({\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right) - \left ({\boldsymbol u}^{n+1}_{1} -{\boldsymbol u}^{n+1}_{2} \right) \right]}_{I_5} \\ &= 0 \, \left(I_1 +I_2 +I_3 +I_4 +I_5 \right)\!. \end{align*} Thanks to the convexity inequality $$\left (|{\boldsymbol b}| {\boldsymbol b} -|{\boldsymbol a}|{\boldsymbol a} \right) \cdot \left({\boldsymbol b} -{\boldsymbol a}\right) \geq 0 \, \forall \left({\boldsymbol a},{\boldsymbol b}\right) \in \mathbb R^d$$, we deduce that $$I_5 \geq 0$$. Consequently, $$\label{I1}|I_1| \leq |I_2| + |I_3| + |I_4|.$$ (4.17) Estimation of $$I_1$$: It comes from Hypothesis (2.1) that $$\label{Est_I1}|I_1| =\; \displaystyle \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \nabla \left({\boldsymbol u}^{n+1}_{i,h} - {\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i} \right) : \nabla {\boldsymbol e}_{i,h}^{n+1}\; \geq \; \displaystyle \nu\sum_{i=1}^2 |{\boldsymbol e}_{i,h}^{n+1} |^2_{1,{\it{\Omega}}_i}.$$ (4.18) Estimation of $$I_2$$: Using Young’s inequality, estimates (4.7) and (4.16), we deduce \begin{align}I_2 = & \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \nabla \left({\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i} - {\boldsymbol u}^{n+1}_{i} \right): \nabla {\boldsymbol e}_{i,h}^{n+1} \notag \\ & \leq c \sum_{i=1}^2 \left |{\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i} - {\boldsymbol u}^{n+1}_{i} \right |_{1,{\it{\Omega}}_i} \; \left |{\boldsymbol e}_{i,h}^{n+1} \right|_{1,{\it{\Omega}}_i}\notag \\ & \leq \displaystyle\frac{c}{\nu} \sum_{i=1}^2 \left |{\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i} - {\boldsymbol u}^{n+1}_{i} \right |^2_{1,{\it{\Omega}}_i} \; + \; \frac{\nu}{4} \sum_{i=1}^2 \left |{\boldsymbol e}_{i,h}^{n+1} \right|^2_{1,{\it{\Omega}}_i} \notag \\ & \leq \frac{c}{\nu}\, h^{2} \, \sum_{i=1}^2 \, |{\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \frac{c}{\nu}\, h^{2} \, \sum_{i=1}^2 \, |p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}\; + \; \frac{\nu}{4} \sum_{i=1}^2 \left |{\boldsymbol e}_{i,h}^{n+1} \right|^2_{1,{\it{\Omega}}_i}\!.\label{Est_I2}\end{align} (4.19) Estimation of $$I_3$$: Using successively the mean value theorem, Hölder inequality, Hypothesis 2.1 and relations (2.2) and (4.9), we obtain \begin{align}I_3 &=\sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left(\alpha_i(k^{n,\varepsilon}_{i,h}) - \alpha_i(k^{n,\varepsilon}_{i}) \right) \nabla {\boldsymbol u}^{n+1}_{i} : \nabla {\boldsymbol e}_{i,h}^{n+1} \notag \\ & \leq c \sum_{i=1}^2 \| k^{n,\varepsilon}_{i,h} - k^{n,\varepsilon}_{i}\|_{L^\infty({\it{\Omega}}_i)} \; |\nabla {\boldsymbol u}^{n+1}_{i}|_{1,{\it{\Omega}}_i} \; \left| {\boldsymbol e}^{n+1}_{i,h}\right|_{1,{\it{\Omega}}_i} \notag \\ & \leq c\, C_\varepsilon \sum_{i=1}^2 | k^{n}_{i,h} - k^{n}_{i}|_{1,{\it{\Omega}}_i} \; |\nabla {\boldsymbol u}^{n+1}_{i}|_{1,{\it{\Omega}}_i} \; \left| {\boldsymbol e}^{n+1}_{i,h}\right|_{1,{\it{\Omega}}_i} \notag \\ & \leq \frac{c \, C^2_\varepsilon}{\nu^3} \; \sum_{i=1}^2 | k^{n}_{i,h} - k^{n}_{i}|^2_{1,{\it{\Omega}}_i} + \frac{\nu}{4} \sum_{i=1}^2 \left| {\boldsymbol e}^{n+1}_{i,h}\right|^2_{1,{\it{\Omega}}_i}\!.\label{Est_I3}\end{align} (4.20) Estimation of $$I_4$$: Using Hölder inequality, the continuity of the injection from $${\boldsymbol H}^{1/2}({\it{\Gamma}})$$ into $$L^3({\it{\Gamma}})^d$$ and the trace operator from $${\boldsymbol H}^{1}({\it{\Omega}}_i)$$ into $${\boldsymbol H}^{1/2}({\it{\Gamma}})$$, it holds \begin{align}I_4 &= \int_{\it{\Gamma}} \left({\left | {\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right|}\, \left ({\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right) - \left | {\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right| \, {\left ({\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right)} \right) \notag \\ & \quad\cdot \left[\left ({\boldsymbol u}^{n+1}_{1} - {\it{\Pi}}_{1,h}\,{\boldsymbol u}^{n+1}_{1} \right) - \left ({\boldsymbol u}^{n+1}_{2} - {\it{\Pi}}_{2,h} \,{\boldsymbol u}^{n+1}_{2} \right) \right] \notag\\ & \leq \left ( \left \| {\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right\|^2_{L^3({\it{\Gamma}})}+ \left \| {\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right \|^2_{L^3({\it{\Gamma}})}\right) \notag \\ & \quad\times \left ( \left \| {\boldsymbol u}^{n+1}_{1} - {\it{\Pi}}_{1,h}\,{\boldsymbol u}^{n+1}_{1} \right\|_{L^3({\it{\Gamma}})} + \left \| {\boldsymbol u}^{n+1}_{2} - {\it{\Pi}}_{2,h} \,{\boldsymbol u}^{n+1}_{2} \right\|_{L^3({\it{\Gamma}})}\right) \notag \\ & \leq c \left ( \left \| {\boldsymbol u}^{n+1}_{1,h} - {\boldsymbol u}^{n+1}_{2,h} \right\|^2_{{\boldsymbol H}^{1/2}({\it{\Gamma}})}+ \left \| {\boldsymbol u}^{n+1}_{1} - {\boldsymbol u}^{n+1}_{2} \right \|^2_{{\boldsymbol H}^{1/2}({\it{\Gamma}})}\right) \notag \\ & \quad\times \left ( \left \| {\boldsymbol u}^{n+1}_{1} - {\it{\Pi}}_{1,h}\,{\boldsymbol u}^{n+1}_{1} \right\|_{{\boldsymbol H}^{1/2}({\it{\Gamma}})}+ \left \| {\boldsymbol u}^{n+1}_{2} - {\it{\Pi}}_{2,h} \,{\boldsymbol u}^{n+1}_{2} \right\|_{{\boldsymbol H}^{1/2}({\it{\Gamma}})}\right) \notag \\ & \leq c \left(| {\boldsymbol u}^{n+1}_{1,h}|^2_{1,{\it{\Omega}}_i} + | {\boldsymbol u}^{n+1}_{2,h}|^2_{1,{\it{\Omega}}_i} + | {\boldsymbol u}^{n+1}_{1}|^2_{1,{\it{\Omega}}_i} + | {\boldsymbol u}^{n+1}_{2}|^2_{1,{\it{\Omega}}_i}\right) \notag \\ & \quad\times \left(\left | {\boldsymbol u}^{n+1}_{1} - {\it{\Pi}}_{1,h}\,{\boldsymbol u}^{n+1}_{1} \right|_{1,{\it{\Omega}}_i} + \left | {\boldsymbol u}^{n+1}_{2} - {\it{\Pi}}_{1,h}\,{\boldsymbol u}^{n+1}_{2} \right|_{1,{\it{\Omega}}_i} \right)\!. \notag \end{align} Thanks to relations (4.7), (4.9) and (4.16), we obtain $$\label{Est_I4}I_4 \leq \; \displaystyle \frac{c}{\nu} \; h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |_{2,{\it{\Omega}}_i}+\frac{c}{\nu} \; h \; \sum_{i=1}^2 | p^{n+1}_{i} |_{1,{\it{\Omega}}_i}.$$ (4.21) Then, estimate (4.14) is obtained easily by combining estimates (4.17)–(4.21) with the interpolation error estimate (4.7). □ 4.2 Analysis of the TKE sequence The purpose of this section is to prove that the errors in TKE $$k^n_i - k^n_{i,h}$$ converge to 0 as $$h$$ tends to 0 and $$n$$ tends to $$\infty$$, for large enough eddy viscosities. To obtain an estimate for the error in the TKEs, the standard choice for the test function in (4.5) and in (3.21) would be $$\ell^{n+1}_{i,h} =k^{n+1}_{i,h} - \mathcal S_{i,h} (k^{n+1}_i)$$. However, in general, this function does not vanish on the boundary $$\partial {\it{\Omega}}_i$$. Hence, it is necessary to introduce the lifting $$\mathcal R_{i,h}$$ and to use the function test $$\phi_{i,h} = \ell^{n+1}_{i,h} - \mathcal R_{i,h}(\ell^{n+1}_{i,h})$$. This choice requires to estimate the norm of the correction term $$\mathcal R_{i,h}(\ell^{n+1}_{i,h})$$. This is done in the following result: Lemma 4.5 Assume that Hypotheses (2.1) and (3.1) hold, and assume that the solution $$({\boldsymbol u}^{n+1}_{i}, p^{n+1}_{i})$$ belongs to $${\boldsymbol H}^{2}({\it{\Omega}}_i)\times H^{1}({\it{\Omega}}_i)$$. Then, the following error estimate holds \begin{align}\label{Est_Rih_k}\left |\mathcal R_{i,h} \left(\ell_{i,h}^{n+1}\right)\right|^2_{1,{\it{\Omega}}_i} &\leq \displaystyle \frac{c\,C^2_\varepsilon}{\nu^3}\left(\displaystyle \left(\nu +1\right)\, h^{2} \sum_{i=1}^2 \left| {\boldsymbol u}^{n+1}_{i} \right|^2_{2,{\it{\Omega}}_i} +h\sum_{i=1}^2 \left| {\boldsymbol u}^{n+1}_{i}\right|_{2,{\it{\Omega}}_i} \right.\notag\\ &\qquad\qquad \left.+ \, \displaystyle h\sum_{i=1}^2 \left| p^{n+1}_{i}\right|_{1,{\it{\Omega}}_i} + \displaystyle \frac{C^2_\varepsilon}{\nu^2} \; \sum_{i=1}^2 \left| k^{n}_{i,h} - k^{n}_{i}\right|^2_{1,{\it{\Omega}}_i}\right)\!.\end{align} (4.22) Proof. Since for all $$n$$, the sequences $$k^n_i$$ and $$k^n_{i,h}$$ are included in $$H^{1}({\it{\Omega}}_i)$$, their trace on $${\it{\Gamma}}$$ belong to $$H^{1/2}({\it{\Gamma}})$$. As both traces on $${\it{\Gamma}}_i$$ vanish, i.e., $$\restriction{k^n_i}{{\it{\Gamma}}_i} \,=\, \restriction{k^n_{i,h}}{{\it{\Gamma}}_i}\,=\,0$$, then, they belong to $$H^{1/2}_{00}({\it{\Gamma}})$$. Furthermore, by (3.16), there exists a positive constant $$c$$ such that: $$\label{Rih_Ell}\left|\mathcal R_{i,h}\left(\ell^{n+1}_{i,h}\right)\right|_{1,{\it{\Omega}}_i} \, \leq \, c\, \left\|\ell^{n+1}_{i,h}\right \|_{H^{1/2}_{00}({\it{\Gamma}})}\!.$$ (4.23) On the other hand, due to Hypothesis 3.1, the trace on $${\it{\Gamma}}$$ of the function $$\ell^{n+1}_{i,h}$$ can be rewritten as \begin{align*}\restriction{\ell^{n+1}_{i,h}}{{\it{\Gamma}}} &= \restriction{k^{n+1}_{i,h}}{{\it{\Gamma}}} - \restriction{\left(\mathcal S_{i,h} \left(k^{n+1}_i \right) \right)}{{\it{\Gamma}}} \\ &= \, \mathcal L_{i,h} \left(\left|{\boldsymbol u}^{n+1,\varepsilon}_{1,h} - {\boldsymbol u}^{n+1,\varepsilon}_{2,h}\right|^2 \right) \, - \, \mathcal L_{i,h} \left(\left|{\boldsymbol u}^{n+1,\varepsilon}_{1} - {\boldsymbol u}^{n+1,\varepsilon}_{2}\right|^2 \right) \\ &= \mathcal L_{i,h}\left(\left|{\boldsymbol u}^{n+1,\varepsilon}_{1,h} - {\boldsymbol u}^{n+1,\varepsilon}_{2,h}\right|^2 - \left|{\boldsymbol u}^{n+1,\varepsilon}_{1} - {\boldsymbol u}^{n+1,\varepsilon}_{2}\right|^2 \right)\!. \end{align*} Next, using (2.2), yielding \begin{align}{3}\|\ell_{i,h}^{n+1}\|_{H^{1/2}_{00}({\it{\Gamma}})} & \leq c \left\| \left|{\boldsymbol u}^{n+1,\varepsilon}_{1,h} - {\boldsymbol u}^{n+1,\varepsilon}_{2,h}\right|^2 - \left|{\boldsymbol u}^{n+1,\varepsilon}_{1} - {\boldsymbol u}^{n+1,\varepsilon}_{2}\right|^2 \right\|_{H^{1/2}_{00}({\it{\Gamma}})} \notag \\ & = c \, \left\|\left [\left({\boldsymbol u}^{n+1,\varepsilon}_{1,h} - {\boldsymbol u}^{n+1,\varepsilon}_{1}\right) -\left({\boldsymbol u}^{n+1,\varepsilon}_{2,h} - {\boldsymbol u}^{n+1,\varepsilon}_{2} \right)\right] \right. \notag \\ & \quad\times \left. \left[\left({\boldsymbol u}^{n+1,\varepsilon}_{1,h} +{\boldsymbol u}^{n+1,\varepsilon}_{1} \right) -\left({\boldsymbol u}^{n+1,\varepsilon}_{2,h} +{\boldsymbol u}^{n+1,\varepsilon}_{2} \right)\right]\right\|_{H^{1/2}_{00}({\it{\Gamma}})} \notag \\ & \leq \, c \, \sum_{i=1}^2 c_i \, \left \|{\boldsymbol u}^{n+1,\varepsilon}_{i,h} -{\boldsymbol u}^{n+1,\varepsilon}_{i}\right\|_{{L^\infty}({\it{\Gamma}})} \; \sum_{i=1}^2 \left \|{\boldsymbol u}^{n+1,\varepsilon}_{i,h} +{\boldsymbol u}^{n+1,\varepsilon}_{i}\right\|_{{L^\infty}({\it{\Gamma}})} \notag \\ & \leq \, c \, \sum_{i=1}^2 \left \|{\boldsymbol u}^{n+1,\varepsilon}_{i,h} -{\boldsymbol u}^{n+1,\varepsilon}_{i}\right\|_{{L^\infty}({\it{\Omega}}_i)} \; \sum_{i=1}^2 \left \|{\boldsymbol u}^{n+1,\varepsilon}_{i,h} +{\boldsymbol u}^{n+1,\varepsilon}_{i}\right\|_{{L^\infty}({\it{\Omega}}_i)}\!. \notag \end{align} According to (4.9) and Poincaré–Friedrichs inequality, we obtain $$\label{LU}\|\ell_{i,h}^{n+1}\|_{H^{1/2}_{00}({\it{\Gamma}})} \leq \, \displaystyle \frac{c \, C_\varepsilon}{\nu} \; \sum_{i=1}^2 \left |{\boldsymbol u}^{n+1}_{i,h} -{\boldsymbol u}^{n+1}_{i}\right|_{1,{\it{\Omega}}_i}\!.$$ (4.24) When adding and subtracting the quantity $${\it{\Pi}}_{i,h}\,{\boldsymbol u}^{n+1}_{i,h}$$, and using (4.23), we obtain \begin{equation*}\left |\mathcal R_{i,h} (\ell_{i,h}^{n+1})\right|^2_{1,{\it{\Omega}}_i} \, \leq \displaystyle \frac{c\, C^2_\varepsilon}{\nu^2} \; \left(\sum_{i=1}^2 \left| {\boldsymbol e}^{n+1}_{i,h} \right|^2_{1,{\it{\Omega}}_i} + \sum_{i=1}^2 \left | {\it{\Pi}}_{i,h} \, {\boldsymbol u}^{n+1}_{i} - {\boldsymbol u}^{n+1}_{i} \right|^2_{1,{\it{\Omega}}_i}\right)\!. \end{equation*} Finally, estimate (4.22) follows, thanks to (4.7), (4.14) and (4.16). □ We may now estimate the test function $$\ell_{i,h}^{n+1}$$: Theorem 4.6 Assume that Hypotheses (2.1) and (3.1) hold, and assume that the solution $$\left({\boldsymbol u}^{n}_{i},p^{n}_{i},k^{n}_{i}\right)$$ belongs to $${\boldsymbol H}^{2}({\it{\Omega}}_i) \times H^{1}({\it{\Omega}}_i) \times H^{2}({\it{\Omega}}_i)$$. Then, the following error estimate holds \begin{align} \nonumber \displaystyle \nu \, \sum_{i=1}^2 \left|\ell_{i,h}^{n+1}\right|^2_{1,{\it{\Omega}}_i}& \leq \; \displaystyle \frac{c \, C_\varepsilon^2}{\nu^6} \; \left(\nu \, C_\varepsilon^2 \, +\, \nu^4 \, + \, C_\varepsilon^2 \, + \, \frac{C_\varepsilon^4}{\nu^2}+ \, \frac{C_\varepsilon^4}{\nu^3}\right) \; \sum_{i=1}^2 \left|k_{i,h}^{n}-k_i^n\right|^2_{1,{\it{\Omega}}_i} \\ & \quad\displaystyle +c\, h^{2} \; \left[\frac{C_\varepsilon^2}{\nu^6} \; \left(\nu \, C_\varepsilon^2 \, +\, \nu^4 \, + \, C_\varepsilon^2 \, + \, \frac{C_\varepsilon^4}{\nu^2}\right) \; \sum_{i=1}^2 \left| k^{n}_{i}\right|^2_{2,{\it{\Omega}}_i}\, + \, \displaystyle \frac{\nu+1}{\nu} \; \sum_{i=1}^2 \left| k^{n+1}_{i}\right|^2_{2,{\it{\Omega}}_i}\right] \; \nonumber \\ &\quad +\, \displaystyle \displaystyle \frac{c\, C^2_\varepsilon}{\nu^5} \, \left(\nu \,+ \, C^2_\varepsilon +\frac{C^2_\varepsilon}{\nu}\right) \biggl[\, \displaystyle \left(\nu +1\right)\, h^{2} \; \sum_{i=1}^2 \left| {\boldsymbol u}^{n+1}_{i} \right|^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 \left| {\boldsymbol u}^{n+1}_{i}\right|_{2,{\it{\Omega}}_i} \biggl]\nonumber \\ &\quad +\, \displaystyle \displaystyle \frac{c\, C^2_\varepsilon}{\nu^5} \, \left(\nu \,+ \, C^2_\varepsilon +\frac{C^2_\varepsilon}{\nu}\right) \biggl[\, \displaystyle \left(\nu +1 \right)\, h^{2} \; \sum_{i=1}^2 \left| p^{n+1}_{i} \right|^2_{1,{\it{\Omega}}_i}\biggl].\label{Est_Ell_h}\end{align} (4.25) Proof. To estimate $$\ell_{i,h}^{n+1}= k_{i,h}^{n+1}-\mathcal S_{i,h} \left(k_{i}^{n+1}\right)$$, we use the difference between equations (4.5) and (3.21), taking as test function $$\phi_{i,h} = \ell_{i,h}^{n+1} \, -\, \mathcal R_{i,h} \left(\ell_{i,h}^{n+1}\right)$$: \begin{align}A &:= \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left[\gamma_i(k^{n,\varepsilon}_{i,h}) \, \nabla \,k^{n+1}_{i,h} \,- \, \gamma_i(k^{n,\varepsilon}_{i}) \, \nabla \,k^{n+1}_{i} \right] \, \cdot \, \nabla \,\phi_{i,h} \notag\\ & = B := \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left[\alpha_i(k^{n,\varepsilon}_{i,h}) \, \nabla {\boldsymbol u}^{n+1}_{i,h} \,: \, \nabla \, {\boldsymbol u}^{n+1,\varepsilon}_{i,h} \,- \, \alpha_i(k^{n,\varepsilon}_{i}) \, \nabla {\boldsymbol u}^{n+1}_{i} \,: \, \nabla {\boldsymbol u}^{n+1,\varepsilon}_{i}\right] \, \phi_{i,h}.\notag \end{align} Adding and subtracting $$\ell_{i,h}^{n+1}$$ in the first factor, then $$A$$ can be rewritten as \begin{align}A &= \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left[\gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \, \nabla \,k^{n+1}_{i,h} \,- \, \gamma_i\left(k^{n,\varepsilon}_{i}\right) \, \nabla \,k^{n+1}_{i} \right] \, \cdot \, \nabla\left (\ell_{i,h}^{n+1} \, -\, \mathcal R_{i,h} \left(\ell_{i,h}^{n+1}\right) \right) \notag \\ &= \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left(\gamma_i\left(k^{n,\varepsilon}_{i,h}\right)- \gamma_i\left(k^{n,\varepsilon}_{i}\right) \right) \nabla k^{n+1}_{i} \cdot \nabla \ell_{i,h}^{n+1} \notag \\ & \quad + \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \nabla\left (\mathcal S_{i,h}\,\left(k^{n+1}_{i}\right) - k^{n+1}_{i} \right) \cdot \nabla \ell_{i,h}^{n+1} \notag \\ &\quad + \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \nabla | \ell_{i,h}^{n+1}|^2 \notag \\ &\quad - \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\biggl( \left(\gamma_i\left(k^{n,\varepsilon}_{i,h}\right) - \gamma_i\left(k^{n,\varepsilon}_{i}\right)\right)\nabla \,k^{n+1}_{i} + \gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \nabla \ell_{i,h}^{n+1}\notag \\ & \qquad\qquad\qquad +\gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \nabla (\mathcal S_{i,h}\,\left(k^{n+1}_{i} - k^{n+1}_{i} \right) \biggr) \cdot \, \nabla \mathcal R_{i,h} \left(\ell_{i,h}^{n+1}\right)\!.\notag \end{align} The relation $$A= B$$ yields \begin{align}\nu \sum_{i=1}^2 |\ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i}&\leq |B| \notag \\ (A_1&:=) + \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left|\left(\gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \, -\, \gamma_i\left(k^{n,\varepsilon}_{i}\right) \right)\, \nabla \,k^{n+1}_{i} \, \cdot \, \nabla \, \ell_{i,h}^{n+1} \right| \notag \\ (A_2&:=) + \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left| \gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \,\nabla \, \left(\mathcal S_{i,h}\,\left(k^{n+1}_{i}\right) \,- \, k^{n+1}_{i} \right) \, \cdot \, \nabla \, \ell_{i,h}^{n+1} \right| \notag \\ (A_3 & :=) + \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left|\left(\gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \, -\, \gamma_i\left(k^{n,\varepsilon}_{i}\right) \right)\, \nabla \,k^{n+1}_{i} \,\cdot \nabla \, \mathcal R_{i,h}\left(\ell_{i,h}^{n+1}\right) \right| \notag \\ (A_4 & :=) +\sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left| \gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \, \nabla \, \ell_{i,h}^{n+1} \, \cdot \, \nabla \, \mathcal R_{i,h}\left(\ell_{i,h}^{n+1}\right) \right| \notag \\ (A_5 & :=) + \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left| \gamma_i\left(k^{n,\varepsilon}_{i,h}\right) \,\nabla \, \left(\mathcal S_{i,h}\,\left(k^{n+1}_{i}\right) \,- \, k^{n+1}_{i} \right) \, \cdot \, \nabla \, \mathcal R_{i,h} \left(\ell_{i,h}^{n+1}\right) \right|\!. \label{Ell_AB}\end{align} (4.26) The next step is to estimate $$(A_j)_{1 \leq j \leq 5}$$ and $$B$$. Estimation of $$A_1$$:   Using successively the mean value theorem, Hölder and Poincaré-Friedrichs inequalities and relations (2.2) and (4.10), there exists positive a constant $$c$$ depending only on $${\it{\Omega}}_i$$ such that \begin{align}A_1 & \leq c \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left|(k^{n,\varepsilon}_{i,h}- k^{n,\varepsilon}_{i}) \nabla \,k^{n+1}_{i} \cdot \nabla \ell_{i,h}^{n+1} \right| \notag \\ & \leq c \sum_{i=1}^2 \| k^{n,\varepsilon}_{i,h} - k^{n,\varepsilon}_{i}\|_{L^\infty({\it{\Omega}}_i)}\| \nabla \,k^{n+1}_{i}\|_{L^2({\it{\Omega}}_i)}| \ell_{i,h}^{n+1}|_{1,{\it{\Omega}}_i} \notag \\ & \leq \, \frac{c \, C_\varepsilon^3}{\nu^2}\; \sum_{i=1}^2 | k^n_{i,h} \, -\, k^n_{i}|_{1,{\it{\Omega}}_i} \; | \ell_{i,h}^{n+1}|_{1,{\it{\Omega}}_i}.\notag \end{align} To simplify the calculations, we introduce a positive number $$\beta$$ which we shall fix later. According to Young’s inequality, this yields for $$\beta>0$$, which we shall fix later \begin{align}A_1 \; \leq \; \frac{\nu}{\beta} \, \sum_{i=1}^2 | \ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i} \, + \, \beta \; \frac{c \, C_\varepsilon^6}{\nu^5}\; \sum_{i=1}^2 | k^n_{i,h} \, -\, k^n_{i}|^2_{1,{\it{\Omega}}_i}.\label{A1}\end{align} (4.27) Estimation of $$A_2$$:   Using (4.8), (4.10) and Young and Cauchy–Schwarz inequalities, \begin{align}A_2 &= \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left| \gamma_i(k^{n,\varepsilon}_{i,h}) \,\nabla \, \left(\mathcal S_{i,h}\,(k^{n+1}_{i}) \,- \, k^{n+1}_{i} \right) \, \cdot \, \nabla \, \ell_{i,h}^{n+1} \right| \notag \\ & \leq \, \frac{\nu}{\beta} \, \sum_{i=1}^2 | \ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i} \, + \, \frac{c\, \beta}{\nu} \, \sum_{i=1}^2 \left |\mathcal S_{i,h}\,(k^{n+1}_{i}) \,- \, k^{n+1}_{i} \right|^2_{1,{\it{\Omega}}_i} \notag \\ & \leq \, \frac{\nu}{\beta} \, \sum_{i=1}^2 | \ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i} \, + \, \beta \, \frac{c}{\nu} \, h^{2} \, \sum_{i=1}^2 | k^{n+1}_{i}|^2_{2,{\it{\Omega}}_i}. \label{A2}\end{align} (4.28) Estimation of $$A_3$$:   The same arguments used to estimate $$A_1$$ and due to (4.22), the following inequalities hold \begin{align}A_3 & \leq \, c \sum_{i=1}^2 \| k^{n,\varepsilon}_{i,h} \, -\, k^{n,\varepsilon}_{i}\|_{L^\infty({\it{\Omega}}_i)} \, | k^{n+1}_{i}|_{1,{\it{\Omega}}_i} \, | \mathcal R_{i,h} (\ell_{i,h}^{n+1})|_{1,{\it{\Omega}}_i} \notag \\ & \leq \, \frac{c \, C^2_\varepsilon}{\nu^2}\; \sum_{i=1}^2 | k^n_{i,h} \, -\, k^n_{i}|_{1,{\it{\Omega}}_i} \; | \mathcal R_{i,h} (\ell_{i,h}^{n+1})|_{1,{\it{\Omega}}_i} \notag \\ &\leq \, \frac{c \, C^2_\varepsilon}{\nu^2}\; \left(\sum_{i=1}^2 | k^n_{i,h} \, -\, k^n_{i}|^2_{1,{\it{\Omega}}_i} \; +\; | \mathcal R_{i,h} (\ell_{i,h}^{n+1})|^2_{1,{\it{\Omega}}_i} \right) \notag \\ & \leq \, \frac{c \, C^4_\varepsilon}{\nu^5}\; \left(\displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i}\right) \notag \\ &\quad + \frac{c \, C^4_\varepsilon}{\nu^5}\; \displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}+ \displaystyle \frac{c\, C^2_\varepsilon}{\nu^5} \; \left(\nu^3 \, + \, C^2_\varepsilon \right) \sum_{i=1}^2 | k^{n}_{i,h} - k^{n}_{i}|^2_{1,{\it{\Omega}}_i}.\label{A3}\end{align} (4.29) Estimation of $$A_4$$:   Using the same procedure, we deduce \begin{align}A_4 & \leq \, c \sum_{i=1}^2 |\ell_{i,h}^{n+1} |_{1,{\it{\Omega}}_i} \, |\mathcal R_{i,h} (\ell_{i,h}^{n+1})|_{1,{\it{\Omega}}_i} \notag \\ &\leq \, \frac{\nu}{\beta} \, \sum_{i=1}^2 | \ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i} \, + \, \frac{c\, \beta}{\nu} \, \sum_{i=1}^2 | \mathcal R_{i,h} (\ell_{i,h}^{n+1})|^2_{1,{\it{\Omega}}_i} \notag \\ & \leq \, \frac{\nu}{\beta} \, \sum_{i=1}^2 | \ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i} \, + \, \displaystyle \frac{c\, C^2_\varepsilon\, \beta}{\nu^4} \, \biggl(\, + \displaystyle \frac{C^2_\varepsilon}{\nu^2} \; \sum_{i=1}^2 | k^{n}_{i,h} - k^{n}_{i}|^2_{1,{\it{\Omega}}_i}\notag \\ &\quad\displaystyle + (\nu +1)\, h^{2} \; \sum_{i=1}^2 | p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}+ (\nu +1)\, h^{2} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i}\biggr).\label{A4}\end{align} (4.30) Estimation of $$A_5$$:   Using the same techniques as previously and the error estimates (4.8) for operator $$\mathcal S_{i,h}$$, we obtain \begin{align}A_5 & \leq \, c \sum_{i=1}^2 |\mathcal S_{i,h}\,(k^{n+1}_{i}) \,- \, k^{n+1}_{i}|_{1,{\it{\Omega}}_i} \, | \mathcal R_{i,h} (\ell_{i,h}^{n+1})|_{1,{\it{\Omega}}_i} \notag \\ &\leq \, \frac{\nu}{\beta} \, \sum_{i=1}^2 |\mathcal S_{i,h}\,(k^{n+1}_{i}) \,- \, k^{n+1}_{i}|^2_{1,{\it{\Omega}}_i} \,+\, \frac{c \beta}{\nu} \,\sum_{i=1}^2| \mathcal R_{i,h} (\ell_{i,h}^{n+1})|^2_{1,{\it{\Omega}}_i}\notag \\ &\leq \, \frac{\nu}{\beta} \,h^{2}\, \sum_{i=1}^2 | k^{n+1}_{i} |^2_{2,{\it{\Omega}}_i} \,+\, \displaystyle \frac{c\, C^2_\varepsilon\, \beta}{\nu^4} \, \biggl(\, \displaystyle \frac{C^2_\varepsilon}{\nu^2} \; \sum_{i=1}^2 | k^{n}_{i,h} - k^{n}_{i}|^2_{1,{\it{\Omega}}_i}\notag \\ &\quad \displaystyle + (\nu +1)\, h^{2} \; \sum_{i=1}^2 | p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}+(\nu +1)\, h^{2} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i}\biggr).\label{A5}\end{align} (4.31) Summing up estimates for the $$(A_j)_{1 \leq j \leq 5}$$ and choosing $$\beta=3$$, for instance, we can write \begin{align}\label{ell_Aj}\displaystyle \nu \, \sum_{i=1}^2 |\ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i}& \leq |B| \;+\; c\, \displaystyle \frac{\nu+1}{\nu} \, h^{2}\, \; \sum_{i=1}^2 | k^{n+1}_{i}|^2_{2,{\it{\Omega}}_i} \notag\\ & \quad+ \displaystyle \frac{c \, C_\varepsilon^2}{\nu^6} \; \left(\nu \, C_\varepsilon^2 \, +\, \nu^4 \, + \, C_\varepsilon^2 \right) \; \sum_{i=1}^2 | k^n_{i,h} \, -\, k^n_{i}|^2_{1,{\it{\Omega}}_i} \notag\\ & \quad+ \displaystyle \displaystyle \frac{c\, C^2_\varepsilon}{\nu^5} \, \left(\nu \,+ \, C^2_\varepsilon \right) \left(\, \displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i} \right)\notag\\ & \quad+ \displaystyle \displaystyle \frac{c\, C^2_\varepsilon}{\nu^5} \, \left(\nu \,+ \, C^2_\varepsilon \right) \, \displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}.\end{align} (4.32) Estimation of $$B$$: Let us write the term $$B$$ as \begin{align}B & = \sum_{i=1}^2 \int_{{\it{\Omega}}_i} \left[\alpha_i(k^{n,\varepsilon}_{i,h}) \, \nabla {\boldsymbol u}^{n+1}_{i,h} \,: \, \nabla \, {\boldsymbol u}^{n+1,\varepsilon}_{i,h} \,- \, \alpha_i(k^{n,\varepsilon}_{i}) \, \nabla {\boldsymbol u}^{n+1}_{i} \,: \, \nabla {\boldsymbol u}^{n+1,\varepsilon}_{i}\right] \, \phi_{i,h}\notag \\ & = \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \, \nabla({\boldsymbol u}^{n+1}_{i,h} - {\boldsymbol u}^{n+1}_{i}) \,:\, \nabla {\boldsymbol u}^{n+1,\varepsilon}_{i,h} \; \phi_{i,h} \notag \\ & \quad + \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\alpha_i(k^{n,\varepsilon}_{i,h}) \, \nabla \, {\boldsymbol u}^{n+1}_{i} \,:\, \nabla ({\boldsymbol u}^{n+1,\varepsilon}_{i,h} \, - \, {\boldsymbol u}^{n+1,\varepsilon}_{i}) \; \phi_{i,h} \notag \\ & \quad+ \sum_{i=1}^2 \int_{{\it{\Omega}}_i}\left(\alpha_i(k^{n,\varepsilon}_{i,h}) \, - \, \alpha_i(k^{n,\varepsilon}_{i}) \right) \, \, \nabla \, {\boldsymbol u}^{n+1}_{i} \,:\, \nabla {\boldsymbol u}^{n+1,\varepsilon}_{i} \; \phi_{i,h} \notag \\ & = \; J_1 + J_2 +J_3. \notag \end{align} The terms $$J_1, J_2$$ and $$J_3$$ can be bounded using $$\phi_{i,h} = \ell_{i,h}^{n+1} - \mathcal R_{i,h} (\ell_{i,h}^{n+1})$$, Young inequality, (2.2), (4.9) and (4.10), as follows: \begin{align}|J_1| + |J_2| & \leq \; \displaystyle \frac{c \, C^2_\varepsilon}{\nu^2} \; \sum_{i=1}^2 |{\boldsymbol u}^{n+1}_{i,h} - {\boldsymbol u}^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}\,+\, |\mathcal R_{i,h} (\ell_{i,h}^{n+1})|^2_{1,{\it{\Omega}}_i}\,+\, \frac{\nu}{4} |\ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i},\notag \\ |J_3| & \leq \; \displaystyle \frac{c \, C^6_\varepsilon}{\nu^8} \; \sum_{i=1}^2 |k^{n}_{i,h} - k^{n}_{i} |^2_{1,{\it{\Omega}}_i}\,+\, |\mathcal R_{i,h} (\ell_{i,h}^{n+1})|^2_{1,{\it{\Omega}}_i}\,+\, \frac{\nu}{4} |\ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i}.\notag \end{align} Then, the following inequality holds, thanks to (4.22) \begin{align}\label{B}|B| &\leq \displaystyle \frac{\nu}{2} |\ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i}\notag\\ &\quad +\, \displaystyle\left(\displaystyle \frac{c \, C^6_\varepsilon}{\nu^8} \,+\, \displaystyle \frac{c \, C^4_\varepsilon}{\nu^5} \right)\; \sum_{i=1}^2 |k^{n}_{i,h} - k^{n}_{i} |^2_{1,{\it{\Omega}}_i}+ \; \displaystyle \frac{C^4_\varepsilon}{\nu^5} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i,h} - {\boldsymbol u}^{n+1}_{i}|^2_{1,{\it{\Omega}}_i} \notag \\ &\quad +\, \displaystyle \frac{c\, C^2_\varepsilon}{\nu^3} \, \left(\, \displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i} \right) \notag\\ & \quad+\, \displaystyle \frac{c\, C^2_\varepsilon}{\nu^3} \, (\nu +1)\, h^{2} \; \sum_{i=1}^2 | p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}.\end{align} (4.33) Combining (4.33) with (4.32) leads to \begin{equation*}\begin{array}{rcl}\displaystyle \nu \, \sum_{i=1}^2 |\ell_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i}&\leq& \displaystyle \frac{c \, C_\varepsilon^2}{\nu^6}\left(\nu \, C_\varepsilon^2 \, +\, \nu^4 \, + \, C_\varepsilon^2 \, + \, \frac{C_\varepsilon^4}{\nu^2}\right) \sum_{i=1}^2 | k^n_{i,h} \, -\, k^n_{i}|^2_{1,{\it{\Omega}}_i}\, +\, \displaystyle \frac{C^4_\varepsilon}{\nu^5} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i,h} - {\boldsymbol u}^{n+1}_{i}|^2_{1,{\it{\Omega}}_i} \\ && +\, \displaystyle \displaystyle \frac{c\, C^2_\varepsilon}{\nu^5} \, \left(\nu \,+ \, C^2_\varepsilon \right) \left(\, \displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i} |^2_{2,{\it{\Omega}}_i}\; + \; \displaystyle h \; \sum_{i=1}^2 | {\boldsymbol u}^{n+1}_{i}|_{2,{\it{\Omega}}_i} \right) \\ && \displaystyle+ \frac{c\, C^2_\varepsilon}{\nu^5} \, \left(\nu \,+ \, C^2_\varepsilon \right) \, \displaystyle (\nu +1)\, h^{2} \; \sum_{i=1}^2 | p^{n+1}_{i} |^2_{1,{\it{\Omega}}_i}+ c\, \displaystyle \frac{\nu+1}{\nu} \, h^{2}\, \; \sum_{i=1}^2 | k^{n+1}_{i}|^2_{2,{\it{\Omega}}_i}. \end{array}\end{equation*} Finally, using (4.14), estimate (4.25) follows.   □ We may now state the following: Theorem 4.7 Assume that Hypotheses 2.1 and 3.1 hold, and assume that the solution $$\left({\boldsymbol u}^{n}_{i},p^{n}_{i},k^{n}_{i}\right)$$ belongs to $${\boldsymbol H}^{2}({\it{\Omega}}_i) \times H^{1}({\it{\Omega}}_i) \times H^{2}({\it{\Omega}}_i)$$. Then, there exists a $$\nu_0>0$$ such that id $$0<\nu_0 <\nu$$, it holds for all $$n=1,2,\cdots$$ \begin{align} \label{Est_Cont_k}&\displaystyle \sum_{i=1}^2 \left | k^{n}_i - k^{n}_{i,h} \right|^2_{1,{\it{\Omega}}_i}\leq \; \displaystyle A_{\varepsilon,\nu}^{n}\, \sum_{i=1}^2 \left | k^{0}_i - k^{0}_{i,h} \right|^2_{1,{\it{\Omega}}_i} +h\, B_{\varepsilon,\nu},\end{align} (4.34) for some constants $$A_{\varepsilon,\nu}\, \in (0,1)$$, $$B_{\varepsilon,\nu}\, >0$$. Proof. By estimates (3.10) and (4.25), \begin{align}\sum_{i=1}^2 \left|k_{i}^{n+1} - k_{i,h}^{n+1} \right|^2_ {1,{\it{\Omega}}_i}\notag & \leq \displaystyle 2 \sum_{i=1}^2 \biggl (\left|k_{i}^{n+1} \, - \, \mathcal S_{i,h}(k_{i}^{n+1}) \right|^2_ {1,{\it{\Omega}}_i} \, + \, \left| \mathcal S_{i,h}(k_{i}^{n+1}) \,-\, k_{i,h}^{n+1} \right|^2_ {1,{\it{\Omega}}_i} \biggl) \notag \\ & \leq \displaystyle 2 \sum_{i=1}^2 |\ell_{i,h}^{n+1} |^2_ {1,{\it{\Omega}}_i} \,+\, c\, h^{2} \; \sum_{i=1}^2 |k_{i}^{n+1}|^2_{1,{\it{\Omega}}_i} \notag \\ & \leq \; \displaystyle A_{\varepsilon,\nu}\, \sum_{i=1}^2 \left | k^{n}_i - k^{n}_{i,h} \right|^2_{1,{\it{\Omega}}_i} +h\,D_{\varepsilon,\nu}\,,\mbox{with}\notag \end{align} \begin{gather*}\displaystyle A_{\varepsilon,\nu}=\frac{c \, C_\varepsilon^2}{\nu^7} \; \left(\nu \, C_\varepsilon^2 \, +\, \nu^4 \, + \, C_\varepsilon^2 \, + \, \frac{C_\varepsilon^4}{\nu^2}+ \, \frac{C_\varepsilon^4}{\nu^3}\right)\!,\\ \displaystyle D_{\varepsilon,\nu}= c\, \max\biggl\{\frac{C_\varepsilon^2}{\nu^6} \; \left(\nu \, C_\varepsilon^2 \, +\frac{C_\varepsilon^2}{\nu}\right)\!, \left (1+\frac{1}{\nu} \right) \biggl\}. \end{gather*} For large enough $$\nu$$, $$A_{\varepsilon,\nu}^n<1$$. Let $$\displaystyle\sigma_n=\sum_{i=1}^2 |k_{i}^{n+1} - k_{i,h}^{n+1}|^2_{1,{\it{\Omega}}_i}$$. Arguing recursively, we deduce $$\sigma_{n} \le A_{\varepsilon,\nu}^n \, \sigma_{0} + h\, D_{\varepsilon,\nu}\, \left (\sum_{k=0}^{n-1} A_{\varepsilon,\nu}^k \right) \le A_{\varepsilon,\nu}^n\, \sigma_{0} + h\, D_{\varepsilon,\nu}\, \frac{A_{\varepsilon,\nu}}{1- A_{\varepsilon,\nu}}.$$ This yields (4.34). □ From this result, it follows: Corollary 4.8 Assume that the hypotheses of Theorem 4.7 hold. Then, the following error estimates hold, \begin{align} \label{Est_up}\displaystyle \sum_{i=1}^2 \left (\left | {\boldsymbol u}^{n}_i - {\boldsymbol u}^{n}_{i,h} \right|^2_{1,{\it{\Omega}}_i}+ \left\| p^{n}_i - p^{n}_{i,h} \right\|^2_{0,{\it{\Omega}}_i} \right) & \leq \; E_{\varepsilon,\nu}\,\displaystyle A_{\varepsilon,\nu}^{n}\, \sum_{i=1}^2 \left | k^{0}_i - k^{0}_{i,h} \right|^2_{1,{\it{\Omega}}_i} \notag \\ &\quad + h\, E_{\varepsilon,\nu}\, B_{\varepsilon,\nu},\end{align} (4.35) for some constant $$E_{\varepsilon,\nu}>0$$, where $$A_{\varepsilon,\nu}$$ and $$B_{\varepsilon,\nu}$$ are the constants appearing in estimate (4.34). Proof. The estimate for the error in velocity in (4.35) follows from estimates (4.4) and (4.34). The estimates for the error in pressures follows from the discrete inf–sup condition and a treatment for the nonlinear boundary terms similar to that of the term $$I_4$$ in the proof of Theorem 4.4, which we omit for brevity.  □ Remark 4.9 The hypothesis that $$\nu$$ (which is a lower bound for the eddy viscosities) is large enough, which seems to be unavoidable by our analysis, however, is not verified by oceanic and atmospheric flows. It would be much closer to reality to weaken this hypothesis to a condition that some average value of the eddy viscosities be large enough. In the numerical experiments reported in Section 5, it is rather this last condition that appears to be sufficient to ensure the convergence of the fixed-point algorithm (3.17)–(3.21). Remark 4.10 The previous error analysis may be extended to higher-order finite elements. Indeed, if spaces $${\boldsymbol X}_{i,h},\,M_{i,h}$$ and $$K_{i,h}$$ are respectively built with finite elements of orders $$s+1, \, s$$ and $$r+1$$ for $$s, r \ge 1$$ then under the hypotheses of Theorem 4.7, if the solution $$\left({\boldsymbol u}^{n}_{i},p^{n}_{i},k^{n}_{i}\right)$$ belongs to $${\boldsymbol H}^{2}({\it{\Omega}}_i) \times H^{1}({\it{\Omega}}_i) \times H^{2}({\it{\Omega}}_i)$$, for $$\nu$$ small enough a similar estimate to (4.35) holds: \begin{align} \label{Est_upsr}\displaystyle \sum_{i=1}^2 \left (\left | {\boldsymbol u}^{n}_i - {\boldsymbol u}^{n}_{i,h} \right|^2_{1,{\it{\Omega}}_i}+ \left \| p^{n}_i - p^{n}_{i,h} \right\|^2_{0,{\it{\Omega}}_i} \right) & \leq \; E_{\varepsilon,\nu}\,\displaystyle A_{\varepsilon,\nu}^{n}\, \sum_{i=1}^2 \left | k^{0}_i - k^{0}_{i,h} \right|^2_{1,{\it{\Omega}}_i} \notag \\ &\quad+ (h^s+h^{2s}+h^{2r})\, E_{\varepsilon,\nu}\, B_{\varepsilon,\nu},\end{align} (4.36) so the overall order of convergence of discretization (3.17)–(3.21) is $$h^{\min\{s/2,r\}}$$ for smooth solutions. 5. Numerical experiments The aim of this section is twofold: we intend to test the convergence order (in $$h$$) of the algorithm (3.17)–(3.21) and also the ability of the discretization to reproduce overall qualitative features of a realistic flow. Actually, we simulate an upwelling oceanic flow generated by an atmospheric cavity flow. Although for analysis purposes we have considered a regularized problem, we perform our numerical tests with the direct discretization of problem (1.1), without any smoothing; that is, we replace $$u_{i,h}^{n+1,\varepsilon}$$ by $$u_{i,h}^{n+1}$$ and $$k_{i,h}^{n,\varepsilon}$$ by $$k_{i,h}^{n}$$ in (3.20) and (3.21). In both cases, we use the with FreeFEM++ code to perform our tests (cf. Hecht, 2012). The solver uses a Taylor–Hood $$\mathcal P_2-\mathcal P_1$$ FEM for the space discretization of velocity–pressure and $$\mathcal P_2$$ FEM for the TKEs equation. At each step, linear systems are obtained and solved using a preconditioned generalized minimal residual method iterative routine (see, for instance, Saad, 2003). 5.1 Accuracy test The first test is aimed at estimating the convergence order (in $$h$$) of the fixed point provided by algorithm (3.17)–(3.21). To do this, we have compared an exact (or very accurate) solution of problem (1.1) with this discrete fixed point, computed as the limit as $$n \to \infty$$ of the solution $$u_{i,h}^{n+1}$$, $$p_{i,h}^{n+1}$$, $$k_{i,h}^{n,\varepsilon}$$. We have performed two tests, one corresponding to a smooth solution and another corresponding to a low-regularity solution. Test 1: Convergence order for smooth solutions. In this test, we have constructed a problem similar to (1.1) with nonhomogeneous data verified by an analytic solution: The domains occupied by the flows are $${\it{\Omega}}_1 =]0,0.5[^3 \qquad \mbox{and} \qquad {\it{\Omega}}_2 =]-0.5,0[^3,$$ and then the interface $${\it{\Gamma}}$$ corresponds to the plane $$z=0$$. The exact solutions $${\boldsymbol u}_{e,i} \,=\, \left(u_{e,ix},u_{e,iy},u_{e,iz} \right)$$, $$p_{e,i}$$ and $$k_{e,i}$$ for $$i=1,2$$ are given by: \begin{eqnarray*} \label{SolAnal1}\left\{ \begin{array}{rcl}u_{e,1x} \,&=&\, \sin(\pi \,x)\, \sin(\pi\,y)\, \sin(\pi\,z)\, - 4xz^3 +y^2,\\ u_{e,1y} &=& \cos(\pi \,x)\, \cos(\pi\,y)\, \sin(\pi\,z)\, + \pi\cos(\pi \,z)y + x^4,\\ u_{e,1z} &=& z^4 - \sin(\pi\,z),\\ p_{e,1} &=& \sin(\pi\,(x+y+z)),\\ k_{e,1} &=& \lambda |{\boldsymbol u}_1 - {\boldsymbol u}_2|^2, \end{array}\right. \end{eqnarray*} \begin{eqnarray*} \label{SolAnal2}\left\{ \begin{array}{rcl}u_{e,2x} &=& \sin(\pi\,y)\, \sin(\pi\,z) \,+yz, \\ u_{e,2y} &=& \cos(\pi\,x)\, + \sin(\pi\,z)\, -x + z^2, \\ u_{e,2z} &=& x^2+y^2, \\ p_{e,2} &=& x+y+z, \\ k_{e,2} &= & \lambda (z^2 +1)\, |{\boldsymbol u}_1 - {\boldsymbol u}_2|^2. \end{array}\right. \end{eqnarray*} Then, it holds $$k_1 \,=\, k_2 \qquad \mbox{on the interface}\; {\it{\Gamma}}.$$ The eddy viscosities and diffusions are: \begin{eqnarray*}\left\{ \begin{array}{rcl}\alpha_i(\ell) &=& \nu \, + \nu_i \, \sqrt{\ell} \\ \gamma_i(\ell) &=& \nu \, + a_i \, \ell. \end{array}\right. \end{eqnarray*} The preceding functions are solution of a modified system (1.1) with nonzero source terms, as follows: Energy equations: $$-\nabla \cdot(\gamma_i(k_i)\nabla k_i) \,-\, \alpha_i(k_i) |\nabla {\boldsymbol u}_i|^2 \;=\; g_i \; \mbox{in}\; {\it{\Omega}}_i.$$ Manning friction law on the interface $${\it{\Gamma}}$$: $$\alpha_i(k_i)\partial_{{\boldsymbol n}_i}{\boldsymbol u}_i - p_i {\boldsymbol n}_i + \kappa ({\boldsymbol u}_i - {\boldsymbol u}_j) | {\boldsymbol u}_i-{\boldsymbol u}_j| = {\boldsymbol h}_i \; \mbox{sur}\; {\it{\Gamma}}.$$ Dirichlet conditions on $$\partial {\it{\Omega}}_i \setminus {\it{\Gamma}}$$: $${\boldsymbol u}_i = {\boldsymbol u}_{e,i}$$ and $$k_i = k_{e,i}$$. The values of the parameters are: $$\lambda =1, \; \nu = 1, \; \nu_1 = 0.1, \; \nu_2 = 0.2, \; a_1 = 0.01, \; a_2 = 0.02\; \mbox{and}\; \kappa = 10^{-3}.$$ We estimate the convergence rate using several grid sizes by \begin{align}\mathcal {\rm{O}}_h &= \displaystyle \frac {{\rm{log}}\left(\frac {\mathcal E_{h_1}} {\mathcal E_{h_2}} \right)} {{\rm{log}}\left(\frac{h_1}{h_2}\right)} \approx p, \qquad \mathcal E_h & := \, \left(\displaystyle \sum_{i=1}^2 \left | k_{i,h} - k_i \right|^2_{1,{\it{\Omega}}_i}\, + \, \left| {\boldsymbol u}_{i,h}- {\boldsymbol u}_{i}\right|^2_{1,{\it{\Omega}}_i} \right)^{\frac{1}{2}}\!.\label{Est_Converg_h}\end{align} (5.1) We present in Tables 1 and 2 the computed errors and estimates of convergence orders for this test. We recover a second-order accuracy in $$H^1$$ norms, better than the one predicted by Theorem 4.7, but also better than the one predicted by the standard finite element estimates (3.8)–(3.10). We thus encounter a superconvergence effect, probably due to the symmetries of the grid. Table 1 Estimated convergence order for smooth solutions: velocities $$h$$ $$| {\boldsymbol u}_{e,1} - {\boldsymbol u}_1|_{1,{\it{\Omega}}_1}$$ $$\| {\boldsymbol u}_{e,2} - {\boldsymbol u}_2\|_{1,{\it{\Omega}}_2}$$ $$\mathcal{\rm{O}}({\boldsymbol u}_{1})-\mathcal{\rm{O}}({\boldsymbol u}_{2})$$ 0.176777 0.0264 0.0112 — 0.117852 0.0118 0.00504 1.979–1.987 0.0883883 0.00668 0.00285 1.986–1.995 0.0707107 0.00428 0.00183 1.99–1.998 0.0589261 0.00298 0.00127 1.992–2.00 0.0505087 0.00219 0.000939 1.994–2.001 0.0441942 0.00167 0.000721 1.994–2.001 0.0392841 0.00132 0.000573 1.995–2.001 $$h$$ $$| {\boldsymbol u}_{e,1} - {\boldsymbol u}_1|_{1,{\it{\Omega}}_1}$$ $$\| {\boldsymbol u}_{e,2} - {\boldsymbol u}_2\|_{1,{\it{\Omega}}_2}$$ $$\mathcal{\rm{O}}({\boldsymbol u}_{1})-\mathcal{\rm{O}}({\boldsymbol u}_{2})$$ 0.176777 0.0264 0.0112 — 0.117852 0.0118 0.00504 1.979–1.987 0.0883883 0.00668 0.00285 1.986–1.995 0.0707107 0.00428 0.00183 1.99–1.998 0.0589261 0.00298 0.00127 1.992–2.00 0.0505087 0.00219 0.000939 1.994–2.001 0.0441942 0.00167 0.000721 1.994–2.001 0.0392841 0.00132 0.000573 1.995–2.001 Table 1 Estimated convergence order for smooth solutions: velocities $$h$$ $$| {\boldsymbol u}_{e,1} - {\boldsymbol u}_1|_{1,{\it{\Omega}}_1}$$ $$\| {\boldsymbol u}_{e,2} - {\boldsymbol u}_2\|_{1,{\it{\Omega}}_2}$$ $$\mathcal{\rm{O}}({\boldsymbol u}_{1})-\mathcal{\rm{O}}({\boldsymbol u}_{2})$$ 0.176777 0.0264 0.0112 — 0.117852 0.0118 0.00504 1.979–1.987 0.0883883 0.00668 0.00285 1.986–1.995 0.0707107 0.00428 0.00183 1.99–1.998 0.0589261 0.00298 0.00127 1.992–2.00 0.0505087 0.00219 0.000939 1.994–2.001 0.0441942 0.00167 0.000721 1.994–2.001 0.0392841 0.00132 0.000573 1.995–2.001 $$h$$ $$| {\boldsymbol u}_{e,1} - {\boldsymbol u}_1|_{1,{\it{\Omega}}_1}$$ $$\| {\boldsymbol u}_{e,2} - {\boldsymbol u}_2\|_{1,{\it{\Omega}}_2}$$ $$\mathcal{\rm{O}}({\boldsymbol u}_{1})-\mathcal{\rm{O}}({\boldsymbol u}_{2})$$ 0.176777 0.0264 0.0112 — 0.117852 0.0118 0.00504 1.979–1.987 0.0883883 0.00668 0.00285 1.986–1.995 0.0707107 0.00428 0.00183 1.99–1.998 0.0589261 0.00298 0.00127 1.992–2.00 0.0505087 0.00219 0.000939 1.994–2.001 0.0441942 0.00167 0.000721 1.994–2.001 0.0392841 0.00132 0.000573 1.995–2.001 Table 2 Estimated convergence order for smooth solutions: energies $$h$$ $$\| k_{1,e} - k_1 \|_{1,{\it{\Omega}}_1}$$ $$\| k_{2,e} - k_2 \|_{1,{\it{\Omega}}_2}$$ $$\mathcal {\rm{O}}(k_{1})-\mathcal{\rm{O}}(k_{2})$$ 0.176777 0.0796 0.126 — 0.117852 0.0355 0.0574 1.987–1.956 0.0883883 0.0200 0.0325 1.995–1.975 0.0707107 0.0128 0.0208 1.998–1.984 0.0589261 0.00890 0.0145 2.000–1.989 0.0505087 0.00654 0.0106 2.001–1.993 0.0441942 0.00500 0.00818 2.001–1.995 0.0392841 0.00395 0.00647 2.001–1.998 $$h$$ $$\| k_{1,e} - k_1 \|_{1,{\it{\Omega}}_1}$$ $$\| k_{2,e} - k_2 \|_{1,{\it{\Omega}}_2}$$ $$\mathcal {\rm{O}}(k_{1})-\mathcal{\rm{O}}(k_{2})$$ 0.176777 0.0796 0.126 — 0.117852 0.0355 0.0574 1.987–1.956 0.0883883 0.0200 0.0325 1.995–1.975 0.0707107 0.0128 0.0208 1.998–1.984 0.0589261 0.00890 0.0145 2.000–1.989 0.0505087 0.00654 0.0106 2.001–1.993 0.0441942 0.00500 0.00818 2.001–1.995 0.0392841 0.00395 0.00647 2.001–1.998 Table 2 Estimated convergence order for smooth solutions: energies $$h$$ $$\| k_{1,e} - k_1 \|_{1,{\it{\Omega}}_1}$$ $$\| k_{2,e} - k_2 \|_{1,{\it{\Omega}}_2}$$ $$\mathcal {\rm{O}}(k_{1})-\mathcal{\rm{O}}(k_{2})$$ 0.176777 0.0796 0.126 — 0.117852 0.0355 0.0574 1.987–1.956 0.0883883 0.0200 0.0325 1.995–1.975 0.0707107 0.0128 0.0208 1.998–1.984 0.0589261 0.00890 0.0145 2.000–1.989 0.0505087 0.00654 0.0106 2.001–1.993 0.0441942 0.00500 0.00818 2.001–1.995 0.0392841 0.00395 0.00647 2.001–1.998 $$h$$ $$\| k_{1,e} - k_1 \|_{1,{\it{\Omega}}_1}$$ $$\| k_{2,e} - k_2 \|_{1,{\it{\Omega}}_2}$$ $$\mathcal {\rm{O}}(k_{1})-\mathcal{\rm{O}}(k_{2})$$ 0.176777 0.0796 0.126 — 0.117852 0.0355 0.0574 1.987–1.956 0.0883883 0.0200 0.0325 1.995–1.975 0.0707107 0.0128 0.0208 1.998–1.984 0.0589261 0.00890 0.0145 2.000–1.989 0.0505087 0.00654 0.0106 2.001–1.993 0.0441942 0.00500 0.00818 2.001–1.995 0.0392841 0.00395 0.00647 2.001–1.998 Test 2: Convergence order for low-regularity solution. We have solved in this test a dimensional flow, with realistic data, used in Chacón Rebollo et al. (2010): \begin{align}\gamma_1(k_1) = 3 \times 10^{-3} + 0.277 \times 10^{-4} \sqrt{k_1}; & \quad \gamma_1(k_2) = 3 \times 10^{-2} + 0.185 \times 10^{-5} \sqrt{k_2}. \notag \\ \alpha_i(\cdot)= \gamma_i(\cdot), \qquad \kappa = 10^{-3}\;{\rm{and}}\;& \lambda = 5 \times 10^{-2}. \notag \end{align} The computational domains we consider here are $${\it{\Omega}}_1 =]0, 2[\times]0,1[\times]0,1[$$ for the ‘atmosphere’ and $${\it{\Omega}}_2 =]0, 2[\times]0,1[\times]-1,0[$$ for the ‘ocean’. We have imposed rigid lid velocity boundary conditions on the top of the atmosphere $$\tilde {\it{\Gamma}}_1$$ (that corresponds to $$z=1$$): $${\boldsymbol u}_1 = {\boldsymbol 0}$$ on $${\it{\Gamma}}_1 / \tilde {\it{\Gamma}}_1$$, $${\boldsymbol u}_1 = (1,0,0)$$ on $$\tilde {\it{\Gamma}}_1$$ and $${\boldsymbol u}_2 = {\boldsymbol 0}$$ on $${\it{\Gamma}}_2$$. We consider homogenous Dirichlet boundary conditions for the TKE on all border $$\partial {\it{\Omega}}_1 \cup \partial {\it{\Omega}}_2$$ and equal to $$\lambda |{\boldsymbol u}_1 -{\boldsymbol u}_2|^2$$ on the interface $${\it{\Gamma}}$$. In this way, a cavity flow is induced in the atmosphere, which drives the oceanic flow through the nonlinear boundary conditions at the interface. The ‘exact’ $${\boldsymbol u}_i$$ and $$k_i$$ are computed with a very fine grid. Table 3 displays the computed convergence orders. These seem to approximate the value $$p\approx 0.25$$. Note that the Dirichlet boundary condition for the velocity $${\boldsymbol u}_1$$ is a piecewise constant function, which belongs to $$H^{1/2-\epsilon}({\it{\Gamma}}_1)$$ for all $$\epsilon >0$$, but not to $$H^{1/2}({\it{\Gamma}}_1)$$ (see Chen, 2016). We may then expect that the velocities have a reduced regularity. In fact, from the error estimates (4.34), the value $$p=0.25$$ would correspond to $${\boldsymbol u}_i \in H^{3/2}({\it{\Omega}}_i)^d$$. Table 3 Estimated convergence order for low-regularity solution Mesh size Order $$\mathcal {\rm{O}}_h$$ $$h$$ — $$h/2$$ 0.12 $$h/4$$ 0.16 $$h/8$$ 0.22 $$h/16$$ 0.23 Mesh size Order $$\mathcal {\rm{O}}_h$$ $$h$$ — $$h/2$$ 0.12 $$h/4$$ 0.16 $$h/8$$ 0.22 $$h/16$$ 0.23 Table 3 Estimated convergence order for low-regularity solution Mesh size Order $$\mathcal {\rm{O}}_h$$ $$h$$ — $$h/2$$ 0.12 $$h/4$$ 0.16 $$h/8$$ 0.22 $$h/16$$ 0.23 Mesh size Order $$\mathcal {\rm{O}}_h$$ $$h$$ — $$h/2$$ 0.12 $$h/4$$ 0.16 $$h/8$$ 0.22 $$h/16$$ 0.23 Using high-order finite element spaces to solve problem (1.1) for low-regularity solutions needs an un-useful computational effort, as the convergence order of the discretization is limited by the low regularity of the solution. Table 4 Number of iterations vs. $$\nu$$ $$\nu$$ 1 0.5 0.1 0.05 $$1e^{-2}$$ $$1e^{-3}$$ $$1e^{-4}$$ $$1e^{-5}$$ $$1e^{-8}$$ Number of iterations 4 5 9 14 18 24 20 20 20 $$\nu$$ 1 0.5 0.1 0.05 $$1e^{-2}$$ $$1e^{-3}$$ $$1e^{-4}$$ $$1e^{-5}$$ $$1e^{-8}$$ Number of iterations 4 5 9 14 18 24 20 20 20 Table 4 Number of iterations vs. $$\nu$$ $$\nu$$ 1 0.5 0.1 0.05 $$1e^{-2}$$ $$1e^{-3}$$ $$1e^{-4}$$ $$1e^{-5}$$ $$1e^{-8}$$ Number of iterations 4 5 9 14 18 24 20 20 20 $$\nu$$ 1 0.5 0.1 0.05 $$1e^{-2}$$ $$1e^{-3}$$ $$1e^{-4}$$ $$1e^{-5}$$ $$1e^{-8}$$ Number of iterations 4 5 9 14 18 24 20 20 20 5.2 Test 3: Dependency of convergence rate upon $$\nu$$ This test aims at determining the convergence and the practical convergence rate of algorithm (3.17)–(3.21) with respect to the laminar viscosity $$\nu$$. To illustrate the effects of the viscosity we have used the analytic solution described in Test 2 computed in tetrahedral mesh, where $$h \approx 0.07$$ with the viscosity $$\nu$$ ranging in the interval $$[10^{-8},1]$$, and we determine the convergence rate $$K$$ of the fixed-point algorithm (3.17)–(3.21) as follows: $$K \,=\, \left(\frac{ \displaystyle \sum_{i=1}^2 \| {{\boldsymbol u}}_{i}^{j+1} - {{\boldsymbol u}}_{i}^{j}\|^2 \,+\, \| k_{i}^{j+1} - k_{i}^{j}\|^2}{\displaystyle\sum_{i=1}^2 \| {{\boldsymbol u}}_{i}^{j} - {{\boldsymbol u}}_{i}^{j-1}\|^2 \,+\, \| k_{i}^{j} - k_{i}^{j-1}\|^2}\right)^{\frac{1}{2}},$$ where the index $$j$$ is the first iteration in the fixed-point algorithm for which $$\left(\displaystyle \sum_{i=1}^2\| {{\boldsymbol u}}_{i}^{j+1} - {{\boldsymbol u}}_{i}^{j}\|^2 \,+\, \| k_{i}^{j+1} - k_{i}^{j}\|^2 \right)^{\frac{1}{2}} \leq \varepsilon = 1e^{-10}.$$ Figure 1 shows that algorithm (3.17)–(3.21) converges with a rather constant rate $$K \simeq 0.55$$ when $$\nu \in [1e^{-8}, 1e^{-3}]$$. This seems to indicate that the eddy viscosities and diffusion remain uniformly bounded from below, thus ensuring a good convergence rate for this fixed-point algorithm. Note that the convergence rate improves as $$\nu$$ increases in the range $$[1e^{-3}, 1]$$. Fig. 1. View largeDownload slide Constant rate $$K$$ with respect to $$\nu$$. Fig. 1. View largeDownload slide Constant rate $$K$$ with respect to $$\nu$$. Also, the errors between the exact solution and the fixed point provided by algorithm (3.17)–(3.21) decrease as $$\nu$$ increases in $$[1e^{-3},1]$$, as predicted by Theorem 4.7, but remains constant for $$\nu \le 1e^{-3}$$, see Figs 2 and 3. This is still consistent with the existence of a uniform lower bound for the eddy viscosities and diffusions as $$\nu$$ decreases. Fig. 2. View largeDownload slide $${\boldsymbol H}^1$$-error between the exact and approximate solutions of velocities with respect to $$\nu$$. Fig. 2. View largeDownload slide $${\boldsymbol H}^1$$-error between the exact and approximate solutions of velocities with respect to $$\nu$$. Fig. 3. View largeDownload slide $${\boldsymbol H}^1$$-error between the exact and approximate solutions of energies with respect to $$\nu$$. Fig. 3. View largeDownload slide $${\boldsymbol H}^1$$-error between the exact and approximate solutions of energies with respect to $$\nu$$. 5.3 Upwelling effects near the shores This test is designed to determine the ability of our iterative algorithm presented above to simulate genuine three-dimensional effects that arise in geophysical flows. Concretely, we test the formation of the upwelling effect due to the interaction between surface wind tension and Coriolis forces. We have considered the following problem that includes transport and Coriolis terms, instead of system (1.1), $$\label{P1-b}\left \{ \begin{array}{rcl}\left({\boldsymbol u}_i \cdot \nabla \right) {\boldsymbol u}_i + \tau \left(-u_{i,y},u_{i,x},0\right) \qquad \qquad&& \\ - \nabla \cdot(\alpha_i(k_i)\nabla {\boldsymbol u}_i) + \nabla p_i &=& {\boldsymbol f}_i \\ \nabla \cdot {\boldsymbol u}_i &=& 0 \\ {\boldsymbol u}_i \nabla k_i -\nabla \cdot(\gamma_i(k_i)\nabla k_i) &=&\alpha_i(k_i) |\nabla {\boldsymbol u}_i|^2, \end{array}\right.$$ (5.2) where the component of the velocity fields are denoted by $${\boldsymbol u}_i = (u_{i,x},u_{i,y},u_{i,z})$$. The term $$\tau \left(-u_{i,y},u_{i,x},0\right)$$ models the Coriolis forces, where the parameter $$\tau$$ depends on the angular velocity of the earth and the latitude. Furthermore, to take into account the inertial effects we have added the convection term $$\left({\boldsymbol u}_i \cdot \nabla \right) {\boldsymbol u}_i$$ in the first equation and the transport term $${\boldsymbol u}_i \nabla k_i$$ in the TKEs equation. The boundary conditions are the same as for system (1.1). We have considered the computational domains $${\it{\Omega}}_1 =]0,\,10^4[\times]0, 5 \cdot10^3[\times]0,\,500[\, (m)$$ for atmosphere and a swimming pool-like domain to model the geometry of the ocean (see Fig. 4), $${\it{\Omega}}_2 = \omega \times \{z = D(x,y); \quad (x,y) \in \omega \}$$, such that Fig. 4. View largeDownload slide Computational domain of ocean. Fig. 4. View largeDownload slide Computational domain of ocean. Horizontal dimensions $$(m)$$:   $$\omega \,= \,]0,\,10^4[\times]0,\, 5 \cdot10^3[$$ Bathimetry $$(m)$$: \begin{equation*}\label{pool}D(x,y) = \left \{ \begin{array}{rclr}-50 \quad &\mbox{if}& 0 \leq x \leq 4 \cdot 10^3 \\ \displaystyle -50 \cdot \frac{5\cdot 10^3 -x}{10^3} -100 \cdot \frac{4\cdot 10^3 -x}{10^3}\quad &\mbox{if}& 4\cdot 10^3 \leq x \leq 5 \cdot 10^3 \\ -100 \quad &\mbox{if}& 5 \cdot 10^3 \leq x \leq 10^4. \end{array}\right. \end{equation*} We use the following set of data: $$\alpha_1(k_1)= \gamma_1(k_1)= 3\cdot 10^-2 + 0,277\cdot 10^{-4} \sqrt{k_1} \,m^2/s$$ $$\alpha_2(k_2)= \gamma_2(k_2)=\nu+ 0,185\cdot 10^{-5} \sqrt{k_2}$$, where $$\nu = (10^{-1},10^{-2},10^{-4}) \,m^2/s$$ $$\lambda = 5\cdot 10^{-2}$$, $$\kappa = 10^{-3}$$, $${\boldsymbol f}_i = {\boldsymbol 0},$$ $$\; \tau = 2 \theta \, \sin(\phi)$$, where $$\theta = 7,3\cdot 10^{-5}\,s^{-1}$$, $$\phi = 45^0\,N$$. The viscosity $$\nu$$ is scaled by the different sizes of $$\omega$$ in the directions $$OX$$, $$OY$$ and $$OZ$$ to take into account the anisotropy of the domain. Also, we have again imposed lid-driven cavity boundary conditions, $${\boldsymbol u}_1 = {\boldsymbol 0}$$ on $${\it{\Gamma}}_1 / \tilde {\it{\Gamma}}_1$$, $${\boldsymbol u}_1 = (-1,0,0)$$ on $$\tilde {\it{\Gamma}}_1$$ and $${\boldsymbol u}_2 = {\boldsymbol 0}$$ on $${\it{\Gamma}}_2$$, where $$\tilde {\it{\Gamma}}_1$$ is the upper face $$z=500$$ of $${\it{\Omega}}_1$$. We consider homogenous Dirichlet boundary conditions for the TKE on all the border $$\partial {\it{\Omega}}_1 \cup \partial {\it{\Omega}}_2$$ and equal to $$\lambda |{\boldsymbol u}_1 -{\boldsymbol u}_2|^2$$ on the interface $${\it{\Gamma}}$$. These settings are chosen to create a driven cavity-like flow in atmosphere domain $${\it{\Omega}}_1$$. The atmospheric flow generates a wind flow at the top of the pool, i.e., the interface air–water and, subsequently, the formation of the upwelling flow besides a lateral wall of the pool. The relatively short dimension of the domain in the cross-wind direction also originates a downwelling flow in the vertical face of the pool opposite to the upwelling. On the other hand, the relatively short dimension of the domain in the wind direction originates a longitudinal recirculation that accelerates due to the bottom ramp, in a direction opposite to the wind. Note that in the representation of the numerical results, the depth and the vertical velocity have been increased by a factor 10 to provide a good visualization. Figure 5 shows the vertical velocity, where Coriolis acceleration effects are apparent. In the Northern Hemisphere, the Earth rotation deviates the flow to its right. Figure 6 represents the velocity profile along a vertical cut of the domain (the plane $$y = 2500$$). We observe a global recirculation of the flow that produces an acceleration along the ramp, in the direction opposite to the wind. The flow presents a quasi-parabolic vertical profile in the less depth part of the domain. Figures 7 and 8 show the projection of the three-dimensional velocity on two planes orthogonal to the wind direction ($$x = 5000,\,x = 8000$$, respectively). We observe not only the upwelling effect on the left of the domain, but also a downwelling effect on the right of the domain. The overall flow is a recirculation transversal to the wind direction. The trajectory of a flow particle describes spirals around an axis parallel to the wind direction. Fig. 5. View largeDownload slide Surface velocity in the ocean. Fig. 5. View largeDownload slide Surface velocity in the ocean. Fig. 6. View largeDownload slide Projection of three-dimensional velocity in the ocean onto plane y = 2500. Fig. 6. View largeDownload slide Projection of three-dimensional velocity in the ocean onto plane y = 2500. Fig. 7. View largeDownload slide Projection of three-dimensional velocity in the ocean onto plane x = 8000. Fig. 7. View largeDownload slide Projection of three-dimensional velocity in the ocean onto plane x = 8000. Fig. 8. View largeDownload slide Projection of three-dimensional velocity in the ocean onto plane x = 5000. Fig. 8. View largeDownload slide Projection of three-dimensional velocity in the ocean onto plane x = 5000. 6. Conclusion In this article, we have analysed a numerical model for the coupling two steady turbulent fluids. The coupling is modeled by the Manning law and the generation of TKE due to shear stress at the interface. Both the viscosity and diffusion depend on the TKE. We have considered an iterative scheme by linearization besides a full finite element discretization of a regularized problem. We have proved that the iterative scheme converges for large enough eddy viscosities to the solution of the continuous problem. Also, numerical tests for smooth solutions are in good agreement with our theoretical results. Furthermore, in these tests, the convergence rate of the fixed-point iteration procedure remains uniformly bounded as the laminar viscosity decreases. This is consistent with the existence of a uniform lower bound for eddy viscosities and diffusions, what opens an interesting research subject from both the theoretical and the numerical viewpoints. Furthermore, some tests for realistic flows show the ability of the discretization introduced to correctly simulate the three-dimensional features associated with the interaction between wind stress induced by atmosphere and Coriolis forces. In future works, we will study the application of the fixed-point method introduced here to evaluate large eddy simulation models similar to the one studied and also to de-couple discretizations of the friction interface boundary condition for the TKE, introduced in Connors et al. (2009). Funding Proyecto de Excelencia (P12-FQM-454), in part; Junta de Andalucía Regional Gand FEDER EU Fund. References Bénilan P. , Boccardo L. , Gallouët T. , Gariepy R. , Pierre M. & Vázquez J. L. ( 1995 ) An $$L^1$$-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) , 22 , 241 – 273 . Bernardi C. , Chacén Rebollo T. , Gémez Mármol M. , Lewandowski R. & Murat F. ( 2004a ) A model for two coupled turbulent fluids. III. Numerical approximation by finite elements. Numer. Math. , 98 , 33 – 66 . Google Scholar Crossref Search ADS Bernardi C. , Chacén Rebollo T. , Lewandowski R. & Murat F. ( 2002 ) A model for two coupled turbulent fluids. 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Published: Oct 16, 2018

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