A priori error analysis and finite element approximations for a coupled model under nonlinear slip boundary conditions (Q6558297)

The authors consider the time-dependent Navier-Stokes problem coupled with the heat equation: \(\frac{\partial u}{\partial t}-\operatorname{div}(2\nu (\theta )Du)+(u\cdot \nabla )u+\nabla p=f\), \(\operatorname{div}u=0\), \(\frac{\partial \theta }{ \partial t}-\operatorname{div}(\mu (\theta )\nabla \theta )+(u\cdot \nabla )\theta =k\) in \( \Omega \times \lbrack 0,T]\), where \(\Omega \) is a connected, bounded and open set in \(\mathbb{R}^{d}\), \(d=2,3\), with a Lipschitz-continuous boundary \( \partial \Omega \) divided in two parts \(S\) and \(\Gamma =\partial \Omega \setminus \overline{S}\), with \(\Gamma \cap \overline{S}=\emptyset \), \(T\) a positive real number, \(u\) the velocity, \(p\) the pressure, \(\theta \) the temperature, \(f\) the external volumic forces, \(k\) the external heat source, \( \nu \) the viscosity coefficient, and \(\mu \) the diffusion coefficient, that are both positive, bounded and depending on the temperature. The initial conditions \(u(x,0)=u_{0}\), \(\theta (x,0)=\theta _{0}\) are imposed in \( \overline{\Omega }\). The Dirichlet conditions \(u(x,t)=0\) on \(\Gamma \times (0,T)\) and \(\theta (x,t)=\theta _{b}\) are imposed on \(\partial \Omega \times (0,T)\), \(\theta _{b}\) being given and non-negative, together with the threshold slip condition formulated through a positive function \( g:S\rightarrow (0,\infty )\) as: if \(\left\vert (\sigma n)_{\tau }\right\vert <g\) then \(u_{\tau }=0\), if \(\left\vert (\sigma n)_{\tau }\right\vert =g\) then \(u_{\tau }\neq 0\) and \(-(\sigma n)_{\tau }=g\frac{u_{\tau }}{ \left\vert u_{\tau }\right\vert }\), on \(S\times (0,T)\). The authors introduce a variational formulation of this problem, they prove that it is equivalent to the above problem, and give a priori estimates, under appropriate hypotheses on the data. \N\NThe first main result proves the existence of a solution \(u\in L^{\infty }(0,T;V(\Omega ))\), \(p\in L^{2}(0,T;L_{0}^{2}(\Omega ))\), \(\theta \in L^{2}(0,T;H_{0}^{1}(\Omega ))\), \( \frac{\partial u}{\partial t}\in L^{2}(0,T;V^{\prime }(\Omega ))\) if \(d=2\), and \(\frac{\partial u}{\partial t}\in L^{4/3}(0,T;V^{\prime }(\Omega ))\) if \( d=3\), where \(V(\Omega )=\{v\in H^{1}(\Omega )\mid v\mid _{\Gamma }=0\), \( v\cdot n\mid _{S}=0\}\). For the proof, the authors introduce a regularizing parameter \(\varepsilon \), and then a Galerkin approximation of the regularized problem, which is proved to have a\ unique solution \((u_{\varepsilon }^{m},\theta _{\varepsilon }^{\ast m})\in L^{2}(0,T;V_{\operatorname{div}}^{m}(\Omega ))\cap L^{\infty }(0,T;L^{2}(\Omega ))\times L^{2}(0,T;W^{m}(\Omega ))\). They prove uniform estimates on this solution\ which allow passing to the limit. \N\NThe second part of the paper presents a space-time discretization of the variational problem, assuming that \(\Omega \) is a polygon when \(d=2\) or a polyhedron when \(d=3\), so that it can be completely meshed by a regular family \(\mathcal{T}_{h}\) of closed and non degenerate triangles or tetrahedra satisfying classical hypotheses. The finite element problem is proved to have at least one solution \((u_{h}^{n},p_{h}^{n},\theta _{h}^{n})\) which verifies bounds. The third main result of the paper proves an \textit {a priori} error estimate for this problem. The authors present the Marchuk-Yanenko operator splitting algorithm and the paper ends with the presentation of two examples: a stationary solution and a time-dependent solution, both in the unit square. The authors also compute the convergence rate for different values of parameters.