Abstract. We consider the Stokes and Navier-Stokes equations with boundary conditions of Dirichlet type on the velocity on one part of the boundary and involving the pressure on the rest of the boundary. We write the variational formulations of such problems. Next we propose a finite element discretization of them and perform the a priori and a posteriori analysis of the discrete problem. Some numerical experiments are presented in order to justify our strategy. 1. Introduction. Most works concerning the Stokes or Navier-Stokes equations deal with Dirichlet boundary conditions on the velocity (also called no-slip conditions); see, for instance, [19] or [32]. However, other types of boundary conditions were suggested in the pioneering paper [4], which was followed by a large number of works on this subject. Among them, the conditions on the normal component of the velocity and the vorticity were thoroughly studied and led to the so-called vorticity-velocitypressure formulation, introduced in [30] and studied in several other papers; see [16], [17], and [7], for instance; their extension to mixed boundary conditions was performed in [8]. However, it seems that the conditions on the tangential components of the velocity and the pressure have less been studied; we refer the reader to [28] and [14] for first works on these topics and also to [5] [31]. Unfortunately this discretization most often relies on finite difference schemes.We wish here to propose a discretization in the case of mixed boundary conditions, Dirichlet conditions on the velocity in part of the boundary, conditions on the tangential components of the velocity and on the pressure on another part, both for the Stokes and Navier-Stokes equations. We first write the variational formulation of these problems and recall their main properties. It can be noted that all conditions on the velocity are prescribed in an essential way, while the boundary condition on the pressure is treated in a natural way. Next, we consider a finite element discretization: In view of the variational formulation, we decide to use the same finite elements as for standard boundary conditions, more precisely the Taylor-Hood element [21]. We

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This paper presents a new algorithm for the numerical solution of the Navier-Stokes equations coupled with the convection-diffusion equation. After establishing convergence of the semidiscrete formulation at each time step, we introduce a new iterative scheme based on a projection method called the coupled prediction scheme. We show that even though the predicted temperature is advected by a velocity prediction which is not necessarily divergence free, the theoretical time accuracy of the global scheme is conserved. From a numerical point of view, this new approach gives a faster and more efficient algorithm compared to the usual fixed-point approaches. Introduction.Heat transfer is an important factor in many fluid dynamics applications. Whenever there is a temperature difference between the fluid and the confining area, heat will be transferred and the flow will be affected in nontrivial ways. Natural convection is such an example in which the driving forces are density variations and gravity (see Jiji [28], for instance). Natural convection flows are observed in different situations such as geophysics, weather, ocean movement and are also exploited in numerous applications such as double-glazed windows, cooling in electronic devices, building insulation, etc.The model is generally described using the Boussinesq approximation. In this approximation, the density of the fluid is assumed to be constant and the gravitational source force (the buoyancy term in the momentum equation) depends on the temperature (Martynenko and Khramtsov [34]).Typically, in the Boussinesq approximation, the coupling between the fluid and the temperature appears through two terms: a source term depending linearly on the temperature, and a convective term based on the velocity of the fluid (see system (2.1)). In this paper we propose a reinforcement of this coupling by adding an explicit dependency to the temperature for the viscosity and the diffusion coefficients. Moreover, since the assumptions on the source term for the momentum equation are not essential (Remark 2.1), we will consider a more general source term. Owing to this departure from the usual Boussinesq equations, the proposed model can be viewed more generally as a thermally coupled Navier-Stokes problem.Thermally coupled incompressible flow problems present two major difficulties requiring special attention: solving the incompressible Navier-Stokes equations on very fine three-dimensional meshes in a reasonable computational time is a difficult task; the strong coupling between the Navier-Stokes and convection-diffusion equations often leads to very complex time dependent dynamics requiring efficient solvers.

In this paper we consider the heat equation coupled with Darcy's law with a nonlinear source term describing heat production due to an exothermic chemical reaction. Existence and uniqueness of a solution are established. Next, a spectral discretization of the problem is presented and thoroughly analysed. Finally, we present some numerical experiments which confirm the interest of the discretization.Résumé: Dans cet article, nous considérons l'équation de la chaleur couplée avec la loi de Darcy avec un terme source non linéaire décrivant la production de chaleur dueà une réaction chimique exothermique. L'existence et l'unicité d'une solution sontétablies. Ensuite, une discrétisation par méthode spectrale du problème est décrite et analysée en détail. Enfin, nous présentons quelques expériences numériques qui confirment l'intérêt de la discrétisation.

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