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The method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some special interface conditions between the three-dimensional and one-dimensional parts. In the case of fluid mechanics this method generates a variational formulation of the velocity with Poiseuille type or Womersley type flow in the tubes at a small distance of the ends. However, the pressure should be then reconstructed using the obtained velocity field. In the present paper the procedure of the reconstruction of pressure is given and justified by the estimates between the exact pressure of the full geometry problem and reconstructed one. Numerical examples apply the method and assess its accuracy depending on the flow regime.

This thesis is devoted to the derivation of a posteriori error estimates for the numerical approximation of fluids flows separated by a free surface. Based on these estimates, error indicators are introduced and adaptive algorithms are proposed to solve the problem with accuracy and low computational costs. We focus on numerical methods that are combinations of anisotropic finite elements and second order methods to advance in time. We split the technical difficulties in the derivation of the error estimates by first studying independent PDEs, and in a second time by gathering the different results to analyse the complete system of equations composed with these latter. The a posteriori error analysis for the approximation of these PDEs will be addressed in a particular and devoted chapter. The last chapter is dedicated to the study of the system describing two fluids flows. In each chapter, we focus on two main objectives. The first is a theoretical analysis and the derivation of error estimates, the second is the description and the implementation of an algorithm to adapt meshes and time steps. Finally, numerical experiments are performed to demonstrate the efficiency of the procedure.

The method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some special interface conditions between the three-dimensional and one-dimensional parts. In the case of fluid mechanics these conditions prescribe a pre-computed Poiseuille-type shape of a solution at the interface, which however are not generalizable to the case with a boundary layer in time. In this work we present a new more general version of the method which considered and justified for the transient Navier-Stokes equations. Although theoretical justification (well posedness, asymptotic analysis) can be shown only for moderate Reynolds numbers, the provided numerical tests show good accuracies for higher values.

Presenta los resultados de la comparación entre las ecuaciones de Stokes y de Navier-Stokes para la simulación del flujo de agua líquida, a condiciones atmosféricas, a través de una placa orificio concéntrica. A partir de los datos experimentales que fueron tomados en el banco de fluidos, se evaluaron las simulaciones de ambas ecuaciones, usando el software libre Freefem++cs, que se basa en el método de los elementos finitos; las variables evaluadas son velocidad y presión en un intervalo de tiempo. Al analizar los resultados obtenidos con las simulaciones y comparar con los datos experimentales se encontró que las ecuaciones de Navier-Stokes representan mejor el sistema que la ecuación de Stokes. This paper presents the results of a comparison between Stokes and Navier-Stokes equations, in order to simulate the flow of liquid water at atmosferic conditions, through a concentric orifice plate. From experimental data taken from the fluids bank, the simulations of both equations were evaluated, using free software Freefem++CS, which is based on the finite elements method. The evaluated variables are velocity and pression in a time interval. When analyzing the results obtained with the simulations and comparing them with the experimental data, it was found that the Navier-Stokes equations represent better the system, than the Stokes equation. Apresenta os resultados da comparação entre as equações de Stokes e de Navier-Stokes para a simulação do fluxo de água líquida, a condições atmosféricas, através de uma placa de orifício concêntrica. A partir dos dados experimentais que foram tomados no banco de fluidos, se avaliaram as simulações de ambas as equações, usando o software livre Freefem++cs, que se baseia no método dos elementos finitos; as variáveis avaliadas são velocidade e pressão em um intervalo de tempo. Ao analisar os resultados obtidos com as simulações e comparar com os dados experimentais encontrou-se que as equações de Navier-Stokes representam melhor o sistema que a equação de Stokes.

The aim of this work is to present some strategies to solve numerically controllability problems for the two-dimensional heat equation, the Stokes equations and the Navier-Stokes equations with Dirichlet boundary conditions. The main idea is to adapt the Fursikov-Imanuvilov formulation, see [A.V. Fursikov, O.Yu. Imanuvilov: Controllability of Evolutions Equations, Lectures Notes Series, Vol. 34, Seoul National University, 1996]; this approach has been followed recently for the onedimensional heat equation by the first two authors. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up near the final time. The associated optimality conditions can be viewed as a differential system in the three variables x1, x2 and t that is second–order in time and fourth–order in space, completed with appropriate boundary conditions. We present several mixed formulations of the problems and, then, associated mixed finite element Lagrangian approximations that are easy to handle. Finally, we exhibit some numerical experiments. Ministerio de Economía y Competitividad Centre national de la recherche scientifique

We investigate the solvability of the instationary Stokes equations with fully inhomogeneous data in \({L^r(0,T;H^{\beta,q}_w(\Omega))}\) , where \({H^{\beta,q}_w(\Omega)}\) is a Bessel-potential space with a Muckenhoupt weight w. Depending on the order of this Bessel-potential space we are dealing with strong solutions or with very weak solutions. Whereas in the context of lowest regularity one obtains solvability with respect to inhomogeneous data by dualization, this is more delicate in the case of higher regularity, where one has to introduce some additional time regularity. As a preparation, we introduce a generalization of the Stokes operator that is appropriate to the context of very weak solutions in weighted Bessel-potential spaces.

We investigate the nonstationary Navier-Stokes equations for an exterior domain $\Omega\subset \bm{R}^3$ in a solution class $L^s (0,T;L^q(\Omega))$ of very low regularity in space and time, satisfying Serrin's condition $\frac{2}{s} + \frac{3}{q} = 1$ but not necessarily any differentiability property. The weakest possible boundary conditions, beyond the usual trace theorems, are given by $u|_{\partial\Omega} = g \in L^s (0,T;W^{-1/q,q}(\partial\Omega))$ , and will be made precise in this paper. Moreover, we suppose the weakest possible divergence condition $k = \div u \in L^s(0,T;L^r(\Omega))$ , where $\frac{1}{3} + \frac{1}{q} = \frac{1}{r}$ .

International audience; This article is devoted to the presentation of a multilevel method using finite differences that is well adapted for solving Stokes and Navier-Stokes problems in primitive variables. We use Uzawa type algorithms to solve the saddle point problems arising from spatial discretization by staggered grids and a semi-explicit temporal scheme. By means of a new change of basis operator, the two-dimensional velocity and pressure fields of an M.A.C mesh are gathered in a hierarchical order, into several grids preserving the M.A.C property on each of them. These new hierarchical unknowns, called Staggered Incremental Unknowns (SIU), allow us to develop techniques which reduce the cost of the resolution of either Stokes or generalized Stokes problems. An experimental estimation of the condition number of the inner matrix is given, and justifies the preconditioning effect of the staggered incremental unknowns.

International audience; We prove for generic steady solutions of the Lattice Boltzmann (LB) models that the variation of the numerical errors is set by specific combinations (called ``magic numbers') of the relaxation rates associated with the symmetric and anti-symmetric collision moments. Given the governing dimensionless physical parameters, such as the Reynolds or Peclet numbers, and the geometry of the computational mesh, the numerical errors remain the same for any change of the transport coefficients only when the ``free' (``kinetic') antisymmetric rates and the boundary rules are chosen properly. The single-relaxation-time (BGK) model has no free collision rate and yields viscosity dependent errors with any boundary scheme for hydrodynamic problems. The simplest and most efficient collision operator for invariant errors is the two-relaxation-times (TRT) model. As an example, this model is able to compute viscosity independent permeabilities for any porous structure. These properties are derived from steady recurrence equations, obtained through linear combinations of the LB evolution equations, in which the equilibrium and non-equilibrium components are directly interconnected via finite-difference link-wise central operators. The explicit dependency of the non-equilibrium solution on the relaxation rates is then obtained. This allows us, first, to confirm the governing role of the ``magic' combinations for steady solutions of the Stokes equation, second, to extend this property to steady solutions of the NavierStokes and anisotropic advectiondiffusion equations, third, to develop a parametrization analysis of the microscopic and macroscopic closure relations prescribed via link-wise boundary schemes.

This work deals with a system of nonlinear parabolic equations arising in turbulence modelling. The unknowns are the N components of the velocity field u coupled with two scalar quantities θ and φ . The system presents nonlinear turbulent viscosity $A(\theta,\varphi)$ and nonlinear source terms of the form $\theta^2|\nabla u|^2$ and $\theta\varphi|\nabla u|^2$ lying in L 1 . Some existence results are shown in this paper, including $L^\infty$ -estimates and positivity for both θ and φ .