We consider the flow of a viscous incompressible fluid in a rigid homogeneous porousmedium provided with mixed boundary conditions. Since the boundary pressure can presenthigh variations, the permeability of the medium also depends on the pressure, so that themodel is nonlinear. A posteriori estimates allow us to omit thisdependence where the pressure does not vary too much. We perform the numerical analysis ofa spectral element discretization of the simplified model. Finally we propose a strategywhich leads to an automatic identification of the part of the domain where the simplifiedmodel can be used without increasing significantly the error.

The penalty method when applied to the Stokes problem provides a very efficient algorithm for solving any discretization of this problem since it gives rise to a system of two equations where the unknowns are uncoupled. For a spectral or spectral element discretization of the Stokes problem, we prove a posteriori estimates that allow us to optimize the penalty parameter as a function of the discretization parameter. Numerical experiments confirm the interest of this technique.

We consider a variational formulation of the three-dimensional Navier–Stokes equations with mixed boundary conditions and prove that the variational problem admits a solution provided that the domain satisfies a suitable regularity assumption. Next, we propose a finite element discretization relying on the Galerkin method and establish a priori and a posteriori error estimates.

From the fundamental laws of elasticity, we write a model for the contact between two membranes and we perform the analysis of the corresponding system of variational inequalities. We propose a finite element discretization of this problem and prove its well-posedness. We also establish a priori and a posteriori error estimates.

As a first draft of a model for a river flowing on a homogeneous porous ground, we consider a system where the Darcy and Stokes equations are coupled via appropriate matching conditions on the interface. We propose a discretization of this problem which combines the mortar method with standard finite elements, in order to handle separately the flow inside and outside the porous medium. We prove a priori and a posteriori error estimates for the resulting discrete problem. Some numerical experiments confirm the interest of the discretization.

This paper deals with the mortar spectral element discretization of two equivalent problems, the Laplace equation and the Darcy system, in a domain which corresponds to a nonhomogeneous anisotropic medium. The numerical analysis of the discretization leads to optimal error estimates and the numerical experiments that we present enable us to verify its efficiency.

We consider the Stokes problem provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We write a variational formulation of this problem with three independent unknowns: the vorticity, the velocity and the pressure. Next we propose a discretization by spectral element methods which relies on this formulation. A detailed numerical analysis leads to optimal error estimates for the three unknowns and numerical experiments confirm the interest of the discretization.

In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive a priori and a posteriori error estimates. We present some numerical experiments that are in good agreement with the analysis.

We analyze a new formulation of the Stokes equations in three-dimensional axisymmetric geometries, relying on Fourier expansion with respect to the angular variable: the problem for each Fourier coefficient is two-dimensional and has six scalar unknowns, corresponding to the vector potential and the vorticity. A spectral discretization is built on this formulation, which leads to an exactly divergence-free discrete velocity. We prove optimal error estimates.

The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.

For the Stokes problem in a two- or three-dimensionalbounded domain, we propose a new mixed finite element discretization which relies ona nonconforming approximation of the velocity and a more accurate approximation of thepressure. We prove that the velocity and pressure discrete spaces are compatible, in thesense that they satisfy an inf-sup condition of Babuška and Brezzi type, and wederive some error estimates.

The standard discretization of the Stokes andNavier–Stokes equations in vorticity and stream function formulation by affine finiteelements is known for its bad convergence. We present here a modified discretization, weprove that the convergence is improved and we establish a priori error estimates.