The hydrostatic approximation of the incompressible 3D stationaryNavier-Stokes equations is widely used in oceanography and other applied sciences. It appears through a limit process due to the anisotropy of the domain in use, an ocean, and it is usually studied as such.We consider in this paper an equivalent formulation to this hydrostatic approximation that includes Coriolis force and an additional pressure term that comes from taking into account the pressure in the state equation for the density. It therefore models a slight dependence of the density upon compression terms. We study this model as an independent mathematical object and prove an existence theorem by means of a mixed variational formulation. The proof uses a family of finite element spaces to discretize the problem coupled with a limit process that yields the solution.We finish this paper with an existence and uniqueness result for the evolutionary linear problem associated to this model. This problem includes the same additional pressure term and Coriolis force.

We derive in this article some models ofCahn-Hilliard equations in nonisotropic media. These models, based onconstitutive equations introduced by Gurtin in [19], take the work ofinternal microforces and also the deformations of the material intoaccount. We then study the existence and uniqueness of solutions andobtain the existence of finite dimensional attractors.

The paper is devoted to analysis of an elliptic-algebraic system ofequations describing heat explosion in a two phase medium filling a star-shaped domain. Three typesof solutions are found: classical, critical andmultivalued. Regularity of solutions is studied as well as theirbehavior depending on the size of the domain and on the coefficient ofheat exchange between the two phases. Critical conditions of existence of solutions are found for arbitrary positive source function.

We propose a new formulation of the 3D Boltzmann non linear operator, without assuming Grad's angular cutoff hypothesis, and for intermolecular laws behaving as 1/rs , with s> 2. It involves natural pseudo differential operators, under a form which is analogous to the Landau operator. It may be used in the study of the associated equations, and more precisely in the non homogeneous framework.

We present an improved, near-optimal hp error estimate for a non-conforming finite element method, called the mortar method (M0). We also present a new hp mortaring technique, called the mortar method (MP), and derive h, p and hp error estimates for it, in the presence of quasiuniform and non-quasiuniform meshes. Our theoretical results, augmented by the computational evidence we present, show that like (M0), (MP) is also a viable mortaring technique for the hp method.

We study the well-posedness of an unsteady fluid-structure interaction problem.We consider a viscous incompressible flow, which is modelled by theNavier-Stokes equations. The structure is a collection of rigid moving bodies. The fluiddomain depends on time and is defined by the position of the structure, itself resultingfrom a stress distribution coming from the fluid. The problem is thennonlinear and the equations we deal with are coupled. We prove its localsolvability in time through two fixed point procedures.

A Legendre spectral collocation method is presented for the solutionof the biharmonic Dirichlet problem on a square. The solution andits Laplacian are approximated using the set of basis functions suggestedby Shen, which are linear combinations of Legendre polynomials. A Schurcomplement approach is used to reduce the resulting linear system to oneinvolving the approximation of the Laplacian of the solution on the twovertical sides of the square. The Schur complement system is solved bya preconditioned conjugate gradient method. The total cost of the algorithmis O(N 3). Numerical results demonstrate the spectral convergence of the method.

We give an analysis of the stability and uniqueness of the simplylaminated microstructure for all three tetragonal to monoclinicmartensitic transformations. The energy density for tetragonal tomonoclinic transformations has four rotationally invariant wells sincethe transformation has four variants. One of these tetragonal tomonoclinic martensitic transformations corresponds to the shearing ofthe rectangular side, one corresponds to the shearing of the squarebase, and one corresponds to the shearing of the plane orthogonal to adiagonal in the square base. We show that the simply laminatedmicrostructure is stable except for a class of special materialparameters. In each case that the microstructure is stable, we deriveerror estimates for the finite element approximation.

In this paper, we construct a combined upwinding and mixed finite element method for the numerical solution of a two-dimensional mean field model of superconducting vortices. An advantage of our method is that it works for any unstructured regular triangulation. A simple convergence analysis is given without resorting to the discrete maximum principle. Numerical examples are also presented.

This work deals with a non linear inverse problem of reconstructingan unknown boundary γ, the boundary conditions prescribed on γ being of Signorini type,by using boundary measurements. The problem is turned into an optimal shape design one, by constructinga Kohn & Vogelius-like cost function, the only minimum of which is proved to be the unknown boundary.Furthermore, we prove that the derivative of this cost function with respect to a direction θ depends only on the state u 0, and not on its Lagrangian derivative u1(θ).