The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly taken into account when the first family of mixed finite elements is used. We, therefore, introduce the second family of mixed finite elements for which a theoretical convergence analysis is presented and error estimates are obtained. A numerical study of the convergence is also considered for a particular object geometry which shows that our theoretical error estimates are optimal.

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, ona solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants onboth the free liquid-liquid and liquid-air interfaces,and the presence of both attractive and repulsive van der Waals forcesin terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure,and a second energy inequality controlling the Laplacianof the liquid heights.We introduce a fully practical finite element approximation of this nonlinear degenerate parabolic system, that satisfies discrete analoguesof these energy inequalities. Finally, we prove convergence of this approximation,and hence existence of a solutionto this nonlinear degenerate parabolic system.

We introduce and analyse a mixed formulation of theMonge-Kantorovich equations, which express optimality conditions forthe mass transportation problem with cost proportional to distance.Furthermore, we introduce and analyse the finite elementapproximation of this formulation using the lowest orderRaviart-Thomas element. Finally, we present some numericalexperiments, where both the optimal transport density and theassociated Kantorovich potential are computed for a coupling problemand problems involving obstacles and regions of cheaptransportation.

We present the convergence analysis of locally divergence-free discontinuous Galerkin methodsfor the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition $\Delta t\sim h^{4/3}$ , we obtain error estimates in L 2 of order $\mathcal{O} (\Delta t^2 + h^{m + 1/2})$ where m is the degree of the local polynomials.

We consider a model for phase separation of a multi-component alloy with non-smooth free energyand a degenerate mobility matrix. In addition to showingwell-posedness and stability bounds for our approximation, we prove convergence in one space dimension. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. We discuss also how our approximation has to be modified in orderto be applicable to a logarithmic free energy.Finally numerical experiments with three components in one and two space dimensions are presented.

In a recent paper [4] we have proposed and analyseda suitable mathematical modelwhich describes the coupling of the Navier-Stokes with theOseen equations.In this paper we propose a numerical solution of the coupledproblem by subdomain splitting.After a preliminary analysis, we prove a convergence result foran iterative algorithm that alternates the solution of the Navier-Stokes problem to the one of the Oseen problem.

The present work is a mathematical analysis of two algorithms, namelythe Roothaan and the level-shifting algorithms, commonly used inpractice to solve the Hartree-Fock equations. The level-shiftingalgorithm is proved to be well-posed and to converge provided the shiftparameter is large enough. On the contrary, cases when the Roothaanalgorithm is not well defined or fails in converging areexhibited. These mathematical results are confronted to numericalexperiments performed by chemists.

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.

Error estimates for the mixed finite element solution of4th order elliptic problems with variable coefficients, which, in the particular case of aniso-/ortho-/isotropic plate bending problems, gives a direct, simultaneous approximation to bending moment tensorfield $\Psi= (\psi_{ij})_{1 \le i,j \le 2}$ and displacement field 'u', have been developed considering the combined effect of boundary approximation and numerical integration.