We introduce a new stable MINI-element pair for incompressible Stokes equations onquadrilateral meshes, which uses the smallest number of bubbles for the velocity. Thepressure is discretized with the P1-midpoint-edge-continuous elements and each component of the velocity field is done withthe standard Q1-conforming elements enriched byone bubble a quadrilateral. The superconvergence in the pressure of the proposed pair isanalyzed on uniform rectangular meshes, and tested numerically on uniform and non-uniformmeshes.

The aim of this work is to introduce and to analyze new algorithms for solving the transport neutronique equation in 2D geometry. These algorithms present the duplicate favors to be, on the one hand faster than some classic algorithms and easily to be implemented and naturally deviced for parallelisation on the other hand. They are based on a splitting of the collision operator holding amount of caracteristics of the transport operator. Some numerical results are given at the end of this work.

The electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable for the full 3D case.

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 aim of this paper is to present how to make a dedicaded computed language polymorphic and multi type, in C++ to solve partial differential equations with the finite element method.The driving idea is to make the language as close as possible to the mathematical notation.

The numerical modeling of the fully developed Poiseuille flow of a Newtonian fluid in a square section with slip yield boundary condition at the wall is presented. The stick regions in outer corners and the slip region in the center of the pipe faces are exhibited. Numerical computations cover the complete range of the dimensionless number describing the slip yield effect, from a full slip to a full stick flow regime. The resolution of variational inequalities describing the flow is based on the augmented Lagrangian method and a finite element method. The localization of the stick-slip transition points is approximated by an anisotropic auto-adaptive mesh procedure. The singular behavior of the solution at the neighborhood of the stick-slip transition point is investigated.

This paper considers Schrödinger operators, and presents a probabilistic interpretation of the variation (or shape derivative) of the Dirichlet groundstate energy when the associated domain is perturbed. This interpretation relies on the distribution on the boundary of a stopped random process with Feynman-Kac weights. Practical computations require in addition the explicit approximation of the normal derivative of the groundstate on the boundary. We then propose to use this formulation in the case of the so-called fixed node approximation of Fermion groundstates, defined by the bottom eigenelements of the Schrödinger operator of a Fermionic system with Dirichlet conditions on the nodes (the set of zeros) of an initially guessed skew-symmetric function. We show that shape derivatives of the fixed node energy vanishes if and only if either (i) the distribution on the nodes of the stopped random process is symmetric; or (ii) the nodes are exactly the zeros of a skew-symmetric eigenfunction of the operator. We propose an approximation of the shape derivative of the fixed node energy that can be computed with a Monte-Carlo algorithm, which can be referred to as Nodal Monte-Carlo (NMC). The latter approximation of the shape derivative also vanishes if and only if either (i) or (ii) holds.

A multilevel preconditioner based on the abstract framework of theauxiliary space method, is developed for the mortar method for thenonconforming P 1 finite element or the lowest orderCrouzeix-Raviart finite element on nonmatching grids. It is shownthat the proposed preconditioner is quasi-optimal in the sense thatthe condition number of the preconditioned system is independent ofthe mesh size, and depends only quadratically on the number ofrefinement levels. Some numerical results confirming the theory arealso provided.

Our studies are motivated by a desire to model long-time simulations of possible scenarios for a waste disposal. Numerical methods aredeveloped for solving the arising systems of convection-diffusion-dispersion-reactionequations, and the received results of several discretizationmethods are presented. We concentrate on linear reaction systems, which can be solved analytically.In the numerical methods, we use large time-steps to achievelong simulation times of about 10 000 years.We propose higher-order discretization methods, which allow us to use large time-steps without losing accuracy.By decoupling of a multi-physical and multi-dimensional equation,simpler physical and one-dimensional equations are obtained and can be discretized with higher-order methods. The results can then be coupled with an operator-splitting method.We discuss benchmark problems given in the literature and real-life applications.We simulate a radioactive waste disposals with underlying flowing groundwater.The transport and reaction simulations for the decay chains are presentedin 2d realistic domains, and we discuss the received results.Finally, we present our conclusions and ideas for further works.

Searching for the optimal partitioning of a domain leads to the use of the adjoint methodin topological asymptotic expansions to know the influence of a domain perturbation on acost function. Our approach works by restricting to local subproblems containing theperturbation and outperforms the adjoint method by providing approximations of higherorder. It is a universal tool, easily adapted to different kinds of real problems and doesnot need the fundamental solution of the problem; furthermore our approach allows toconsider finite perturbations and not infinitesimal ones. This paper provides theoreticaljustifications in the linear case and presents some applications with topologicalperturbations, continuous perturbations and mesh perturbations. This proposed approach canalso be used to update the solution of singularly perturbed problems.

Three non-overlapping domain decomposition methods are proposed for thenumericalapproximation of time-harmonic Maxwell equations with damping (i.e., in a conductor). Foreach method convergence is proved and, for the discrete problem, the rate ofconvergenceof the iterative algorithm is shown to be independent of the number ofdegrees of freedom.

We consider a domain decomposition method for some unsteadyheat conduction problem in composite structures.This linear model problem is obtained by homogenization of thin layersof fibres embedded into some standard material.For ease of presentation we consider the case of two space dimensions only.The set of finite element equations obtained by the backward Euler schemeis parallelized in a problem-oriented fashion by some noniterative overlappingdomain splitting method,eventually enhanced by inexpensive local iterationsto reduce the overlap.We present a detailed convergence analysis of this algorithmwhich is particularly well appropriate to handle fibre layersof nonlinear material.Special emphasis is to take into account the specific regularity propertiesof the present mathematical model.Numerical experiments show the reliability of the theoretical predictions.

Using the approach in [5] for analysingtime discretization error and assuming more regularity on the initial data, we improve onthe error bound derived in [2] for a fully practical piecewise linear finite element approximation with a backward Euler time discretization of a model for phase separation of a multi-component alloy withnon-smooth free energy.

The superconsistent collocation method, which is based on acollocation grid different from the one used to represent thesolution, has proven to be very accurate in the resolution ofvarious functional equations. Excellent results can be alsoobtained for what concerns preconditioning. Some analysis andnumerous experiments, regarding the use of finite-differencespreconditioners, for matrices arising from pseudospectralapproximations of advection-diffusion boundary value problems, arepresented and discussed, both in the case of Legendre andChebyshev representation nodes.

We analyze an isoparametric finite element method to compute the vibration modes of a plate, modeled by Reissner-Mindlin equations, in contact with a compressible fluid, described in terms of displacement variables. To avoid locking in the plate, we consider a low-order method of the so called MITC (Mixed Interpolation of Tensorial Component) family on quadrilateral meshes. To avoid spurious modes in the fluid, we use a low-order hexahedral Raviart-Thomas elements and a non conforming coupling is used on the fluid-structure interface. Applying a general approximation theory for spectral problems, under mild assumptions, we obtain optimal order error estimates for the computed eigenfunctions, as well as a double order for the eigenvalues. These estimates are valid with constants independent of the plate thickness. Finally, we report several numerical experiments showing the behavior of the methods.

We introduce in this paper some elements for the mathematical and numerical analysis of algebraic turbulence models foroceanic surface mixing layers. In these models the turbulent diffusions are parameterized by means of the gradient Richardson number, that measures the balance between stabilizing buoyancy forces and destabilizing shearing forces. We analyze the existence and linear exponential asymptotic stability of continuous and discrete equilibria states. We also analyze the well-posedness of a simplified model, by application of the linearization principle for non-linear parabolic equations. We finally present some numerical tests for realistic flows in tropical seas that reproduce the formation of mixing layers in time scales of the order of days, in agreement with the physics of the problem. We conclude that the typical mixing layers are transient effects due to the variability of equatorial winds. Also, that these states evolve to steady states in time scales of the order of years, under negative surface energy flux conditions.

We investigate the approximationof the evolution of compact hypersurfaces of $\mathbb{R}^N$ depending, not only on terms of curvature of the surface, but alsoon non local terms such as the measure of the set enclosedby the surface.

In this work, we analyze hierarchic hp-finite element discretizations of the full, three-dimensional plate problem. Based on two-scale asymptotic expansion of the three-dimensional solution, we give specific mesh design principles for the hp-FEM which allow to resolve the three-dimensional boundary layer profiles at robust, exponential rate.We prove that, as the plate half-thickness ε tends to zero, the hp-discretization is consistent with the three-dimensional solution to any power of ε in the energy norm for the degree $p={\cal O}(\left|{\log \varepsilon}\right|)$ and with ${\cal O}({p^4})$ degrees of freedom.

We present here a discretization of a nonlinear obliquederivative boundary value problem for the heat equation in dimensiontwo. This finite difference scheme takes advantages of thestructure of the boundary condition, which can be reinterpreted as aBurgers equation in the space variables. This enables to obtain anenergy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of thisproblem and a numerical study of the stability of the scheme, whichappears to be in good agreement with the theory.

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.