Our aim here is to study the thermal diffusion phenomenon in a forced convective flow. A system of nonlinear parabolic equations governs the evolution of the mass fractions in multicomponent mixtures. Some existence and uniqueness results are given under suitable conditions onstate functions. Then, we present a numerical scheme based on a "mixed finite element"method adapted to a finite volume scheme, of which we give numerical analysis. In a last part, we apply an homogenization technique to the studied equations in order to obtain an efficient modelling of Soret effect and adsorption in a porous medium at a macroscopic scale.

We propose, analyze, and compare several numerical methods for the computation of the deformation of a pressurized martensitic thin film. Numerical results have been obtained for the hysteresis of the deformation as the film transforms reversibly from austenite to martensite.

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.

In this paper, we study how solutions to elliptic problems withperiodically oscillating coefficients behave inthe neighborhood of the boundary of a domain. We extend theresults known for flat boundaries to domains with curved boundariesin the case of a layered medium. This is done by generalizing thenotion of boundary layer and by defining boundary correctors whichlead to an approximation of order ε in the energy norm.

The phenomenon of roll waves occurs in a uniform open-channelflow down an incline, when the Froude number is above two. The goal of this paper is to analyze the behavior of numerical approximations to a model roll wave equation ut + uux = u,u(x,0) = u0(x), which arises as a weakly nonlinear approximation of the shallow waterequations. The main difficulty associated with the numerical approximation ofthis problem is its linear instability. Numerical round-off errorcan easily overtake the numerical solution and yields false roll wavesolution at the steady state.In this paper, we first study the analytic behavior of the solution to the abovemodel. We then discuss the numerical difficulty, and introduce a numericalmethod that predicts precisely the evolution and steady state of itssolution. Various numerical experiments are performed to illustratethe numerical difficulty and the effectiveness of the proposed numericalmethod.

We discretize the nonlinear Schrödinger equation,with Dirichlet boundary conditions, by a linearlyimplicit two-step finite element method which conserves the L 2 norm. We prove optimal order a priori error estimates in the L 2 and H 1 norms, under mild mesh conditions for two and three space dimensions.

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 paper we are interested in bilateral obstacle problems for quasilinear scalar conservation laws associated with Dirichlet boundary conditions. Firstly, we provide a suitable entropy formulation which ensures uniqueness. Then, we justify the existence of a solution through the method of penalization and by referring to the notion of entropy process solution due to specific properties of bounded sequences in L ∞. Lastly, we study the behaviour of this solution and its stability properties with respect to the associated obstacle functions.

In this paper we extend recent work on the detection of inclusions using electrostatic measurements to the problem of crack detection in a two-dimensional object. As in the inclusion case our method is based on a factorization of the difference between two Neumann-Dirichlet operators. The factorization possible in the case of cracks is much simpler than that for inclusions and the analysis is greatly simplified. However, the directional information carried by the crack makes the practical implementation of our algorithm more computationally demanding.

We present the combination of a state control and shape design approachesfor the optimization of micro-fluidic channels used for sample extraction andseparation of chemical species existing in a buffer solution.The aim is to improve the extraction and identification capacities of electroosmotic micro-fluidic devices by avoiding dispersion of the extracted advected band.

In this review we discuss bifurcation theory in a Banach space setting using the singularity theory developed by Golubitsky and Schaeffer to classify bifurcation points. The numerical analysis of bifurcation problems is discussed and the convergence theory for several important bifurcations is described for both projection and finite difference methods. These results are used to provide a convergence theory for the mixed finite element method applied to the steady incompressible Navier–Stokes equations. Numerical methods for the calculation of several common bifurcations are described and the performance of these methods is illustrated by application to several problems in fluid mechanics. A detailed description of the Taylor–Couette problem is given, and extensive numerical and experimental results are provided for comparison and discussion.

The Delaunay triangulation of a finite point set is a central theme in computational geometry. It finds its major application in the generation of meshes used in the simulation of physical processes. This paper connects the predominantly combinatorial work in classical computational geometry with the numerical interest in mesh generation. It focuses on the two- and three-dimensional case and covers results obtained during the twentieth century.

Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.

Radial basis function methods are modern ways to approximate multivariate functions, especially in the absence of grid data. They have been known, tested and analysed for several years now and many positive properties have been identified. This paper gives a selective but up-to-date survey of several recent developments that explains their usefulness from the theoretical point of view and contributes useful new classes of radial basis function. We consider particularly the new results on convergence rates of interpolation with radial basis functions, as well as some of the various achievements on approximation on spheres, and the efficient numerical computation of interpolants for very large sets of data. Several examples of useful applications are stated at the end of the paper.

This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation $c_t + \nabla \cdot ( {\bf u}f(c)) - \nabla \cdot (D \nabla c) + \lambda c = 0$ .The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L 1-norm,independent of the diffusion parameter D. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability of the theoretical results.

We deal with boundary layers and quasi-neutral limits in the drift-diffusion equations. We first show that this limit is unique and determined by a system of two decoupled equations with given initial and boundary conditions. Then we establish the boundary layer equations and prove the existence and uniqueness of solutions with exponential decay. This yields a globally strong convergence (with respect to the domain) of the sequence of solutions and an optimal convergence rate $O(\varepsilon^\frac{1}{2})$ to the quasi-neutral limit in L 2.

By using an inductive procedure we prove that the Galerkin finite element approximations of electromagnetic eigenproblems modelling cavity resonators by elements of any fixed order of either Nedelec's edge element family on tetrahedral meshes are convergent and free of spurious solutions. This result is not new but is proved under weaker hypotheses, which are fulfilled in most of engineering applications. The method of the proofis new, instead, and shows how families of spurious-freeelements can be systematically constructed. The tools here developed are used to define a new family of spurious-freeedge elements which, in some sense, are complementary to those defined in 1986 by Nedelec.

The paper deals with the application of a non-conforming domaindecomposition methodto the problem of the computation of induced currents in electric engineswith moving conductors.The eddy currents model is considered as a quasi-staticapproximation of Maxwellequations and we study its two-dimensional formulation with either themodified magnetic vector potential or the magnetic field as primary variable.Two discretizations are proposed, the first one based on curved finiteelementsand the second one based on iso-parametric finite elements in both thestatic and movingparts. The coupling is obtained by means of the mortar element method(see [CITE])and the approximation on the whole domain turns out to be non-conforming.In bothcases optimal error estimates are provided. Numerical tests are then proposed in the case of standard first order finiteelements to test the reliability and precision of the method. An applicationof the method to study the influence of the free part movement on thecurrents distribution is also provided.

We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a"rough"coefficient function k(x). We show that the Engquist-Osher (and hence all monotone)finite difference approximations convergeto the unique entropy solution of the governing equation if, among other demands, k' is in BV, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion equations as well as new convergence results for their finite difference approximations. In the inviscid case, we also provide a rate of convergence. Our convergence proofs are based on deriving a series of a priori estimates and using a general Lp compactness criterion.

We present a new methodology for the numerical resolution of the hydrodynamicsof incompressible viscid newtonian fluids. It is based on the Navier-Stokesequations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurationstypical to the motion of biological structures in viscous fluids.Although the method is applicable to three dimensions, we address herein detail only the two dimensional case. We provide numerical data forsome test cases to which we apply the computational scheme.