We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ut = Δu, vt = Δv in Ω x (0,T); fully coupled by the boundary conditions $\frac{\partial u}{\partial\eta} = u^{p_{11}}v^{p_{12}}$ , $\frac{\partial v}{\partial\eta} = u^{p_{21}}v^{p_{22}}$ on ∂Ω x (0,T), where Ω is a bounded smooth domain in ${\mathbb{R}}^d$ . We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation (U,V). We prove that if U blows up in finite time then V can fail to blow up if and only if p 11 > 1 and p 21 < 2(p 11 - 1), which is the same condition as the one for non-simultaneous blow-up in the continuous problem. Moreover, we find that if the continuous problem has non-simultaneous blow-up then the same is true for the discrete one. We also prove some results about the convergence of the scheme and the convergence of the blow-up times.

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.

Degenerate parabolic variational inequalities with convection are solved bymeans of a combined relaxation method and method of characteristics. Themathematical problem is motivated by Richard's equation, modelling theunsaturated – saturated flow in porous media. By means of the relaxationmethod we control the degeneracy. The dominance of the convection iscontrolled by the method of characteristics.

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.

Dorsey, Di Bartolo and Dolgert (Di Bartolo et al., 1996; 1997) have constructed asymptotic matched solutions at order two for the half-space Ginzburg-Landau model, in the weak-κ limit. These authors deduceda formal expansion for the superheating field in powers of $\kappa^{\frac{1}{2}}$ up to order four, extending the formula by De Gennes (De Gennes, 1966) and the two terms in Parr's formula (Parr, 1976). In this paper, we construct asymptotic matched solutions at all orders leading to a complete expansion in powers of $\kappa^{\frac{1}{2}}$ for the superheating field.

This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation illustrate the behavior of minimum effort controls.

This paper deals with the non-conservative coupling of two one-dimensional barotropic Euler systems at an interface at x = 0. The closure pressure laws differ in the domains x < 0 and x > 0, and a Dirac source term concentrated at x = 0 models singular pressure losses. We propose two numerical methods. The first one relies on ghost state reconstructions at the interface while the second is based on a suitable relaxation framework. Both methods satisfy a well-balanced property for stationary solutions. In addition, the second method preserves mass conservation and exactly restores the prescribed singular pressure drops for both unsteady and steady solutions.

This article is devoted to the presentation of a new contact algorithm for bodies undergoing finite deformations. We only address the kinematic aspect of the contact problem, that is the numerical treatment of the non-intersection constraint. In consequence, mechanical aspects like friction, adhesion or wear are not investigated and we restrict our analysis to the simplest frictionless case. On the other hand, our method allows us to treat contacts and self-contacts, thin or non-thin structures in a single setting.

We study preconditioning techniques for discontinuous Galerkin discretizations of isotropic linear elasticity problems in primal (displacement) formulation. We propose subspace correction methods based on a splitting of the vector valued piecewise linear discontinuous finite element space, that are optimal with respect to the mesh size and the Lamé parameters. The pure displacement, the mixed and the traction free problems are discussed in detail. We present a convergence analysis of the proposed preconditioners and include numerical examples that validate the theory and assess the performance of the preconditioners.

Stability and asymptotic stability of the solutions of impulsivenonlinear pa ra bo lic equationsare studied via the method of differential inequalities.

We consider a parabolic 2D Free Boundary Problem, with jump conditions at the interface. Its planar travelling-wave solutions are orbitally stableprovided the bifurcation parameter $u_*$ does not exceed a critical value $u_{*}^{c}$ . The latter isthe limit of a decreasing sequence $(u_{*}^{k})$ of bifurcation points. The paper deals with the study of the 2D bifurcated branchesfrom the planar branch, for small k. Our technique is based on the elimination of the unknown front, turning the problem into a fully nonlinear one, to which we canapply the Crandall-Rabinowitz bifurcation theorem for a local study.We point out that the fully nonlinear reformulation of the FBP can also serve to developefficient numerical schemes in view of global information, such as techniques based on arc length continuation.

In this work, the quasistatic thermoviscoelastic thermistor problem isconsidered. The thermistor model describes the combination of the effects due tothe heat, electrical current conduction and Joule's heat generation. The variationalformulation leads to a coupled system of nonlinear variational equations for whichthe existence of a weak solution is recalled.Then, a fully discrete algorithm is introduced based on the finite elementmethod to approximate the spatial variable and an Euler scheme to discretizethe time derivatives. Error estimates are derived and, under suitableregularity assumptions, the linear convergence of the scheme is deduced.Finally, some numerical simulations are performed in order to show the behaviourof the algorithm.

In this paper, we present some interesting connections between anumber of Riemann-solver free approaches to the numerical solutionof multi-dimensional systems of conservation laws. As a main part,we present a new and elementary derivation of Fey's Method ofTransport (MoT) (respectively the second author's ICE version ofthe scheme) and the state decompositions which form the basis of it.The only tools that we use are quadrature rules applied to themoment integral used in the gas kinetic derivation of the Eulerequations from the Boltzmann equation, to the integration in timealong characteristics and to space integrals occurring in the finitevolume formulation. Thus, we establish a connection between theMoT approach and the kinetic approach. Furthermore,Ostkamp's equivalence result between her evolution Galerkin schemeand the method of transport is lifted up from the level ofdiscretizations to the level of exact evolution operators,introducing a new connection between the MoT and theevolution Galerkin approach. At the same time, we clarifysome important differences between these two approaches.

We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C.Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006)717–724.; X. Blanc, C. Le Bris and P.-L. Lions, J. Math. Pures Appl.88 (2007) 34–63.]. The equation under consideration is a standardlinear elliptic equation in divergence form, where the highly oscillatory coefficient isthe composition of a periodic matrix with a stochastic diffeomorphism. The homogenizedlimit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions,C. R. Acad. Sci. Série I 343 (2006) 717–724.]. We firstestablish, in the one-dimensional case, a convergence result (with an explicit rate) onthe residual process, defined as the difference between the solution to the highlyoscillatory problem and the solution to the homogenized problem. We next return to themultidimensional situation. As often in random homogenization, the homogenized matrix isdefined from a so-called corrector function, which is the solution to a problem set on theentire space. We describe and prove the almost sure convergence of an approximationstrategy based on truncated versions of the corrector problem.

In this article, we wish to investigate the behavior of a two-layer k - εturbulence model from the mathematical point of view, as this model is useful for the near-wall treatment in numerical simulations.First, we explain the difficulties inherent in themodel. Then, we present a new variable θ that enables the mathematical study. Due to a problem of definition of the turbulentviscosity on the wall boundary, we consider an alternative version of the original equation. We show that some physical aspectsof the model are preserved by the new formulation, and in particular, we show how the physicists can help us to provethe existence of a solution of our problem. Finally, we are interested in the Navier-Stokes equations coupled with the modified turbulencemodel and we show that the alternative model may be preferred to the original one, because of its good properties(existence of a solution of the coupled problems).

In this paper we analyze the consistency, the accuracy and some entropy properties ofparticle methods with remeshing in the case of a scalar one-dimensional conservation law.As in [G.-H. Cottet and L. Weynans, C. R. Acad. Sci. Paris, Ser. I343 (2006) 51–56] we re-write particle methods with remeshing inthe finite-difference formalism. This allows us to prove the consistency of these methods,and accuracy properties related to the accuracy of interpolation kernels. Cottet and Magnidevised recently in [G.-H. Cottet and A. Magni, C. R. Acad. Sci. Paris, Ser. I347 (2009) 1367–1372] and [A. Magni and G.-H. Cottet, J.Comput. Phys. 231 (2012) 152–172] TVD remeshing schemes forparticle methods. We extend these results to the nonlinear case with arbitrary velocitysign. We present numerical results obtained with these new TVD particle methods for theEuler equations in the case of the Sod shock tube. Then we prove that with these new TVDremeshing schemes the particle methods converge toward the entropy solution of the scalarconservation law.

We describe both the classical Lagrangian and the Eulerian methods for first order Hamilton–Jacobi equations of geometric optic type. We then explain the basic structure of the softwareand how new solvers/models can be added to it. A selection of numerical examples are presented.

This paper is devoted to the spectral analysis of a non elliptic operator A , deriving from the study of superconducting micro-strip lines.Once a sufficient condition for the self-adjointness of operator A has been derived, we determine its continuous spectrum. Then, we show that A is unbounded from below and that it has a sequence of negative eigenvalues tending to -∞. Using the Min-Max principle, a characterization ofits positive eigenvalues is given. Thanks to this characterization, some conditions on the geometrical (large width) and physical (large dielectric permittivityin modulus) properties of the strip that ensure the existence of positive eigenvalues are derived. Finally, we analyze the asymptotic behavior of the eigenvaluesof A as the dielectric permittivity of the strip goes to -∞.

This special volume of the ESAIM Journal, Mathematical Modelling and Numerical Analysis,contains a collection of articles on probabilistic interpretations of some classes of nonlinear integro-differential equations.The selected contributions deal with a wide range of topics in applied probability theory and stochastic analysis, with applications in a variety of scientific disciplines, includingphysics, biology, fluid mechanics, molecular chemistry, financial mathematics and bayesian statistics. In this preface, we provide a brief presentation of the main contributions presented in this special volume. We have also included an introduction to classic probabilistic methods and a presentation of the more recent particle methods, with a synthetic picture of their mathematical foundations and their range of applications.

In the case of an elastic strip we exhibit two properties ofdispersion curves λn,n ≥ 1, that were not pointed outpreviously. We show cases where λ'n(0) = λ''n(0) = λ'''n(0) = 0 and we point out that these curves are not automatically monotoneous on ${\mathbb{R}}_{+}$ . The non monotonicity was an open question (see [2],for example) and, for the first time, we give a rigourous answer. Recall thecharacteristic property of the dispersion curves: {λn(p);n ≥ 1} isthe set of eigenvalues of A p , counted with their multiplicity. Theoperators Ap , $p\in{\mathbb{R}}$ , are the reduced operators deduced from the elasticoperator A using a partial Fourier transform. The second goal of this article is the introduction of a dispersion relationD(p,λ) = 0 in a general framework, and not only for a homogeneous situation(in this last case the relation is explicit). Recall that a dispersionrelation isan implicit equation the solutions of which are eigenvalues of A p . The mainproperty of the function D that we build is the following one: themultiplicity of an eigenvalue λ of A p is equal to the multiplicity ithas as a root of D(p,λ) = 0. We give also some applications.