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.

Bhattacharya and Kohn have used small-strain (geometrically linear)elasticity to analyze the recoverable strains of shape-memory polycrystals.The adequacy of small-strain theory is open to question, however, since someshape-memory materials recover as much as 10 percent strain. This paperprovides the first progress toward an analogous geometrically nonlineartheory. We consider a model problem, involving polycrystals madefrom a two-variant elastic material in two space dimensions. The lineartheory predicts that a polycrystal with sufficient symmetry can have norecoverable strain. The nonlinear theory corrects this to the statement thata polycrystal with sufficient symmetry can have recoverable strain nolarger than the 3/2 power of the transformation strain. This result isin a certain sense optimal. Our analysis makes use of Fritz John'stheory of deformations with uniformly small strain.

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 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.

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 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.

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.

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.

In this article, we prove convergence of the weakly penalized adaptive discontinuousGalerkin methods. Unlike other works, we derive the contraction property for variousdiscontinuous Galerkin methods only assuming the stabilizing parameters are large enoughto stabilize the method. A central idea in the analysis is to construct an auxiliarysolution from the discontinuous Galerkin solution by a simple post processing. Based onthe auxiliary solution, we define the adaptive algorithm which guides to the convergenceof adaptive discontinuous Galerkin methods.

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.

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 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.

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.

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 -∞.

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).

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.

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.