In this work we consider a solid body $\Omega\subset{\Bbb R}^3$ constituted by anonhomogeneous elastoplastic material, submitted to a density of body forces $\lambda f $ and a density of forces $\lambda g$ acting on the boundary where the real $\lambda $ is theloading parameter.The problem is to determine, in the case of an unbounded convex of elasticity, the Limitload denoted by $\bar{\lambda}$ beyond which there is a break of the structure. The case of a bounded convex of elasticity is done in [El-Fekih and Hadhri, RAIRO: Modél. Math. Anal. Numér. 29 (1995) 391–419]. Then assuming that the convex of elasticity at the point x of Ω, denotedby K(x), is written in the form of $\mbox{K}^D (x) + {\BbbR}\mbox{I}$ , I is the identity of ${{\Bbb R}^9}_{sym}$ , and thedeviatoric component $\mbox{K}^D$ is bounded regardless of x $\in\Omega$ , we show under the condition “Rot f $\not=0$ or g is not colinear to the normal on a part of the boundary of Ω", that theLimit Load $\bar{\lambda}$ searched is equal to the inverse ofthe infimum of the gauge of the Elastic convex translated bystress field equilibrating the unitary load corresponding to $\lambda =1$ ; moreover we show that this infimum is reached in asuitable function space.

We study the theoretical and numericalcoupling of two hyperbolic systems of conservation laws at a fixed interface. As already proven in the scalar case, the couplingpreserves in a weak sense the continuity of the solution at the interfacewithout imposing the overall conservativity of the coupled model. We develop a detailed analysis of the coupling inthe linear case. In the nonlinear case, we either use a linearized approach or a coupling method based on the solution of a Riemann problem. We discuss both approaches in the case of the coupling of two fluid models at a material contact discontinuity, the models being the usual gas dynamics equations with different equations ofstate. We also study the coupling of two-temperature plasma fluid models and illustrate the approach by numericalsimulations.

Tuning the alternating Schwarz method to theexterior problems is the subject of this paper.We present the original algorithmand we propose a modification of it, so that the solution of the subproblem involving the condition at infinityhas an explicit integral representation formulas while the solutionof the other subproblem, set in a bounded domain,is approximated by classical variational methods. We investigate many of the advantages of the newSchwarz approach: a geometrical convergence rate,an easy implementation, a substantial economyin computational costs anda satisfactory accuracy in the numerical resultsas well as their agreement with the theoretical statements.

In this paper, we are concerned with a kind of Signorinitransmission problem in a unbounded domain. A variationalinequality is derived when discretizing this problem by coupledFEM-BEM. To solve such variational inequality, an iterativemethod, which can be viewed as a variant of the D-N alternativemethod, will be introduced. In the iterative method, the finiteelement part and the boundary element part can be solvedindependently. It will be shown that the convergence speed of thisiteration is independent of the mesh size. Besides, a combinationbetween this method and the steepest descent method is alsodiscussed.

We present and analyze an interior penalty method for the numerical discretization of the indefinitetime-harmonic Maxwell equations in mixed form.The method is based on the mixed discretization of the curl-curl operator developedin [Houston et al.,J. Sci. Comp.22 (2005) 325–356]and can be understood as a non-stabilized variantof the approach proposed in [Perugia et al.,Comput. Methods Appl. Mech. Engrg.191 (2002) 4675–4697].We show the well-posedness of this approach andderive optimal a priori error estimates in the energy-normas well as the L 2-norm. The theoretical results areconfirmed in a series of numerical experiments.

Finite element approximation for degenerate parabolic equations is considered. We propose a semidiscrete scheme provided with order-preserving and L 1 contraction properties, making use of piecewise linear trial functions and the lumping mass technique. Those properties allow us to apply nonlinear semigroup theory, and the wellposedness and stability in L 1 and L ∞, respectively, of the scheme are established. Under certain hypotheses on the data, we also derive L 1 convergence without any convergence rate.The validity of theoretical results is confirmed by numerical examples.

We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the L2-gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute numerically these solutions by a moving-mesh method which allows us to deal with the singularity at the origin. We show numerical evidence of the convergence of the method.

In order to describe a solid which deforms smoothly in some region, butnon smoothly in some other region, many multiscale methods have recentlybeen proposed. They aim at coupling an atomistic model (discretemechanics) with a macroscopic model (continuum mechanics).We provide here a theoretical ground for such a coupling in aone-dimensional setting. We briefly study the general case of a convexenergy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case. In thelatter situation, we prove that the discretization needs to account inan adequate way for the coexistence of a discrete model and a continuousone. Otherwise, spurious discretization effects may appear.We provide a numerical analysis of the approach.

The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented Lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas.