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.

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.

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 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.

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.

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.

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.

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.

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 mathematical theory of the passage from compressible to incompressible fluid flow is reviewed.

The purpose of this work is to study an example of low Mach (Froude) number limit of compressible flows when the initial density (height) is almost equal to a function depending on x.This allows us to connect the viscous shallow water equation and the viscous lake equations.More precisely, we study this asymptotic with well prepared data in a periodic domain looking at the influence of the variability of the depth. The result concerns weak solutions.In a second part, we discuss the general low Mach number limit for standard compressible flows given in P.–L. Lions' book that means with constant viscosity coefficients.

In the second part of the paper, we compare the solutions producedin the framework of the conference “Mathematical and numericalaspects of low Mach number flows” organized by INRIA and MAB inPorquerolles, June 2004, to the reference solutions described inPart 1. We make some recommendations on how to produce goodquality solutions, and list a number of pitfalls to be avoided.

This paper is devoted to the numerical simulation of wavebreaking. It presents the results of a numerical workshop that washeld during the conference LOMA04. The objective is to compareseveral mathematical models (compressible or incompressible) andassociated numerical methods to compute the flow field during awave breaking over a reef. The methods will also be compared withexperiments.

Preconditioners for hyperbolic systems are numerical artifacts to accelerate the convergence to a steady state.In addition, the preconditioner should also be included in the artificial viscosity or upwinding terms to improve the accuracy of the steady state solution. For time dependent problemswe use a dual time stepping approach. The preconditioner affects the convergence rate and the accuracy of the subiterations within each physical time step. We considertwo types of local preconditioners:Jacobi and low speed preconditioning.We can express the algorithm in several sets of variableswhile using only the conservation variables for the flux terms.We compare the effect of these various variable setson the efficiency and accuracy of the scheme.

The first part of this paper reviews the single time scale/multiplelength scale low Mach number asymptotic analysis by Klein (1995, 2004). This theory explicitly reveals the interaction of small scale,quasi-incompressible variable density flows with long wave linearacoustic modes through baroclinic vorticity generation and asymptoticaccumulation of large scale energy fluxes. The theory is motivated byexamples from thermoacoustics and combustion. In an almost obvious way specializations of this theory to a singlespatial scale reproduce automatically the zero Mach number variabledensity flow equations for the small scales, and the linear acousticequations with spatially varying speed of sound for the large scales. Following the same line of thought we show how a large number ofwell-known simplified equations of theoretical meteorology canbe derived in a unified fashion directly from the three-dimensionalcompressible flow equations through systematic (low Mach number) asymptotics. Atmospheric flows are, however, characterized by several singularperturbation parameters that appear in addition to the Mach number,and that are defined independently of any particular length or timescale associated with some specific flow phenomenon. These are theratio of the centripetal acceleration due to the earth's rotation vs.the acceleration of gravity, and the ratio of the sound speed vs. therotational velocity of points on the equator. To systematicallyincorporate these parameters in an asymptotic approach, we couple themwith the square root of the Mach number in a particular distinguished sothat we are left with a single small asymptotic expansion parameter,ε. Of course, more familiar parameters, such as the Rossby andFroude numbers may then be expressed in terms of ε as well. Next we consider a very general asymptotic ansatz involvingmultiple horizontal and vertical as well as multiple time scales.Various restrictions of the general ansatz to only one horizontal, onevertical, and one time scale lead directly to the family of simplifiedmodel equations mentioned above. Of course, the main purpose of the general multiple scales ansatz isto provide the means to derive true multiscale models which describeinteractions between the various phenomena described by the members ofthe simplified model family. In this context we will summarize a recentsystematic development of multiscale models for the tropics (with Majda).

We propose a Diphasic Low Mach Number (DLMN) system for the modelling of diphasic flows without phase change at low Mach number, system which is an extension of the system proposed by Majda in [Center of Pure and Applied Mathematics, Berkeley, report No. 112] and [Combust. Sci. Tech.42 (1985) 185–205] for low Mach number combustion problems. This system is written for a priori any equations of state. Under minimal thermodynamic hypothesis which are satisfied by a large class of generalized van der Waals equations of state, we recover some natural properties related to the dilation and to the compression of bubbles. We also propose an entropic numerical scheme in Lagrangian coordinates when the geometry is monodimensional and when the two fluids are perfect gases. At last, we numerically show that the DLMN system may become ill-posed when the entropy of one of the two fluids is not a convex function.

The calculation of sound generation and propagation in low Mach number flows requires serious reflections on the characteristics of the underlying equations. Although the compressible Euler/Navier-Stokes equations cover all effects, an approximation via standard compressible solvers does not have the ability to represent acoustic waves correctly. Therefore, different methods have been developed to deal with the problem. In this paper, three of them are considered and compared to each other. They are the Multiple Pressure Variables Approach (MPV), the Expansion about Incompressible Flow (EIF) and a coupling method via heterogeneous domain decomposition. In the latter approach, the non-linear Euler equations are used in a domain as small as possible to cover the sound generation, and the locally linearized Euler equations approximated with a high-order scheme are used in a second domain to deal with the sound propagation. Comparisons will be given in constructionprinciples as well as implementational effort and computational costs on actual numerical examples.