In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive a priori and a posteriori error estimates. We present some numerical experiments that are in good agreement with the analysis.

We study a time-delay regularization of the anisotropic diffusion model for image denoising of Perona and Malik [IEEE Trans. Pattern Anal. Mach. Intell12 (1990) 629–639], which has been proposed by Nitzberg and Shiota [IEEE Trans. Pattern Anal. Mach. Intell14 (1998) 826–835]. In the two-dimensional case, we show the convergence of a numerical approximation and the existence of a weak solution. Finally, we show some experiments on images.

The chronotherapy concept takes advantage of the circadian rhythm of cells physiology in maximising a treatment efficacy on its target while minimising its toxicity on healthy organs. The object of the present paper is to investigate mathematically and numerically optimal strategies in cancer chronotherapy. To this end a mathematical model describing the time evolution of efficiency and toxicity of an oxaliplatin anti-tumour treatment has been derived. We then applied an optimal control technique to search for the best drug infusion laws.The mathematical model is a set of six coupled differentialequations governing the time evolution of both the tumour cell population(cells of Glasgow osteosarcoma, a mouse tumour) and the mature jejunalenterocyte population, to be shielded from unwanted side effectsduring a treatment by oxaliplatin. Starting from known tumour and villi populations, and a time dependent freeplatinum Pt (the active drug) infusion law being given,the mathematical model allows to compute the time evolution of both tumour andvilli populations. The tumour population growth is based on Gompertz law and the Pt anti-tumour efficacy takes into account the circadian rhythm. Similarly the enterocyte population is subject to a circadian toxicityrhythm. The model has been derived using, as far as possible, experimental data.We examine two different optimisation problems. The eradication problem consists in finding the drug infusion law able to minimise the number of tumour cells while preserving a minimal level for the villi population. On the other hand, the containment problem searches for a quasi periodic treatment able to maintain the tumour population at the lowest possible level, while preserving the villi cells. The originality of these approaches is that the objective and constraint functions we use are L ∞ criteria. We are able to derive their gradients with respect to the infusion rate and then to implement efficient optimisation algorithms.

This article discusses the numerical approximation oftime dependent Ginzburg-Landau equations. Optimal error estimates which are robust with respectto a large Ginzburg-Landau parameter are established for asemi-discrete in time and a fully discrete approximationscheme. The proofs rely on an asymptotic expansion of the exact solution and a stability resultfor degree-one Ginzburg-Landau vortices. The error boundsprove that degree-one vortices can be approximated robustlywhile unstable higher degree vortices are critical.

In this work we consider a solid body $\Omega\subset{\Bbb R}^3$ constituted by a nonhomogeneous 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 the loading parameter. The problem is to determine, in the case of an unbounded convex of elasticity, the Limit load 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, denoted by K(x), is written in the form of $\mbox{K}^D (x) + {\Bbb R}\mbox{I}$, I is the identity of ${{\Bbb R}^9}_{sym}$, and the deviatoric 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 the Limit Load $\bar{\lambda}$ searched is equal to the inverse of the infimum of the gauge of the Elastic convex translated by stress field equilibrating the unitary load corresponding to $\lambda =1$; moreover we show that this infimum is reached in a suitable function space.

In this paper we present a methodology for constructing accurateand efficient hybrid central-upwind (HCU) type schemes forthe numerical resolution of a two-fluid model commonly used by thenuclear and petroleum industry. Particularly, we propose a methodwhich does not make use of any information about theeigenstructure of the Jacobian matrix of the model. The two-fluid model possesses a highly nonlinear pressure law. From the mass conservation equations we develop an evolutionequation which describes how pressure evolves in time. By applyinga quasi-staggered Lax-Friedrichs type discretization for thispressure equation together with a Modified Lax-Friedrich typediscretization of the convective terms, we obtain a central typescheme which allows to cope with the nonlinearity (nonlinearpressure waves) of the two-fluid model in a robust manner.Then, in order to obtain an accurate resolution of mass fronts, weemploy a modification of the convective mass fluxes by hybridizingthe central type mass flux components with upwind type components.This hybridization is based on a splitting of the mass fluxes intocomponents corresponding to the pressure and volume fractionvariables, recovering an accurate resolution of a contactdiscontinuity. In the numerical simulations, the resulting HCU scheme givesresults comparable to an approximate Riemann solver while beingsuperior in efficiency. Furthermore, the HCU scheme yields betterrobustness than other popular Riemann-free upwind schemes.

We consider the classical Interpolating Moving Least Squares (IMLS)interpolant as defined by Lancaster and Šalkauskas [Math. Comp.37 (1981) 141–158] andcompute the first and second derivative of this interpolant at the nodes of agiven grid with the help of a basic lemma on Shepard interpolants. We comparethe difference formulae with those defining optimal finite difference methods anddiscuss their deviation from optimality.

We describe a constructive algorithm for obtaining smoothsolutions of a nonlinear, nonhyperbolic pair of balance lawsmodeling incompressible two-phase flow in one space dimension andtime. Solutions are found as stationary solutions of a relatedhyperbolic system, based on the introduction of an artificial timevariable.As may be expected for such nonhyperbolic systems, in general thesolutions obtained do not satisfy both components of the giveninitial data. This deficiency may be overcome, however, byintroducing an alternative “solution" satisfying both componentsof the initial data and an approximate form of a correspondinglinearized system.

In this work we consider the dual-primal Discontinuous Petrov–Galerkin (DPG) method for the advection-diffusion model problem. Since in the DPG method both mixed internal variables are discontinuous,a static condensation procedure can becarried out, leading to a single-field nonconformingdiscretization scheme. For this latter formulation,we propose a flux-upwind stabilization technique to deal with the advection-dominated case. The resulting scheme is conservative and satisfies a discrete maximum principle under standard geometrical assumptions on the computational grid. A convergence analysis isdeveloped, proving first-order accuracy of the method in a discrete H 1-norm, and the numerical performance of the scheme is validated on benchmark problems with sharp internal and boundary layers.

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.

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

A simple dynamical problem involving unilateral contact and dry friction of Coulomb type is considered as an archetype. We are concerned with the existence and uniqueness of solutions of the system with Cauchy data. In the frictionless case, it is known [Schatzman, Nonlinear Anal. Theory, Methods Appl.2 (1978) 355–373] that pathologies of non-uniqueness can exist, even if all the data are of class C ∞. However, uniqueness is recovered provided that the data are analytic [Ballard, Arch. Rational Mech. Anal.154 (2000) 199–274]. Under this analyticity hypothesis, we prove theexistence and uniqueness of solutions for the dynamical problem with unilateral contact and Coulomb friction, extending [Ballard, Arch. Rational Mech. Anal.154 (2000) 199–274] to the case where Coulomb friction is added to unilateral contact.

We address here mathematical models related to the Laser-Plasma Interaction. After a simplified introduction to the physical background concerning the modelling of the laser propagation and its interaction with a plasma, we recall some classical results about the geometrical optics in plasmas. Then we deal with the well known paraxial approximation of the solution of the Maxwell equation; we state a coupling model between the plasma hydrodynamics and the laser propagation. Lastly, we consider the coupling with the ion acoustic waves which has to be taken into account to model the so called Brillouin instability. Here, besides the macroscopic density and the velocity of the plasma, one has to handle the space-time envelope of the main laser wave, the space-time envelope of the stimulated Brillouin backscattered laser wave and the space envelope of the Brillouin ion acoustic waves. Numerical methods are also described to deal with the paraxial model and the three-wave coupling system related to the Brillouin instability.

In this survey paper, we are concerned with the zero Mach number limitfor compressible viscous flows. For the sake of (mathematical) simplicity, we restrict ourselves to the case of barotropicfluids and we assume that the flow evolves in the whole space or satisfies periodic boundary conditions. We focus on the case of ill-prepared data. Hence highly oscillating acoustic waves are likely to propagate through the fluid. We nevertheless statethe convergence to the incompressible Navier-Stokes equations when the Mach number ϵ goes to 0. Besides, it is shown that the global existence for the limit equations entails the global existence for the compressible model with small ϵ. The reader is referred to [R. Danchin, Ann. Sci. Éc. Norm. Sup. (2002)] for the detailed proof in the whole space case, and to [R. Danchin, Am. J. Math.124 (2002) 1153–1219] for the case of periodic boundary conditions.

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

When analysing general systems of PDEs, it is important first to find the involutive form of the initial system.This is because the properties of the system cannot in general be determined if the system is not involutive.We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive formof the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution ofseveral flow equations with the aim of showing the impact of the involutive form of the systems in simplifying numerical schemes.

In this paper we solve the time-dependent incompressible Navier-Stokesequations by splitting the non-linearity and incompressibility, andusing discontinuous or continuous finite element methods in space. Weprove optimal error estimates for the velocity and suboptimalestimates for the pressure. We present some numerical experiments.

A Discontinuous Galerkin method is used for to thenumerical solution of the time-domain Maxwell equations onunstructured meshes. The method relies on the choice of local basisfunctions, a centered mean approximation for the surface integralsand a second-order leap-frog scheme for advancing in time. The methodis proved to be stable for cases with either metallic or absorbingboundary conditions, for a large class of basis functions. A discrete analog of the electromagnetic energy is conserved formetallic cavities. Convergence is proved for $\mathbb{P}_k$ Discontinuous elements on tetrahedral meshes, as well as a discretedivergence preservation property. Promising numerical examples withlow-order elements show the potential of the method.

The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equationin $\mathbb{R}^d$ , d=2 or 3,using backward Euler's scheme. For this discretization, we derive a residual indicator, which usea spatial residual indicator based on thejumps of normal and tangential derivatives of the nonconformingapproximation and a time residual indicator based on the jump of broken gradients at each time step.Lower and upper bounds form the main results with minimal assumptions on the mesh.Numerical experiments and a space-time adaptive algorithm confirm the theoretical predictions.