A sparse algebraic multigrid method is studied as a cheap and accurateway to compute approximations of Schur complements of matricesarising from the discretization of some symmetric and positive definitepartial differential operators. The construction of such a multigrid isdiscussed and numerical experiments are used to verify the propertiesof the method.

We present a Gause type predator–prey model incorporating delay due to response of prey population growth to density and gestation. The functional response of predator is assumed to be of Holling type II. In absence of prey, predator has a density dependent death rate. Sufficient criterion for uniform persistence is derived. Conditions are found out for which system undergoes a Hopf–bifurcation.

We derive a constitutive law for the myocardium from the description of both the geometrical arrangement ofcardiomyocytes and their individual mechanical behaviour. We model a set of cardiomyocytes by a quasiperiodic discretelattice of elastic bars interacting by means of moments. We work in a large displacement framework and we use a discretehomogenization technique. The macroscopic constitutive law is obtained through the resolution of anonlinear self-equilibrum system of the discrete lattice reference cell.

We consider a family of shell finite elements with quadratic displacementsacross the thickness. These elements are very attractive, but compared to standard general shell elements they face another sourceof numerical locking in addition to shear and membrane locking. Thisadditional locking phenomenon – that we call “pinching locking” – is thesubject of this paper and we analyse a numerical strategy designed to overcomethis difficulty. Using a model problem in which only this specific sourceof locking is present, we are able to obtain error estimatesindependent of the thickness parameter, which shows that pinching lockingis effectively treated. This is also confirmed by some numerical experimentsof which we give an account.

In this paper we show how abstract physical requirements are enoughto characterize the classical collision kernels appearing in kinetic equations. In particular Boltzmann and Landau kernels are derived.

We establish an asymptotic representation formula for the steady state voltageperturbations caused by low volume fraction internal conductivityinhomogeneities. This formula generalizes and unifies earlierformulas derived for special geometries and distributions of inhomogeneities.

Our purpose is to estimate numerically the influence of particles on the global viscosity of fluid–particle mixtures. Particles are supposed to rigid, and the surrounding fluid is newtonian. The motion of the mixture is computed directly, i.e. all the particle motions are computed explicitly. Apparent viscosity, based on the force exerted by the fluid on the sliding walls, is computed at each time step of the simulation.In order to perform long–time simulations and still control the solid fraction, we assume periodicity of the flow in the shear direction.Direct simulations are based on the so–called Arbitrary Lagrangian Eulerian approach, which we adapted to make it suitable to periodic domains.As a first step toward modelling of interacting red cells in the blood, we propose a simple model of circular particles submitted to an attractive force which tends to form aggregates.

The main goal of this article is to establish a priori and a posteriori error estimates for the numerical approximation of some non linear ellipticproblems arising in glaciology. The stationary motion of a glacier is givenby a non-Newtonian fluid flow model which becomes, in a firsttwo-dimensional approximation, the so-called infinite parallel sided slabmodel. The approximation of this model is made by a finite element methodwith piecewise polynomial functions of degree 1. Numerical results show thatthe theoretical results we have obtained are almost optimal.

We consider a simple model for the immune systemin which virus are able to undergo mutations and are in competitionwith leukocytes. These mutations are related to several other concepts which havebeen proposed in the literature like those of shape or ofvirulence – a continuous notion. For a given species, the system admits aglobally attractive critical point. We prove that mutations do not affect thispicture for small perturbations and under strong structural assumptions.Based on numerical and theoretical arguments, we also examine how, releasing these assumptions, the system can blow-up.

The present paper deals with a finite element approximation of partial differential equations when the domain is decomposed into sub-domains which are meshed independently. The method we obtain is never conforming because the continuity constraints on the boundary of the sub-domains are not imposed strongly but only penalized. We derive a selection rule for the penalty parameter which ensures a quasi-optimal convergence.

In this paper, we consider a 2D mathematical modelling of the verticalcompaction effect in a water saturated sedimentary basin. This model isdescribed by the usual conservation laws, Darcy's law, the porosity as afunction of the vertical component of the effective stress and theKozeny-Carman tensor, taking into account fracturation effects. This modelleads to study the time discretization of a nonlinear system ofpartial differential equations. The existence is obtained by a fixed-pointargument. The uniqueness proof, by Holmgren's method, leads to work out a linear, strongly coupled, system of partial differential equations andboundary conditions.

A hyperelastic constitutive law, for use in anatomically accurate finite element models ofliving structures, is suggested for the passive and the active mechanical properties of incompressiblebiological tissues. This law considers the passive and active states as a same hyperelastic continuummedium, and uses an activation function in order to describe the whole contraction phase.The variational and the FE formulations are also presented, and the FE code has been validatedand applied to describe the biomechanical behavior of a thick-walled anisotropic cylinder underdifferent active loading conditions.

We consider in this paper the problem of finding appropriate models for Large Eddy Simulations of turbulent incompressible flows from a mathematical point of view. The Smagorinsky model is analyzed and the vorticity formulation of the Navier–Stokes equations is used to explore more efficient subgrid-scale models as minimal regularizations of these equations. Two classes of variants of the Smagorinsky model emerge from this approach: a model based on anisotropic turbulent viscosity and a selective model based on vorticity angles. The efficiency of these models is demonstrated by comparisons with reference results on decaying turbulence experiments.

The aim of this paper is to present a finite volume kinetic method to compute the transport of a passive pollutant by a flow modeled by the shallow water equations using a new time discretization that allows large time steps for the pollutant computation. For the hydrodynamic part the kinetic solver ensures – even in the case of a non flat bottom – the preservation of the steady state of a lake at rest, the non-negativity of the water height and the existence of an entropy inequality. On an other hand the transport computation ensures the conservation of pollutant mass, a non-negativity property and a maximum principle for the concentration of pollutant and the preservation of discrete steady states associated with the lake at rest equilibrium. The interest of the developed method is to preserve these theoretical properties with a scheme that allows to disconnect the hydrodynamic time step – related to a classical CFL condition – and the transport one – related to a new CFL condition – and further the hydrodynamic calculation and the transport one. The CPU time is very reduced and we can easily solve different transport problems with the same hydrodynamic solution without large storage. Moreover the numerical results exhibit a better accuracy than with a classical method especially when using 1D or 2D regular grids.