We present a domain decomposition theory on an interface problemfor the linear transport equation between a diffusive and a non-diffusive region.To leading order, i.e. up to an error of the order of the mean free path in thediffusive region, the solution in the non-diffusive region is independent of thedensity in the diffusive region. However, the diffusive and the non-diffusive regionsare coupled at the interface at the next order of approximation. In particular, ouralgorithm avoids iterating the diffusion and transport solutions as is done in mostother methods — see for example Bal–Maday (2002). Our analysis is based instead on an accurate description of the boundarylayer at the interface matching the phase-space density of particles leaving thenon-diffusive region to the bulk density that solves the diffusion equation.

In this paper we outline the hyperbolic system of governing equationsdescribing one-dimensional blood flow in arterial networks. Thissystem is numerically discretised using a discontinuous Galerkinformulation with a spectral/hp element spatial approximation. Weapply the numerical model to arterial networks in theplacenta. Starting with a single placenta we investigate the velocity waveformin the umbilical artery and its relationship with the distalbifurcation geometry and the terminal resistance. We then present resultsfor the waveform patterns and the volume fluxes throughout a simplifiedmodel of the arterial placental network in a monochorionic twin pregnancy with an arterio-arterial anastomosis and an arterio-venous anastomosis. Theeffects of varying the time period of the two fetus' heart beats, increasing the input flux of one fetus and the role ofterminal resistance in the network are investigated and discussed. The results show that the main features of the in vivo, physiological waves are captured by thecomputational model and demonstrate the applicability of themethods to the simulation of flows in arterial networks.

When two miscible fluids, such as glycerol (glycerin) and water,are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients existduring some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical modelconsisting of the diffusion equation with convective terms and ofthe Navier-Stokes equations with the Korteweg stress.We prove the global existence and uniqueness of the solution for the associated initial-boundary value problem in a two-dimensional bounded domain.We study the longtime behavior of the solution and show that it convergesto the uniform composition distribution with zero velocity field.We also present numerical simulations of miscible drops and show howtransient interfacial phenomena can change their shape.

The aim of this paper is to find estimates of the Green's function of stationary discrete shock profiles and discrete boundary layers of the modified Lax–Friedrichs numerical scheme, by using techniques developed by Zumbrun and Howard [CITE] in the continuous viscous setting.

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.

In this note, we propose and analyse a method for handlinginterfaces between non-matching grids based on an approachsuggested by Nitsche (1971) for the approximation ofDirichlet boundary conditions. The exposition is limited toself-adjoint elliptic problems, using Poisson's equation as amodel. A priori and a posteriori error estimates are given. Somenumerical results are included.

In this paper, the main purpose is to reveal what kind of qualitative dynamicalchanges a continuous age-structured model may undergo as continuous reproduction is replaced withan annual birth pulse. Using the discrete dynamical system determined by the stroboscopic map we obtain an exact periodic solution of system with density-dependent fertility and obtain the threshold conditions for its stability. We also present formal proofs of the supercritical flip bifurcation at the bifurcation as well as extensive analysis of dynamics in unstableparameter regions. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behavior of the single species model with birth pulses are very complex, including small-amplitude annual oscillations, large-amplitude multi-annual cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allowsfor a period-doubling route to chaos. Finally, we discuss the effects of generation delay on stability of positiveequilibrium (or positive periodic solution), and show that generation delay is found to act both as a destabilizing and a stabilizing effect.

In this work we introduce an accurate solver for theShallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the non-homogeneous case, in such a way that all equilibria are computed at least with second order accuracy. We perform several tests for relevant flows showing the performance of our scheme.

We recently derived a very general representation formulafor the boundary voltage perturbations caused by internalconductivity inhomogeneities of low volume fraction (cf. Capdeboscq and Vogelius (2003)). In this paper we show how thisrepresentation formula may be used to obtain veryaccurate estimates for the size of the inhomogeneitiesin terms of multiple boundary measurements. As demonstrated by our computational experiments, these estimates are significantly better than previously known (single measurement) estimates,even for moderate volume fractions.

This paper presents a model based on spectral hyperviscosity for the simulation of 3D turbulent incompressible flows. One particularity of this model is that the hyperviscosity is active only at the short velocity scales, a feature which is reminiscent of Large Eddy Simulation models. We propose a Fourier–Galerkin approximation of the perturbedNavier–Stokes equations and we show that, as the cutoff wavenumbergoes to infinity, the solution of the modelconverges (up to subsequences) to a weak solution which is dissipativein the sense defined by Duchon and Robert (2000).

Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions.It is shown that they are equivalent to certain active set strategies. Global and local super-linear convergence areproved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty versionis used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an L∞ estimate for the penalized solutions. Unilateral as well as bilateral problems are considered.

Vasculogenesis and angiogenesis are two different mechanisms for bloodvessel formation. Angiogenesis occurs when new vessels sprout from pre-existing vasculature in response to external chemical stimuli. Vasculogenesis occurs via the reorganization of randomly distributed cells into a blood vessel network. Experimental modelsof vasculogenesis have suggested that the cells exert traction forcesonto the extracellular matrix and that these forces may playan important role in the network forming process.In order to study the role of the mechanical and chemical forcesin both of these stages of blood vessel formation, we present amathematical model which assumes that (i) cells exert traction forcesonto the extracellular matrix, (ii) the matrix behaves as a linearviscoelastic material, (iii) the cells move along gradients ofexogenously supplied chemical stimuli (chemotaxis) and (iv) these stimuli diffuse or are uptaken by the cells.We study the equations numerically, present an appropriate finite difference scheme and simulate the formation of vascular networks in a plane. Our results compare very well with experimental observations and suggest that spontaneous formation of networks can be explained via a purely mechanical interaction betweencells and the extracellular matrix. We find that chemotaxis alone is not a sufficient force to stimulate formation of pattern. Moreover, duringvessel sprouting, we find that mechanical forces can help in the formationof well defined vascular structures.

The limit behavior of the solutions of Signorini's type-likeproblems in periodically perforated domains with periodε is studied. The main feature of this limit behaviour isthe existence of a critical size of the perforations thatseparates different emerging phenomena as ε → 0. In the critical case, it is shown that Signorini's problemconverges to a problem associated to a new operator whichis the sum of a standard homogenized operator and an extra zeroorder term (“strange term”) coming from the geometry; itsappearance is due to the special size of the holes. The limitproblem captures the two sources of oscillations involved in thiskind of free boundary-value problems, namely, those arising fromthe size of the holes and those due to the periodic inhomogeneityof the medium. The main ingredient of the method used in the proofis an explicit construction of suitable test functions whichprovide a good understanding of the interactions between the abovementioned sources of oscillations.

The paper deals with the numerical resolution of the convection-diffusion system which arises when modeling combustion for turbulent flow. The considered model is of compressible turbulent reacting type where the turbulence-chemistry interactions are governed by additional balance equations. The system of PDE's, that governs such a model, turns out to be in non-conservation form and usual numerical approaches grossly fail in the capture of viscous shock layers. Put in other words, classical finite volume methods induce large errors when approximated the convection-diffusion extracted system. To solve this difficulty, recent works propose a nonlinear projection scheme based on cancellation phenomenon of relevant dissipation rates of entropy. Unfortunately, such a property never holds in the present framework. The nonlinear projection procedures are thus extended.

We consider the inverse problem of determining a crack submitted to a non linear impedance law. Identifiability and local Lipschitz stability results are proved for both the crack and the impedance.

We show that it is possible to construct a class of entropicschemes for the multicomponent Euler system describing a gas or fluidhomogeneous mixture at thermodynamic equilibrium by applying a relaxation technique. Afirst order Chapman–Enskog expansion shows that the relaxed systemformally converges when the relaxation frequencies go to the infinitytoward a multicomponent Navier–Stokes system with the classical Fick andNewton laws, with a thermal diffusion which can be assimilated to a Soret effect in the case of a fluid mixture,and with also a pressure diffusion or a density diffusion respectively for a gas or fluid mixture. We also discuss on the link between the convexity of the entropies of each species and the existence of the Chapman–Enskog expansion.

In this work, we address the numerical solution of fluid-structure interaction problems. This issue is particularly difficulty to tackle when the fluid and the solid densities are of the same order, for instance as it happens in hemodynamic applications, since fully implicit coupling schemes are required to ensure stability of the resulting method. Thus, at each time step, we have to solve a highly non-linear coupled system, since the fluid domain depends on the unknown displacement of the structure. Standard strategies for solving this non-linear problems, are fixed point based methods such as Block-Gauss-Seidel (BGS) iterations. Unfortunately, these methods are very CPU time consuming and usually show slow convergence. We propose a modified fixed-point algorithm which combines the standard BGS iterations with a transpiration formulation. Numerical experiments show the great improvement in computing time with respect to the standard BGS method.

We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix–Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for which the CR element is not stable in that it does not fulfill a discrete Korn's inequality, the discontinuous framework naturally suggests the appearance of (weakly consistent) stabilization terms. Thus, a stabilized version of the CR element, which does not lock, can be used for both compressible and (nearly) incompressible elasticity. Numerical results supporting these assertions are included. The analysis directly extends to higher order elements and three spatial dimensions.

We start from a mathematical model which describes the collective motion of bacteria taking into account the underlying biochemistry. This model was first introduced by Keller-Segel [13]. A new formulation of the system of partial differential equations is obtained by the introduction of a new variable (this new variable is similar to the quasi-Fermi level in the framework of semiconductor modelling). This new system of P.D.E. is approximated via a mixed finite element technique. The solution algorithm is then described and finally we give some preliminary numerical results. Especially our method is well adapted to compute the concentration of bacteria.

Models of two phase flows in porous media, used in petroleumengineering, lead to a system of two coupled equations with ellipticand parabolic degenerate terms, and two unknowns,the saturation and the pressure.For the purpose of their approximation, a coupled scheme, consisting ina finite volume method together witha phase-by-phase upstream weighting scheme, is used in the industrial setting.This paper presents a mathematical analysis of this coupled scheme, first showingthat it satisfies some a priori estimates:the saturation is shown to remain in a fixed interval, anda discrete L2(0,T;H1 (Ω)) estimate is proved for both the pressureand a function of the saturation. Thanks to these properties,a subsequence of the sequence of approximate solutions is shown toconverge to a weak solutionof the continuous equationsas the size of the discretization tends to zero.