The phenomenon of roll waves occurs in a uniform open-channelflow down an incline, when the Froude number is above two. The goal of this paper is to analyze the behavior of numerical approximations to a model roll wave equation ut + uux = u,u(x,0) = u0(x), which arises as a weakly nonlinear approximation of the shallow waterequations. The main difficulty associated with the numerical approximation ofthis problem is its linear instability. Numerical round-off errorcan easily overtake the numerical solution and yields false roll wavesolution at the steady state.In this paper, we first study the analytic behavior of the solution to the abovemodel. We then discuss the numerical difficulty, and introduce a numericalmethod that predicts precisely the evolution and steady state of itssolution. Various numerical experiments are performed to illustratethe numerical difficulty and the effectiveness of the proposed numericalmethod.

This paper presents anextension to stabilized methods of the standard technique for thenumerical analysis of mixed methods. We prove that the stability of stabilizedmethods follows from an underlying discrete inf-sup condition, plus a uniformseparation property between bubble and velocity finite element spaces. We applythe technique introduced to provethe sta bi li ty of stabilized spectral element methods so asstabilized solution of the primitive equations of the ocean.

We analyze a two species discrete predator-prey model inwhich the prey disperses between two patches of a heterogeneous environmentwith barriers and the mature predator disperses between thepatches with no barrier. By using the discrete dynamical system generated by a monotone, concavemaps for subcommunity of prey, we obtain the subcommunity of preyexists an equilibrium which attracts all positive solutions, andusing the stability trichotomy results on the monotone andcontinuous operator, we obtain some sufficient conditions for thepermanence of species. These results are applied to the modelswith rational growth functions and exponential growth functions.We also present numerical examples to illustrate the dynamiccomplexity of systems.

We consider a Boltzmann equation for inelastic particles on the line and prove existence and uniqueness for the solutions.

We consider a nonlinear second order elliptic boundary value problem (BVP)in a bounded domain $\Omega\subset {\mathbb R}^N$ witha nonlocal boundary condition. A Dirichlet BC containing an unknown additive constant,accompanied with a nonlocal (integral) Neumann side condition is prescribed at some boundary part Γn . The rest of the boundary is equipped with Dirichlet or nonlinear Robin type BC. The solution is found via linearization. We design a robust and efficient approximation scheme. Error estimates for the linearization algorithm are derived inL 2(Ω),H 1(Ω) andL ∞(Ω) spaces.

We deal with boundary layers and quasi-neutral limits in the drift-diffusion equations. We first show that this limit is unique and determined by a system of two decoupled equations with given initial and boundary conditions. Then we establish the boundary layer equations and prove the existence and uniqueness of solutions with exponential decay. This yields a globally strong convergence (with respect to the domain) of the sequence of solutions and an optimal convergence rate $O(\varepsilon^\frac{1}{2})$ to the quasi-neutral limit in L 2.

The paper deals with some mixed finite element methods on a classof anisotropic meshes based on tetrahedra and prismatic (pentahedral)elements. Anisotropic localinterpolation error estimates are derived in some anisotropic weighted Sobolevspaces. As particularapplications, the numerical approximation by mixed methods of the Laplace equationin domainswith edges is investigated where anisotropic finiteelement meshes are appropriate. Optimal error estimates are obtained using some anisotropic regularity results of thesolutions.

Our aim here is to study the thermal diffusion phenomenon in a forced convective flow. A system of nonlinear parabolic equations governs the evolution of the mass fractions in multicomponent mixtures. Some existence and uniqueness results are given under suitable conditions onstate functions. Then, we present a numerical scheme based on a "mixed finite element"method adapted to a finite volume scheme, of which we give numerical analysis. In a last part, we apply an homogenization technique to the studied equations in order to obtain an efficient modelling of Soret effect and adsorption in a porous medium at a macroscopic scale.

The aim of this paper is to investigate the stabilityof boundary layers which appear in numerical solutionsof hyperbolic systems of conservation laws in one spacedimension on regular meshes. We prove stability under a sizecondition for Lax Friedrichs type schemes and inconditionnalstability in the scalar case. Examples of unstable boundary layersare also given.

In this paper we study a free boundary problem appearing inelectromagnetism and its numerical approximation by means ofboundary integral methods. Once the problem is written in aequivalent integro-differential form, with the arcparametrization of the boundary as unknown, we analyse it inthis new setting. Then we consider Galerkin and collocationmethods with trigonometric polynomial and spline curves asapproximate solutions.

We present a new methodology for the numerical resolution of the hydrodynamicsof incompressible viscid newtonian fluids. It is based on the Navier-Stokesequations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurationstypical to the motion of biological structures in viscous fluids.Although the method is applicable to three dimensions, we address herein detail only the two dimensional case. We provide numerical data forsome test cases to which we apply the computational scheme.

The goal of this paper is to construct a first-order upwind schemefor solving the system of partial differential equations governing theone-dimensional flow of two superposed immiscible layers of shallow waterfluids. This is done by generalizing a numerical scheme presented byBermúdez and Vázquez-Cendón [3, 6, 27] for solving one-layer shallow water equations, consisting in a Q-scheme with a suitable treatment of the source terms.The difficulty in the two layer system comes from the coupling terms involving some derivatives of the unknowns.Due to these terms, a numerical scheme obtained by performing the upwinding of each layer, independently from the other one, can be unconditionally unstable.In order to define a suitable numerical scheme with global upwinding,we first consider an abstract system that generalizes the problem under study.This system is not a system of conservation laws but, nevertheless,Roe's method can be applied to obtain an upwind scheme based on Approximate Riemann State Solvers. Following this, we present some numerical tests to validate the resulting schemes and to highlight the fact that, in general, numerical schemes obtained by applying a Q-scheme to each separate conservation law of the system do not yield good results. First, a simple system of coupled Burgers' equations is considered. Then, the Q-scheme obtained is applied to the two-layer shallow water system.

We consider a model for phase separation of a multi-component alloy with non-smooth free energyand a degenerate mobility matrix. In addition to showingwell-posedness and stability bounds for our approximation, we prove convergence in one space dimension. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. We discuss also how our approximation has to be modified in orderto be applicable to a logarithmic free energy.Finally numerical experiments with three components in one and two space dimensions are presented.

We present the combination of a state control and shape design approachesfor the optimization of micro-fluidic channels used for sample extraction andseparation of chemical species existing in a buffer solution.The aim is to improve the extraction and identification capacities of electroosmotic micro-fluidic devices by avoiding dispersion of the extracted advected band.

We study a family of non linear schemes for the numerical solution oflinear advection on arbitrary grids in several space dimension. A proof of weak convergence of the family of schemes is given, based on a new Longitudinal Variation Diminishing (LVD) estimate.This estimate is a multidimensional equivalent to the well-known TVD estimate in one dimension. The proof uses a corollary of the Perron-Frobenius theorem applied to a generalized Harten formalism.

In this note we give sharp lower bounds for a non-convex functional when minimised over the space of functions that are piecewise affine on a triangular grid and satisfyan affine boundary condition in the second lamination convex hull of the wells of the functional.

By using an inductive procedure we prove that the Galerkin finite element approximations of electromagnetic eigenproblems modelling cavity resonators by elements of any fixed order of either Nedelec's edge element family on tetrahedral meshes are convergent and free of spurious solutions. This result is not new but is proved under weaker hypotheses, which are fulfilled in most of engineering applications. The method of the proofis new, instead, and shows how families of spurious-freeelements can be systematically constructed. The tools here developed are used to define a new family of spurious-freeedge elements which, in some sense, are complementary to those defined in 1986 by Nedelec.

The aim of this article is to give a regularization method for an unilateral obstacle problem with obstacle ψ and second member f, which generalizes the one established by the authors of [4] in case of null obstacle and a second member is equal to constant 1.

In this article, we consider the initial value problem which is obtained after a space discretization (with space step h) of the equations governing the solidification process of a multicomponent alloy. We propose a numerical scheme to solve numerically this initial value problem. We prove an error estimate which is not affected by the step size h chosen in the space discretization. Consequently, our scheme provides global convergence without any stability condition between h and the time step size τ. Moreover, it is not of excessive algorithmic complexity since it does not require more than one resolution of a linear system at each time step.

We consider the identification of a distributed parameter in an ellipticvariational inequality. On the basis of an optimal control problemformulation, the application of a primal-dual penalizationtechnique enables us to prove the existenceof multipliers giving a first order characterization of the optimal solution.Concerning the parameter we consider differentregularity requirements. For the numerical realization we utilize a complementarity function,which allows us to rewrite the optimality conditions as a set of equalities.Finally, numerical results obtained from a least squares type algorithmemphasize the feasibility of our approach.