The purpose of this article is the analysis and the development of Eulerian multi-fluid models to describe the evolution of the mass density of evaporating liquid sprays.First, the classical multi-fluid model developed in [Laurent and Massot, Combust. Theor. Model.5 (2001) 537–572] is analyzed in the framework of an unsteady configuration without dynamical nor heating effects, where the evaporation process is isolated, since it is a key issue.The classical multi-fluid method consists then in a discretization of the droplet size variable into cells called sections.This analysis provides a justification of the “right” choice for this discretization to obtain a first order accurate and monotone scheme, with no restrictive CFL condition.This result leads to the development of a class of methods of arbitrary high order accuracy through the use of moments on the droplet surface in each section and a Godunov type method.Moreover, an extension of the two moments method is proposed which preserves the positivity and limits the total variation.Numerical results of the multi-fluid methods are compared to examine their capability to accurately describe the mass density in the spray with a small number of variables.This is shown to be a key point for the use of such methods in realistic flow configurations.


We consider a special configuration of vorticity that consists of a pair ofexternally tangent circular vortex sheets, each having a circularly symmetric core of bounded vorticity concentric to the sheet, and each core precisely balancing the vorticity mass of the sheet. This configuration is a stationary weak solution of the 2D incompressible Euler equations. We propose to perform numerical experiments to verify that certain approximations of this flow configuration converge to a non-stationary weak solution. Preliminary simulations presented here suggest this isindeed the case. We establish a convergence theorem for the vortex blob method that applies to this problem. This theorem and the preliminary calculations we carried out support the existence of two distinct weak solutions with the same initial data.


This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of ${\mathbb R}^2$ with a fractal boundary. Both the Laplace and Helmholtz equations are studied. A generalized Neumann boundary condition is imposed on the fractal boundary.Sobolev spaces on this domain are studied. In particular, extension and trace results are obtained. These results enable the investigation of the variational formulation of the above mentioned boundary value problems. Next, for homogeneous Neumann conditions, the emphasis is placed on transparent boundary conditions, which allow the computationof the solutions in the subdomains obtained by stopping the geometric construction after a finite number of steps. The proposed methods and algorithmswill be used numerically in forecoming papers.

In this paper, we study a Zakharov system coupled to an electrondiffusion equation in order to describe laser-plasma interactions. Starting fromthe Vlasov-Maxwell system, we derive a nonlinear Schrödinger like system which takes into account the energy exchanged between the plasma waves and the electrons via Landau damping. Two existence theorems are established in a subsonic regime. Using a time-splitting, spectral discretizations for the Zakharov system and a finite difference scheme for the electron diffusion equation, we perform numerical simulations and show how Landau damping works quantitatively.

We present a new stabilized mixed finite element method for the linear elasticity problem in $\mathbb{R}^2$ . Theapproach is based on the introduction of Galerkin least-squares terms arising from the constitutive andequilibrium equations, and from the relation defining the rotation in terms of the displacement. We show thatthe resulting augmented variational formulation and the associated Galerkin scheme are well posed, and thatthe latter becomes locking-free and asymptotically locking-free for Dirichlet and mixed boundary conditions,respectively. In particular, the discrete scheme allows the utilization of Raviart–Thomas spaces of lowestorder for the stress tensor, piecewise linear elements for the displacement, and piecewise constants for therotation. In the case of mixed boundary conditions, the essential one (Neumann) is imposed weakly, whichyields the introduction of the trace of the displacement as a suitable Lagrange multiplier. This trace isthen approximated by piecewise linear elements on an independent partition of the Neumann boundary whose meshsize needs to satisfy a compatibility condition with the mesh size associated to the triangulation of thedomain. Several numerical results illustrating the good performance of the augmented mixed finite elementscheme in the case of Dirichlet boundary conditions are also reported.

The Discontinuous Galerkin Time Domain (DGTD) methods are now popular for the solution of wave propagation problems. Able to deal with unstructured, possibly locally-refined meshes, they handleeasily complex geometries and remain fully explicit with easy parallelization and extension to high orders of accuracy. Non-dissipative versions exist, where some discrete electromagnetic energy is exactly conserved. However, the stability limit of the methods, related to the smallest elements in the mesh, calls for the construction of local-time stepping algorithms. These schemes have already been developed for N-body mechanical problems and are known as symplectic schemes. They are applied here to DGTD methods on wave propagation problems.

This paper derives upper and lower bounds for the $\ell^p$ -conditionnumber of the stiffness matrix resulting from the finite elementapproximation of a linear, abstract model problem. Sharp estimates interms of the meshsize h are obtained. The theoretical results areapplied to finite element approximations of elliptic PDE's invariational and in mixed form, and to first-order PDE's approximatedusing the Galerkin–Least Squares technique or bymeans of a non-standard Galerkin technique in L 1(Ω). Numerical simulations are presented to illustrate thetheoretical results.

In this work we derive a posteriori error estimates basedon equations residuals for the heat equation with discontinuousdiffusivity coefficients. The estimates are based on a fully discretescheme based on conforming finite elements in each time slab andon the A-stable θ-scheme with 1/2 ≤ θ ≤ 1.Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo40 (2003) 195–212] it is easyto identify a time-discretization error-estimatorand a space-discretization error-estimator. In this work we introduce a similarsplitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math.72 (1996) 313–348; Petzoldt, Adv. Comput. Math.16 (2002) 47–75] we have upper and lower boundswhose ratio is independent of any meshsize, timestep, problem parameter and its jumps.

Singularly perturbed reaction-diffusionproblems exhibit in general solutions with anisotropic features,e.g. strong boundary and/or interior layers.This anisotropy is reflected in a discretization by using mesheswith anisotropic elements. The quality of the numerical solutionrests on the robustness of the a posteriori error estimator withrespect to both, the perturbation parameters of the problemand the anisotropy of the mesh. The equilibrated residual method has been shown to provide one of the most reliable error estimates for the reaction-diffusion problem. Its modification suggested byAinsworth and Babuška has been proved to be robust for the case of singular perturbation. In the present work we investigate the modified method on anisotropic meshes. The method in the form of Ainsworth and Babuška is shown here to fail on anisotropic meshes. We suggest a new modification based on the stretching ratios of the mesh elements. The resulting error estimator is equivalent tothe equilibrated residual method in the case of isotropic meshesand is proved to be robust on anisotropic meshes as well. Among others, the equilibrated residual method involves the solution of an infinite dimensional local problem on each element. In practical computations an approximate solution to this local problem was successfully computed. Nevertheless, up to now no rigorous analysis has been done showing the appropriateness of any computable approximation. This demands special attention since an improper approximate solution to the local problem can be fatal for the robustness of the whole method. In the present work we provide one of the desired approximations. We prove that the method is not affected by the approximate solution of the local problem.

In this paper, we propose a new diffuse interface model for the study of three immiscible component incompressible viscous flows. The model is based on the Cahn-Hilliard free energy approach. The originality of our study lies in particular in the choice of the bulk free energy. We show that one must take care of this choice in order for the model to give physically relevant results. More precisely, we give conditions for the model to be well-posed and to satisfy algebraically and dynamically consistency properties with the two-component models. Notice that our model is also able to cope with some total spreading situations.We propose to take into account the hydrodynamics of the mixture by coupling our ternary Cahn-Hilliard system and the Navier-Stokes equation supplemented by capillary force terms accounting for surface tension effects between the components. Finally, we present some numerical results which illustrate our analysis and which confirm that our model has a better behavior than other possible similar models.

In this paper we develop a residual based a posteriori error analysis for an augmentedmixed finite element method applied to the problem of linear elasticity in the plane.More precisely, we derive a reliable and efficient a posteriori error estimator for thecase of pure Dirichlet boundary conditions. In addition, several numericalexperiments confirming the theoretical properties of the estimator, andillustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution, are also reported.

This work represents a first step towards the simulation of the motion of water in a complex hydrodynamic configuration, such asa channel network or a river delta, by means of a suitable“combination” of different mathematicalmodels. In this framework a wide spectrum of space and time scales is involveddue to the presence of physical phenomena ofdifferent nature. Ideally, moving from a hierarchy of hydrodynamic models, one should solvethroughout the whole domain the most complex model (with solution $u_{\rm{fine}}$ ) to accuratelydescribe all the physical features of the problem at hand. In our approach instead,for a user-defined output functional ${\cal F}$ , we aim to approximate,within a prescribed tolerance τ, the value ${\cal F}(u_{\rm{fine}})$ by means of the quantity ${\cal F}(u_{\rm{adapted}})$ , $u_{\rm{adapted}}$ being the so-called adapted solution solving thesimpler models on most of the computational domain while confining the complex ones only on a restricted region. Moving fromthe simplified setting where only two hydrodynamic models, fine and coarse, are considered, weprovide an efficient tool able to automatically selectthe regions of the domain where the coarsemodel rather than the fine one are to be solved, while guaranteeing $|{\cal F}(u_{\rm{fine}})-{\cal F}(u_{\rm{adapted}})|$ below the tolerance τ.This goal is achieved via a suitable a posteriori modeling error analysisdeveloped in the framework of a goal-oriented theory.We extend the dual-based approach provided in [Braack and Ern,Multiscale Model Sim.1 (2003) 221–238],for steady equations to the case of a generic time-dependent problem.Then this analysis is specialized to the case we areinterested in, i.e. the free-surface flows simulation, by emphasizing the crucial issue of the time discretization for both the primal and the dualproblems. Finally, inthe last part of the paper a widespread numerical validation is carried out.

In this work we design a new domain decomposition method for the Euler equations in 2 dimensions. The starting point is the equivalence with a third order scalar equation to whom we can apply an algorithm inspired from the Robin-Robin preconditioner for the convection-diffusion equation [Achdou and Nataf,C. R. Acad. Sci. Paris Sér. I325 (1997) 1211–1216]. Afterwards we translate it into an algorithm for the initial system and prove that at the continuous level and for a decomposition into 2 sub-domains, it converges in 2 iterations. This property cannot be conserved strictly at discrete level and for arbitrary domain decompositions but we still have numerical results which confirm a very good stability with respect to the various parameters of the problem (mesh size, Mach number, ...).

We present and analyse in this paper a novel colocated Finite Volume scheme for the solution of the Stokes problem.It has been developed following two main ideas.On one hand, the discretization of the pressure gradient term is built as the discrete transposed of the velocity divergence term, the latter being evaluated using a natural finite volume approximation; this leads to a non-standard interpolation formula for the expression of the pressure on the edges of the control volumes.On the other hand, the scheme is stabilized using a finite volume analogue to the Brezzi-Pitkäranta technique.We prove that, under usual regularity assumptions for the solution (each component of the velocity in H2(∞) and pressure in H1(∞)), the scheme is first order convergent in the usual finite volume discrete H1 norm and the L2 norm for respectively the velocity and the pressure, provided, in particular, that the approximation of the mass balance flux is of second order.With the above-mentioned interpolation formulae, this latter condition is satisfied only for particular meshes: acute angles triangulations or rectangular structured discretizations in two dimensions, and rectangular parallelepipedic structured discretizations in three dimensions.Numerical experiments confirm this analysis and show, in addition, a second order convergence for the velocity in a discrete L2 norm.

For a plate subject to stress boundary condition, the deformationdetermined by the Reissner–Mindlin plate bending model could bebending dominated, transverse shear dominated, or neither(intermediate), depending on the load. We show that theReissner–Mindlin model has a wider range of applicability thanthe Kirchhoff–Love model, but it does not always converge to theelasticity theory. In the case of bending domination, both the twomodels are accurate. In the case of transverse shear domination,the Reissner–Mindlin model is accurate but the Kirchhoff–Lovemodel totally fails. In the intermediate case, while theKirchhoff–Love model fails, the Reissner–Mindlin solution alsohas a relative error comparing to the elasticity solution, whichdoes not decrease when the plate thickness tends to zero. We alsoshow that under the conventional definition of the resultantloading functional, the well known shear correction factor 5/6in the Reissner–Mindlin model should be replaced by 1.Otherwise, the range of applicability of the Reissner–Mindlinmodel is not wider than that of Kirchhoff–Love's.

We study the existence of spatial periodic solutions for nonlinearelliptic equations $- \Delta u \, + \, g(x,u(x)) = 0, \;x \in {\mathbb R}^N$ where g is a continuous function, nondecreasing w.r.t. u. Wegive necessary and sufficient conditions for the existence ofperiodic solutions. Some cases with nonincreasing functions gare investigated as well. As an application we analyze themathematical model of electron beam focusing system and we provethe existence of positive periodic solutions for the envelopeequation. We present also numerical simulations.

This paper proposes and analyzes a BEM-FEM scheme to approximatea time-harmonic diffusion problem in the plane with non-constantcoefficients in a bounded area. The model is set as a Helmholtztransmission problem with adsorption and with non-constantcoefficients in a bounded domain. We reformulate the problem as afour-field system. For the temperature and the heat flux we usepiecewise constant functions and lowest order Raviart-Thomaselements associated to a triangulation approximating the boundeddomain. For the boundary unknowns we take spaces of periodicsplines. We show how to transmit information from the approximateboundary to the exact one in an efficient way and provewell-posedness of the Galerkin method. Error estimates areprovided and experimentally corroborated at the end of the work.

We discuss best N-term approximation spaces for one-electron wavefunctions $\phi_i$ and reduced density matrices ρemerging from Hartree-Fock and density functional theory. The approximation spaces $A^\alpha_q(H^1)$ for anisotropicwavelet tensor product bases have been recently characterized by Nitsche in terms of tensor product Besov spaces. We have used the norm equivalence of these spaces to weighted $\ell_q$ spaces of wavelet coefficients toproof that both $\phi_i$ and ρ are in $A^\alpha_q(H^1)$ for all $\alpha > 0$ with $\alpha = \frac{1}{q} - \frac{1}{2}$ . Our proof is based on the assumption that the $\phi_i$ possess an asymptotic smoothness property at the electron-nuclear cusps.

In this paper, by use of affine biquadratic elements, we constructand analyze a finite volume element scheme for elliptic equations onquadrilateral meshes. The scheme is shown to be of second-order inH 1-norm, provided that each quadrilateral in partition is almosta parallelogram. Numerical experiments are presented to confirm theusefulness and efficiency of the method.

We consider mathematical models describing dynamics of an elastic beam which is clamped at its left end to a vibrating support and which can move freely at its right end between two rigid obstacles. We model the contact with Signorini's complementary conditions between the displacement and the shear stress. For this infinite dimensional contact problem, we propose a family of fully discretized approximations and their convergence is proved. Moreover some examples of implementation are presented. The results obtained here are also valid in the case of a beam oscillating between two longitudinal rigid obstacles.