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 deal with an inverse scattering problem whose aim is to determine the thicknessvariation of a dielectric thin coating located on a conducting structure of unknown shape.The inverse scattering problem is solved through the application of the GeneralizedImpedance Boundary Conditions (GIBCs) which contain the thickness, curvature as well asmaterial properties of the coating and they have been obtained in the previous work [B.Aslanyürek, H. Haddar and H.Şahintürk, Wave Motion 48 (2011)681–700] up to the third order with respect to the thickness. After proving uniquenessresults for the inverse problem, the required total field as well as its higher orderderivatives appearing in the GIBCs are obtained by the analytical continuation of themeasured data to the coating surface through the single layer potential representation.The resulting system of non-linear differential equations for the unknown coatingthickness is solved iteratively via the Newton−Raphson method after expanding thethickness function in a series of exponentials. Through the simulations it has been shownthat the approach is effective under the validity conditions of the GIBCs.

In this paper, we construct a combined upwinding and mixed finite element method for the numerical solution of a two-dimensional mean field model of superconducting vortices. An advantage of our method is that it works for any unstructured regular triangulation. A simple convergence analysis is given without resorting to the discrete maximum principle. Numerical examples are also presented.

In this paper we present an analysis of the partial differential equations thatdescribe the Chemical Vapor Infiltration (CVI) processes. The mathematical modelrequires at least two partial differential equations, one describing thegas phase and one corresponding to the solid phase.A key difficulty in the process is the long processing times that are typicallyrequired. We address here the issue of optimization and show that we can chooseappropriate pressure and temperature to minimize these processing times.

This paper deals with modeling the passivebehavior of skeletal muscle tissue includingcertain microvibrations at the cell level. Our approach combines a continuum mechanics model with large deformation and incompressibility at the macroscale with chains of coupled nonlinear oscillators.The model verifies that an externally appliedvibration at the appropriate frequency is able to synchronize microvibrations in skeletal muscle cells.From the numerical analysis point of view, one faces here a partial differential-algebraic equation (PDAE) that after semi-discretization in space by finite elements possessesan index up to three, depending on certain physicalparameters. In this context, the consequences forthe time integration as well as possible remedies are discussed.

In this article we introduce an adaptive multi-levelmethod in space and time for convection diffusion problems. The scheme is based on a multi-level spatial splitting and the use of differenttime-steps. The temporal discretization relies on the characteristics method. We derive an a posteriori error estimate and design a correspondingadaptive algorithm. The efficiency of the multi-level method is illustrated by numerical experiments,in particular for a convection-dominated problem.

A finite element code, called ZéBuLoN was parallelised some years ago. This code is entirely written using an object oriented framework (C++ is the support language). The aim of this paper is to present some problems which arose during the parallelization, and some innovative solutions. Especially, a new concept of message passing is presented which allows to take into account SMP machines while still using the parallel virtual machine abstraction.

The compatibility of unsynchronized interleaved uniform sampling with Sigma-Deltaanalog-to-digital conversion is investigated. Let f be a bandlimitedsignal that is sampled on a collection of N interleaved grids {kT + Tn} k ∈ Zwith offsets \hbox{$\{T_n\}_{n=1}^N\subset [0,T]$}$\mathrm{\{}{\mathit{T}}_{\mathit{n}}{\mathrm{\}}}_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathrm{\subset }\mathrm{[}\mathrm{0}\mathit{,T}\mathrm{]}$. If the offsets Tn arechosen independently and uniformly at random from  [0,T]  and if thesample values of f are quantized with a first order Sigma-Deltaalgorithm, then with high probability the quantization error \hbox{$|f(t) - \widetilde{f}(t)|$}$\mathrm{\left|}\mathit{f}\mathrm{\right(}\mathit{t}\mathrm{)}\mathrm{-}\begin{array}{}{}_{\mathrm{􏽥}}\\ \mathit{f}\end{array}\mathrm{(}\mathit{t}\mathrm{\left)}\mathrm{\right|}$is at most of orderN-1log N.

Let Φ : H → R be a C 2 function on a real Hilbert space and ∑ ⊂ H x R the manifold defined by ∑ := Graph (Φ). We study the motion of a material point with unit mass, subjected to stay on Σ and which moves under the action of the gravity force(characterized by g>0), the reaction force and the friction force ( $\gamma>0$ is the friction parameter). For any initial conditions at time t=0, we prove the existence of a trajectory x(.) defined on R +. We are then interested in the asymptotic behaviour of the trajectories when t → +∞. More precisely, we prove the weak convergence of the trajectories when Φ is convex. When Φ admits a strong minimum, we show moreover that the mechanical energy exponentially decreases to its minimum.

For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss–Seidel type which are performed only on basis functions associated with newly created edges/nodal points or those edges/nodal points where the support of the corresponding basis function has changed during the refinement process. The adaptive mesh refinement is based on Dörfler marking for residual-type a posteriori error estimators and the newest vertex bisection strategy. Using the abstract Schwarz theory of multilevel iterative schemes, quasi-optimal convergence of the LMM is shown, i.e., the convergence rates are independent of mesh sizes and mesh levels provided the coarsest mesh is chosen sufficiently fine. The theoretical findings are illustrated by the results of some numerical examples.

In this paper, a multi-parameter error resolutiontechnique is applied into a mixed finite element method for theStokes problem. By using this technique and establishing a multi-parameter asymptotic error expansion for the mixed finite element method, an approximation of higheraccuracy is obtained by multi-processor computers in parallel.

We propose and analyze a domain decomposition method on non-matching gridsfor partial differential equations with non-negativecharacteristic form. No weak or strong continuity of the finiteelement functions, their normal derivatives, or linearcombinations of the two is imposed across the boundaries of the subdomains.Instead, we employ suitable bilinear forms defined on the commoninterfaces, typical of discontinuous Galerkinapproximations.We prove an error bound which is optimal with respect to the mesh–size andsuboptimal with respect to the polynomial degree.Our analysis is valid for arbitrary shape–regular meshes and arbitrarypartitions into subdomains.Our method can be applied to advective, diffusive, and mixed–type equations,as well,and is well-suited for problems coupling hyperbolic and elliptic equations.We present some two-dimensionalnumerical results that support our analysis for the case of linear finite elements.

The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly taken into account when the first family of mixed finite elements is used. We, therefore, introduce the second family of mixed finite elements for which a theoretical convergence analysis is presented and error estimates are obtained. A numerical study of the convergence is also considered for a particular object geometry which shows that our theoretical error estimates are optimal.

This paper deals with the solution of problems involving partial differential equations in  $\mathbb{R}^3$ . For three dimensional case, methods are useful if they require neither domain boundary regularity nor regularity for the exact solution of the problem. A new domain decomposition method is therefore presented whichuses low degree finite elements. The numerical approximation of the solution is easy, and optimal error bounds are obtained according to suitablenorms.

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.


In recent years several papers have been devoted to stabilityand smoothing properties in maximum-norm offinite element discretizations of parabolic problems.Using the theory of analytic semigroups it has been possibleto rephrase such properties as bounds for the resolventof the associated discrete elliptic operator. In all thesecases the triangulations of the spatial domain has beenassumed to be quasiuniform. In the present paper weshow a resolvent estimate, in one and two space dimensions,under weaker conditions on the triangulations than quasiuniformity.In the two-dimensional case, the bound for the resolvent containsa logarithmic factor.


The aim of this paper is to present how to make a dedicaded computed language polymorphic and multi type, in C++ to solve partial differential equations with the finite element method.The driving idea is to make the language as close as possible to the mathematical notation.

The electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable for the full 3D case.

We derive in this article some models ofCahn-Hilliard equations in nonisotropic media. These models, based onconstitutive equations introduced by Gurtin in [19], take the work ofinternal microforces and also the deformations of the material intoaccount. We then study the existence and uniqueness of solutions andobtain the existence of finite dimensional attractors.

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, ona solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants onboth the free liquid-liquid and liquid-air interfaces,and the presence of both attractive and repulsive van der Waals forcesin terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure,and a second energy inequality controlling the Laplacianof the liquid heights.We introduce a fully practical finite element approximation of this nonlinear degenerate parabolic system, that satisfies discrete analoguesof these energy inequalities. Finally, we prove convergence of this approximation,and hence existence of a solutionto this nonlinear degenerate parabolic system.