This paper is concerned with the coupling of two models for the propagation of particles in scattering media. The first model is a linear transport equation of Boltzmann type posed in the phase space (position and velocity). It accurately describes the physics but is very expensive to solve. The second model is a diffusion equation posed in the physical space. It is only valid in areas of high scattering, weak absorption, and smooth physical coefficients, but its numerical solution is much cheaper than that of transport. We are interested in the case when the domain is diffusive everywhere except in some small areas, for instance non-scattering or oscillatory inclusions. We present a natural coupling of the two models that accounts for both the diffusive and non-diffusive regions. The interface separating the models is chosen so that the diffusive regime holds in its vicinity to avoid the calculation of boundary or interface layers. The coupled problem is analyzed theoretically and numerically. To simplify the presentation, the transport equation is written in the even parity form. Applications include, for instance, the treatment of clear or spatially inhomogeneous regions in near-infra-red spectroscopy, which is increasingly being used in medical imaging for monitoring certain properties of human tissues.

In this paper, we are interested in finding the optimal shapeof a magnet. The criterion to maximize is the jump of theelectromagnetic field between two different configurations.We prove existence of an optimal shape into a natural classof domains. We introduce a quasi-Newton type algorithm whichmoves the boundary. This method is very efficient to improvean initial shape. We give some numerical results.

This paper deals with the diffusion limit of a kinetic equation where thecollisions are modeled by a Lorentz type operator. The main aim is to construct adiscrete scheme to approximate this equation which gives for any value of theKnudsen number, and in particular at the diffusive limit, the right discretediffusion equation with the same value of the diffusion coefficient as in thecontinuous case. We are also naturally interested with a discretization whichcan be used with few velocity discretization points, in order to reduce the cost ofcomputation.

Les méthodes sans maillage emploient une interpolation associée à un ensemble de particules : aucune information concernant la connectivité ne doit être fournie.Un des atouts de ces méthodes est que la discrétisationpeut être enrichie d'une façon très simple, soit en augmentant le nombre de particules (analogue à lastratégie de raffinement h), soit en augmentant l'ordre de consistance (analogueà la stratégie de raffinement p). Néanmoins, le coût du calcul des fonctionsd'interpolation est très élevé et ceci représente un inconvénient vis-à-visdes éléments finis. Cet article présente une interpolation mixte élémentsfinis-particules qui résulte de la généralisation de plusieurs travaux dans cedomaine. La formulation de cette interpolation mixte est valable pour n'importe quel ordre de consistance. Dans ce contexte, onénonce un estimateur d'erreur a priori dont la démonstrationse base dans les propriétés de l'interpolation mixte.Ce résultat permet d'étudier la convergence de laméthode d'enrichissement et d'établir les stratégies de raffinement del'interpolation qui permettent d'atteindre une solutionavec une précision satisfaisante.

In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping(given in terms of a boundary integral operator) to solve linear exterior transmission problems inthe plane. As a model we consider a second order elliptic equation in divergence form coupled withthe Laplace equation in the exterior unbounded region. We show that the resulting mixed variationalformulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derivethe usual Cea error estimate and the corresponding rate of convergence. In addition, we develop twodifferent a-posteriori error analyses yielding explicit residual and implicit Bank-Weiser typereliable estimates, respectively. Several numerical results illustrate the suitability of theseestimators for the adaptive computation of the discrete solutions.

In this work, we investigate the PerfectlyMatched Layers (PML) introduced by Bérenger [3] for designing efficient numerical absorbing layers in electromagnetism.We make a mathematical analysis of this model, first via a modalanalysis with standard Fourier techniques, then via energytechniques. We obtain uniform in time stability results (that makeprecise some results known in the literature) and state some energydecay results that illustrate the absorbing properties of themodel. This last technique allows us to prove the stability of theYee's scheme for discretizing PML's.

We consider a dynamical one-dimensional nonlinear von Kármán model for beamsdepending on a parameter ε > 0 and studyits asymptotic behavior for t large, as ε → 0. Introducing appropriate dampingmechanisms we show that the energy of solutionsof the corresponding damped models decayexponentially uniformly with respect to theparameter ε. In order for this to be true thedamping mechanism has to have the appropriatescale with respect to ε. In the limit as ε → 0 we obtain damped Berger–Timoshenko beam modelsfor which the energy tends to zero exponentiallyas well. This is done both in the case ofinternal and boundary damping. We address the sameproblem for plates with internal damping.

This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitean form. This eigenvalue problem is discretized by a finite element method on graded meshes. Based on regularity results for the eigensolutions estimates for the finite element error are derived both for the eigenvalues and the eigensolutions. Finally, some numerical results are presented.

For a nonconforming finite element approximation of an elliptic model problem, we propose a posteriori error estimates in the energy norm which use as an additive term the “post-processing error” between the original nonconforming finite element solution and an easy computable conforming approximation of that solution.Thus, for the error analysis, the existing theory from the conformingcase can be used together with some simple additional arguments.As an essential point, the property is exploited that the nonconformingfinite element space contains as a subspace a conforming finite element space of first order. This property is fulfilled for many knownnonconforming spaces. We prove local lower and global upper a posteriori error estimates for an enhanced error measure which is the discretization error in the discrete energy norm plus the error of the best representation of the exact solution by a function in the conforming space used for thepost-processing. We demonstrate that the idea to use a computed conforming approximation ofthe nonconforming solution can be applied also to derive an a posteriorierror estimate for a linear functional of the solution which representssome quantity of physical interest.

This contribution gives an overview of current research in applying object oriented programming to scientific computing at the computational mechanics laboratory (LABMEC) at the school of civil engineering – UNICAMP. The main goal of applying object oriented programming to scientific computing is to implement increasingly complex algorithms in a structured manner and to hide the complexity behind a simple user interface. The following areas are current topics of research and documented within the paper: hp-adaptive finite elements in one-, two- and three dimensions with the development of automatic refinement strategies, multigrid methods applied to adaptively refined finite element solution spaces and parallel computing.

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.

In this paper, we prove the convergence of the current defined from the Schrödinger-Poisson system with the presence of a strong magnetic field toward a dissipative solution of the Euler equations.

This article is devoted to the numerical study of a flame ball model, derived by Joulin, which obeys to a singular integro-differential equation. The numerical scheme that we analyze here, is based upon a one step method, and we are interested in its long-time behaviour. We recover the same dynamics as in the continuous case: quenching, or stabilization of the flame, depending on heat losses, and an energy input parameter.

In the present work the symmetrized sequential-parallel decomposition methodof the third degree precision for the solution of Cauchy abstract problemwith an operator under a split form, is presented. The third degreeprecision is reached by introducing a complex coefficient with the positivereal part. For the considered schema the explicit a priori estimation isobtained.

A modal synthesis method to solve the elastoacoustic vibration problemis analyzed. A two-dimensional coupled fluid-solid system is considered;the solid is described by displacement variables, whereas displacement potential is used for the fluid. A particular modal synthesis leading to a symmetric eigenvalue problem is introduced. Finite element discretizations with Lagrangian elements are considered for solving the uncoupled problems.Convergence for eigenvalues and eigenfunctions is proved, error estimates are given, and numerical experiments exhibiting the good performance of the method are reported.

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.

We prove the convergence of a finite volumemethod for a noncoercive linear elliptic problem, with right-handside in the dual space of the natural energy space of the problem.

The C++// language (pronounced C++parallel) was designed and implemented with the aim of importingreusability into parallel and concurrentprogramming, in the framework of a mimd model.From a reduced set of rather simple primitives,comprehensive and versatile libraries are defined.In the absence of any syntactical extension,the C++// user writes standard C++ code.The libraries are themselvesextensible by the final users, making C++// an open system. Two specific techniques to improve performances ofa distributed object language such as C++// arethen presented: Shared-on-Read and Overlapping of Communicationand Computation.The appliance of those techniques is guided by the programmer ata very high-level of abstraction, so the additional work to yieldthose good performance improvements is kept to the minimum.

We show that the Maxwell equations in the low frequency limit, in a domain composed of insulatingand conducting regions, has a saddle point structure, wherethe electric field in the insulating region is the Lagrangemultiplier that enforces the curl-free constraint on the magnetic field.We propose a mixed finite element techniquefor solving this problem, and we show that, under mild regularityassumption on the data, Lagrange finite elements can be usedas an alternative to edge elements.

In this paper we consider the Maxwell resolvent operator and its finite elementapproximation. In this framework it is natural the use of the edge elementspaces and to impose the divergence constraint in a weaksense with the introduction of a Lagrange multiplier, followingan idea by Kikuchi [14].We shall review some of the known properties for edge elementapproximations and prove some new result. In particular we shall prove auniform convergence in the L 2 norm for the sequence of discrete operators.These results, together with a general theory introduced by Brezzi, Rappaz andRaviart [8], allow an immediate proof of convergence for thefinite element approximation of the time-harmonicMaxwell system.