The parareal in time algorithm allows for efficient parallel numerical simulations oftime-dependent problems. It is based on a decomposition of the time interval intosubintervals, and on a predictor-corrector strategy, where the propagations over eachsubinterval for the corrector stage are concurrently performed on the different processorsthat are available. In this article, we are concerned with the long time integration ofHamiltonian systems. Geometric, structure-preserving integrators are preferably employedfor such systems because they show interesting numerical properties, in particularexcellent preservation of the total energy of the system. Using a symmetrization procedureand/or a (possibly also symmetric) projection step, we introduce here several variants ofthe original plain parareal in time algorithm [L. Baffico, et al. Phys. Rev. E66 (2002) 057701; G. Bal and Y. Maday, A parareal timediscretization for nonlinear PDE’s with application to the pricing of an American put, inRecent developments in domain decomposition methods, Lect.Notes Comput. Sci. Eng. 23 (2002) 189–202; J.-L. Lions, Y. Madayand G. Turinici, C. R. Acad. Sci. Paris, Série I 332 (2001)661–668.] that are better adapted to the Hamiltonian context. These variants arecompatible with the geometric structure of the exact dynamics, and are easy to implement.Numerical tests on several model systems illustrate the remarkable properties of theproposed parareal integrators over long integration times. Some formal elements ofunderstanding are also provided.

The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we prove that three greedy algorithms converge; the last algorithm, based on the use of an a posteriori estimator, is the approach actually employed in the calculations.

In this article, we provide a priori error estimates for the spectral andpseudospectral Fourier (also called planewave) discretizations of theperiodic Thomas-Fermi-von Weizsäcker (TFW) model and for the spectraldiscretization of the periodic Kohn-Shammodel, within the local density approximation (LDA). These modelsallow to compute approximations of the electronic ground state energy and densityof molecular systems in the condensed phase. The TFW model is strictlyconvex with respect to the electronic density, and allows for acomprehensive analysis. This is not the case for the Kohn-Sham LDAmodel, for which the uniqueness of the ground state electronic densityis not guaranteed. We prove that, for any local minimizer $\Phi^0$ of the Kohn-Sham LDA model, and under a coercivity assumption ensuring the local uniqueness of this minimizer up to unitary transform, the discretized Kohn-Sham LDA problem has a minimizer in the vicinity of $\Phi^0$ for large enough energy cut-offs, and that this minimizer is unique up to unitary transform. We then derive optimal a priori error estimates for the spectral discretization method.


In a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations, the construction of lower bounds for the coercivity and inf-supstability constants is essential. In [Huynh et al., C. R. Acad.Sci. Paris Ser. I Math.345 (2007) 473–478], the authors presented an efficientmethod, compatible with an off-line/on-line strategy, where the on-line computation is reduced tominimizing a linear functional under a few linear constraints. These constraints depend on nested sets of parameters obtained iteratively using a greedy algorithm. We improve here this method so that it becomes more efficient and robust due to two related properties: (i) the lower bound isobtained by a monotonic process with respect to the size of the nested sets; (ii) less eigen-problems need to be solved. This improved evaluation of the inf-sup constant is then used to consider a reduced basis approximation of a parameter dependent electromagnetic cavity problem both for the greedy construction of the elements of the basis and the subsequent validation of the reduced basis approximation. The problem we consider has resonance features for some choices of the parameters that are well captured by the methodology.


In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameterdependence to problems involving (a) nonaffine dependence on theparameter, and (b) nonlinear dependence on the field variable.The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computationaldecomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateralreduced-basis approximation space, and (ii) a stable and inexpensiveinterpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in eachinstance, we discuss the reduced-basis approximation and the associated offline-online computationalprocedures. Numerical results are presented to assess our approach.

The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations. The basic idea is to first decomposethe computational domain into a series of subdomains that are deformationsof a few reference domains (or generic computational parts). Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation,but solved for different choices of deformations of the referencesubdomains and mapped onto the reference shape.The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mappedfrom the generic computational part to the actual computational part.We extend earlier work in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to satisfying the inf-sup condition,to a posteriori error estimation, and to “gluing” the local solutions together in the multidomain case.

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.

Many problems in quantumchemistry deal with the computation of fundamental or excited states of molecules and lead to the resolution of eigenvalue problems. One of the major difficulties in these computations lies in the very largedimension of the systems to be solved. Indeed these eigenfunctions dependon 3n variables where n stands for the number of particles (electrons and/or nucleari) in the molecule. In order to diminish the size of the systems to be solved, the chemists have proposed manyinteresting ideas. Among those stands the adiabatic variable method; we present in this paper a mathematical analysis of thisapproximation and propose, in particular, an a posteriori estimate thatmight allow for verifying the adiabaticity hypothesis that is done on some variables; numerical simulations that support thea posteriori estimators obtained theoretically are also presented.

The paper deals with the application of a non-conforming domaindecomposition methodto the problem of the computation of induced currents in electric engineswith moving conductors.The eddy currents model is considered as a quasi-staticapproximation of Maxwellequations and we study its two-dimensional formulation with either themodified magnetic vector potential or the magnetic field as primary variable.Two discretizations are proposed, the first one based on curved finiteelementsand the second one based on iso-parametric finite elements in both thestatic and movingparts. The coupling is obtained by means of the mortar element method(see [CITE])and the approximation on the whole domain turns out to be non-conforming.In bothcases optimal error estimates are provided. Numerical tests are then proposed in the case of standard first order finiteelements to test the reliability and precision of the method. An applicationof the method to study the influence of the free part movement on thecurrents distribution is also provided.

We study the well-posedness of an unsteady fluid-structure interaction problem.We consider a viscous incompressible flow, which is modelled by theNavier-Stokes equations. The structure is a collection of rigid moving bodies. The fluiddomain depends on time and is defined by the position of the structure, itself resultingfrom a stress distribution coming from the fluid. The problem is thennonlinear and the equations we deal with are coupled. We prove its localsolvability in time through two fixed point procedures.

Linear Force-free (or Beltrami) fields are three-components divergence-free fields solutions of the equation curlB = α B, where α is a real number.Such fields appear in many branches of physics like astrophysics, fluid mechanics, electromagnetics and plasma physics. In this paper,we deal with some related boundary value problems in multiply-connected bounded domains, in half-cylindrical domains and in exterior domains.