The numerical approximation of parametric partial differential equations is acomputational challenge, in particular when the number of involved parameter is large.This paper considers a model class of second order, linear, parametric, elliptic PDEs on abounded domain D with diffusion coefficients depending on the parametersin an affine manner. For such models, it was shown in [9, 10] that under very weak assumptionson the diffusion coefficients, the entire family of solutions to such equations can besimultaneously approximated in the Hilbert space V = H01(D) by multivariate sparse polynomials in the parametervector y with a controlled number N of terms. Theconvergence rate in terms of N does not depend on the number ofparameters in V, which may be arbitrarily large or countably infinite,thereby breaking the curse of dimensionality. However, these approximation results do notdescribe the concrete construction of these polynomial expansions, and should thereforerather be viewed as benchmark for the convergence analysis of numerical methods. Thepresent paper presents an adaptive numerical algorithm for constructing a sequence ofsparse polynomials that is proved to converge toward the solution with the optimalbenchmark rate. Numerical experiments are presented in large parameter dimension, whichconfirm the effectiveness of the adaptive approach.

We consider a Vlasov-Fokker-Planck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. We also prove a propagation of chaos property for an associated particle system, and give rates on the approximation of the solution by the particle system. Finally, a transportation inequality for the distribution of the particle system leads to quantitative deviation bounds on the approximation of the equilibrium solution of the equation by an empirical mean of the particles at given time.

We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such equations arise from the theory of semiconductors and are composed of two continuity equations coupled with a Poisson equation. In the case that the continuity equations are non degenerate, we prove the convergence of the scheme and then the existence of solutions to the problem. The key point of the proof relies on the construction of an approximate gradient of the electric potential which allows us to deal with coupled terms in the continuity equations. Finally, a numerical example is given to show the efficiency of the scheme.

The aim of this paper is to develop a crowd motion model designed to handle highly packed situations. The model we propose rests on two principles: we first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other people. The actual velocity is then computed as the projection of the spontaneous velocity onto the set of admissible velocities (i.e. velocities which do not violate the non-overlapping constraint). We describe here the underlying mathematical framework, and we explain how recent results by J.F. Edmond and L. Thibault on the sweeping process by uniformly prox-regular sets can be adapted to handle this situation in terms of well-posedness. We propose a numerical scheme for this contact dynamics model, based on a prediction-correction algorithm. Numerical illustrations are finally presented and discussed.

The construction of reduced order models for dynamical systems usingproper orthogonal decomposition (POD) is based on the informationcontained in so-called snapshots. These provide the spatialdistribution of the dynamical system at discrete time instances.This work is devoted to optimizing the choice of these timeinstances in such a manner that the error between the POD-solutionand the trajectory of the dynamical system is minimized. First andsecond order optimality systems are given. Numerical examplesillustrate that the proposed criterion is sensitive with respect tothe choice of the time instances and further they demonstrate thefeasibility of the method in determining optimal snapshot locationsfor concrete diffusion equations.

The Mumford-Shah functional for image segmentation is an original approachof the image segmentation problem, based on a minimal energy criterion. Itsminimization can be seen as a free discontinuity problem and is based onΓ-convergence and bounded variation functions theories. Some newregularization results, make possible to imagine a finite element resolutionmethod. In a first time, the Mumford-Shah functional isintroduced and some existing results are quoted. Then, adiscrete formulation for the Mumford-Shah problem is proposedand its Γ-convergence is proved. Finally, somenumerical results, computed from both artificial and realimages are presented and discussed.

We consider the system of partial differential equations governingthe one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in general. Here we use entropy inequalities in order to control the stability. We introduce a stable well-balanced time-splitting scheme for the two-layer shallow water system that satisfies a fully discrete entropy inequality. In contrast with Roe type solvers, it does not need the computation of eigenvalues, which is not simple for the two-layer shallow water system. The solver has the property to keep the water heights nonnegative, and to be able to treat vanishing values.

Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite and is assembled cellwise from local operators. We analyze two schemes depending on whether the potential degrees of freedom are attached to the vertices or to the cells of the primal mesh. We derive new functional analysis results on the discrete gradient that are the counterpart of the Sobolev embeddings. Then, we identify the two design properties of the local discrete Hodge operators yielding optimal discrete energy error estimates. Additionally, we show how these operators can be built from local nonconforming gradient reconstructions using a dual barycentric mesh. In this case, we also prove an optimal L2-error estimate for the potential for smooth solutions. Links with existing schemes (finite elements, finite volumes, mimetic finite differences) are discussed. Numerical results are presented on three-dimensional polyhedral meshes.

This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation $c_t + \nabla \cdot ( {\bf u}f(c)) - \nabla \cdot (D \nabla c) + \lambda c = 0$ .The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L 1-norm,independent of the diffusion parameter D. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability of the theoretical results.

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.

We start from a mathematical model which describes the collective motion of bacteria taking into account the underlying biochemistry. This model was first introduced by Keller-Segel [13]. A new formulation of the system of partial differential equations is obtained by the introduction of a new variable (this new variable is similar to the quasi-Fermi level in the framework of semiconductor modelling). This new system of P.D.E. is approximated via a mixed finite element technique. The solution algorithm is then described and finally we give some preliminary numerical results. Especially our method is well adapted to compute the concentration of bacteria.


We prove stability and derive error estimates for the recently introduced central discontinuous Galerkin method, in the context of linear hyperbolic equations with possibly discontinuous solutions. A comparison between the centraldiscontinuous Galerkin method and the regular discontinuousGalerkin method in this context is also made.Numerical experiments are provided to validate the quantitativeconclusions from the analysis.

This paper addresses the derivation of new second-kind Fredholm combined field integral equations for the Krylov iterative solution of tridimensional acoustic scattering problems by a smooth closed surface. These integralequations need the introduction of suitable tangential square-root operators to regularize the formulations. Existence and uniqueness occur for these formulations. They can be interpreted asgeneralizations of the well-known Brakhage-Werner [A. Brakhage and P. Werner, Arch. Math.16 (1965) 325–329] and Combined Field Integral Equations (CFIE) [R.F. Harrington and J.R. Mautz, Arch. Elektron. Übertragungstech (AEÜ)32 (1978) 157–164].Finally, some numerical experiments are performed to test their efficiency.


We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed.Both the construction and analysis of the proposed domain decomposition methods are presented in a unified framework. For symmetric schemes, it is shown that the condition number of the preconditioned system is of order O(H/h), where H and h are the mesh sizes of the coarse and fine grids respectively, which are assumed to be nested. For non-symmetric schemes, we show by numerical computations that the Eisenstat et al. [SIAM J. Numer. Anal.20 (1983) 345–357] GMRES convergence theory, generally used in the analysis of Schwarz methods for non-symmetric problems, cannot be applied even if the numerical results show that the GMRES applied to the preconditioned systems converges in a finite number of steps and the proposed preconditioners seem to be scalable.Extensive numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods are presented.


We present a sparse grid/hyperbolic cross discretization for many-particle problems.It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization.Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derivedthat, in the best case, result in complexities and error estimates which are independent of the number of particles.Furthermore we introduce an additional constraint which gives antisymmetric sparse grids which are suited to fermionic systems. We apply the antisymmetric sparse grid discretization to the electronic Schrödinger equationand compare costs, accuracy, convergence rates and scalability with respect to the number of electrons present in the system.


In this paper we extend recent work on the detection of inclusions using electrostatic measurements to the problem of crack detection in a two-dimensional object. As in the inclusion case our method is based on a factorization of the difference between two Neumann-Dirichlet operators. The factorization possible in the case of cracks is much simpler than that for inclusions and the analysis is greatly simplified. However, the directional information carried by the crack makes the practical implementation of our algorithm more computationally demanding.

The Ultra Weak Variational Formulation (UWVF) of the Helmholtz equationprovides a variational framework suitable for discretization using plane wave solutionsof an appropriate adjoint equation. Currently convergence of the method is only proved on the boundary of the domain. However substantial computational evidence exists showing that the method also converges throughout the domain of the Helmholtz equation. In this paper we exploit the fact that the UWVF is essentially an upwind discontinuous Galerkin method to prove convergence of the solution in the special case where there is no absorbing medium present. We also provide some other estimates in the case when absorption is present, and give some simple numerical results to test the estimates. We expect that similar techniques can be used to prove error estimates for the UWVF applied to Maxwell's equations and elasticity.

The Mumford-Shah functional, introduced to study image segmentation problems, is approximated in the sense of vergence by a sequence ofintegral functionals defined on piecewise affine functions.

This paper is concerned with the numerical approximation of Cauchy problems for one-dimensional nonconservative hyperbolic systems. The theory developed by Dal Maso et al. [J. Math. Pures Appl.74 (1995) 483–548] is used in order to define the weak solutions of the system: an interpretation of the nonconservative products as Borel measures is given, based on the choice of a family of paths drawn in the phase space. Even if the family of paths can be chosen arbitrarily, it is natural to require this family to satisfy some hypotheses concerning the relation of the paths with the integral curves of the characteristic fields. The first goal of this paper is to investigate the implications of three basic hypotheses of this nature. Next, we show that, when the family of paths satisfies these hypotheses, Godunov methods can be written in a natural form that generalizes their classical expression for systems of conservation laws. We also study the well-balance properties of these methods. Finally, we provethe consistency of the numerical scheme with the definition of weak solutions: we prove that, under hypothesis of bounded total variation, if the approximations provided by a Godunov method based on a family of paths converge uniformly to some function as the mesh is refined, then this function is a weak solution (related to that family of paths) of the nonconservative system. We extend this result to a family of numerical schemes based on approximate Riemann solvers.

In this survey paper, we are concerned with the zero Mach number limitfor compressible viscous flows. For the sake of (mathematical) simplicity, we restrict ourselves to the case of barotropicfluids and we assume that the flow evolves in the whole space or satisfies periodic boundary conditions. We focus on the case of ill-prepared data. Hence highly oscillating acoustic waves are likely to propagate through the fluid. We nevertheless statethe convergence to the incompressible Navier-Stokes equations when the Mach number ϵ goes to 0. Besides, it is shown that the global existence for the limit equations entails the global existence for the compressible model with small ϵ. The reader is referred to [R. Danchin, Ann. Sci. Éc. Norm. Sup. (2002)] for the detailed proof in the whole space case, and to [R. Danchin, Am. J. Math.124 (2002) 1153–1219] for the case of periodic boundary conditions.