We present a phase field approach to wetting problems, related tothe minimization of capillary energy. We discuss in detail boththe Γ-convergence results on which our numerical algorithmare based, and numerical implementation. Two possible choices ofboundary conditions, needed to recover Young's law for the contactangle, are presented. We also consider an extension of theclassical theory of capillarity, in which the introduction of adissipation mechanism can explain and predict the hysteresis ofthe contact angle. We illustrate the performance of the model byreproducing numerically a broad spectrum of experimental results:advancing and receding drops, drops on inclined planes andsuperhydrophobic surfaces.

A conceptual numerical strategy for rate-independent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The novelty is that we obtain convergence ofsubsequences of space-time discretizations even in case where the limitproblem does not have a unique solution and we need noadditional assumptions on higher regularity of the limit solution.The variety of general perspectives thus obtained is illustrated on several specific examples: plasticity with isotropic hardening, damage, debonding, magnetostriction, and two models of martensitic transformation in shape-memory alloys.

We consider the magnetic induction equation for the evolution of a magnetic field in a plasma where the velocity is given. The aim is to design a numerical scheme which also handles the divergence constraint in a suitable manner. We design and analyze an upwind scheme based on the symmetrized version of the equations in the non-conservative form. The scheme is shown to converge to a weak solution of the equations. Furthermore, the discrete divergence produced by the scheme is shown to be bounded. We report several numerical experiments that show that the stable upwind scheme of this paper is robust.

Semi-smooth Newton methods for elliptic equations with gradient constraints are investigated. The one- and multi-dimensional cases are treated separately. Numerical examples illustrate the approach and as well as structural features of the solution.

The powerful Hamilton-Jacobi theory is used for constructing regularizations and error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method for optimal design and reconstruction: the first, analytical, step is to regularize the Hamiltonian; next the solution to its stationary Hamiltonian system, a nonlinear partial differential equation, is computed with the Newton method. The method is efficient for designs where the Hamiltonian function can be explicitly formulated and when the Jacobian is sparse, but becomes impractical otherwise (e.g. for non local control constraints). An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the Hamiltonian and its finite dimensional regularization along the solution path and its L2 projection, i.e. not on the difference of the exact and approximate solutions to the Hamiltonian systems.

A multilevel preconditioner based on the abstract framework of theauxiliary space method, is developed for the mortar method for thenonconforming P 1 finite element or the lowest orderCrouzeix-Raviart finite element on nonmatching grids. It is shownthat the proposed preconditioner is quasi-optimal in the sense thatthe condition number of the preconditioned system is independent ofthe mesh size, and depends only quadratically on the number ofrefinement levels. Some numerical results confirming the theory arealso provided.

This paper deals with the numerical solution of nonlinear Black-Scholes equation modeling European vanilla call option pricing under transaction costs. Using an explicit finite difference scheme consistent with the partial differential equation valuation problem, a sufficient condition for the stability of the solution is given in terms of the stepsize discretization variables and the parameter measuring the transaction costs. This stability condition is linked to some properties of the numerical approximation of the Gamma of the option, previously obtained. Results are illustrated with numerical examples.

We state a superconvergence result for the lowest order Raviart-Thomas approximation of eigenvalue problems. It is known that a similar superconvergence result holds for the mixed approximation of Laplace problem; here we introduce a new proof, since the one given for the source problem cannot be generalized in a straightforward way to the eigenvalue problem. Numerical experiments confirm thesuperconvergence property and suggest that it also holds for the lowest orderBrezzi-Douglas-Marini approximation.

In this paper, we present an abstract framework which describes algebraically the derivation of order conditions independently of the nature of differential equations considered or the type of integrators used to solve them. Our structure includes a Hopf algebra of functions, whose properties are used to answer several questions of prime interest in numerical analysis. In particular, we show that, under some mild assumptions, there exist integrators of arbitrarily high orders for arbitrary (modified) vector fields.

This study concerns some asymptotic models used to compute the flow outside and inside fractures in a bidimensional porous medium. The flow is governed by the Darcy law both in the fractures and in the porous matrix with large discontinuities in the permeability tensor. These fractures are supposed to have a small thickness with respect to the macroscopic length scale,so that we can asymptotically reduce them to immersed polygonal faultinterfaces and the model finally consists in a coupling between a2D elliptic problem and a 1D equation on the sharp interfaces modelling the fractures.A cell-centered finite volume scheme on general polygonal meshes fitting the interfacesis derived to solve the set of equations with the additional differential transmission conditions linking both pressure and normal velocityjumps through the interfaces.We prove the convergence of the FV scheme for any set of data and parameters of the models and derive existence and uniqueness of the solution to the asymptotic models proposed. The models are then numerically experimented for highly or partially immersed fractures.Some numerical results are reported showing different kinds of flowsin the case of impermeable or partially/highly permeable fractures. The influence of the variation of the aperture of the fractures is also investigated. The numericalsolutions of the asymptotic models are validated by comparing them to the solutions of the global Darcy model or to some analytic solutions.

From the fundamental laws of elasticity, we write a model for the contact between two membranes and we perform the analysis of the corresponding system of variational inequalities. We propose a finite element discretization of this problem and prove its well-posedness. We also establish a priori and a posteriori error estimates.

This paper is concerned with the analysis and implementation of spectral Galerkinmethods for a class of Fokker-Planck equations that arisesfrom the kinetic theory of dilute polymers. A relevant feature of the class of equationsunder consideration from the viewpoint of mathematical analysis and numerical approximation isthe presence of an unbounded drift coefficient, involving a smooth convex potential U that is equal to +∞ alongthe boundary ∂D of the computational domain D.Using a symmetrization of the differential operator based on the Maxwellian M corresponding to U,which vanishes along ∂D, we remove the unbounded drift coefficient at the expenseof introducing a degeneracy, through M, in the principal part of the operator.The general class of admissible potentials consideredincludes the FENE (finitely extendible nonlinear elastic) model.We show the existence of weak solutions to the initial-boundary-value problem, and develop a fully-discretespectral Galerkin method for such degenerate Fokker-Planck equationsthat exhibits optimal-order convergence in the Maxwellian-weighted H1 norm on D.In the case of the FENE model, we also discuss variants of these analytical results when the Fokker-Planck equation is subjected to an alternative class of transformations proposed by Chauvière and Lozinski; these map the original Fokker-Planck operator with an unbounded drift coefficient into Fokker-Planck operators with unbounded drift and reaction coefficients, that have improved coercivity properties in comparison with theoriginal operator. The analytical results are illustrated by numerical experiments for the FENE model in two space dimensions.

In long-time numerical integration of Hamiltonian systems,and especially in molecular dynamics simulation,it is important that the energy is well conserved. For symplecticintegrators applied with sufficiently small step size, thisis guaranteed by the existence of a modifiedHamiltonian that is exactly conserved up to exponentially smallterms. This article is concerned with the simplifiedTakahashi-Imada method, which is a modificationof the Störmer-Verlet method that is as easy to implement buthas improved accuracy. This integrator is symmetric andvolume-preserving, but no longer symplectic. We study itslong-time energy conservation and give theoreticalarguments, supported by numerical experiments, whichshow the possibility of a drift in the energy (linear or like a random walk).With respect to energy conservation, this article provides empiricaland theoretical data concerning the importance of using a symplecticintegrator.

We derive a posteriori error estimates for singularlyperturbed reaction–diffusion problems which yield a guaranteedupper bound on the discretization error and are fully and easilycomputable. Moreover, they are also locally efficient and robust inthe sense that they represent local lower bounds for the actualerror, up to a generic constant independent in particular of thereaction coefficient. We present our results in the framework ofthe vertex-centered finite volume method but their nature isgeneral for any conforming method, like the piecewise linear finiteelement one. Our estimates are based on a H(div)-conformingreconstruction of the diffusive flux in the lowest-orderRaviart–Thomas–Nédélec space linked with mesh dual to the originalsimplicial one, previously introduced by the last author in thepure diffusion case. They also rely on elaborated Poincaré,Friedrichs, and trace inequalities-based auxiliary estimatesdesigned to cope optimally with the reaction dominance. In order tobring down the ratio of the estimated and actual overall energyerror as close as possible to the optimal value of one,independently of the size of the reaction coefficient, we finallydevelop the ideas of local minimizations of the estimators by localmodifications of the reconstructed diffusive flux. The numericalexperiments presented confirm the guaranteed upper bound,robustness, and excellent efficiency of the derived estimates.

This paper is concerned with the numerical approximation of the solutions of a two-fluid two-pressure model used in the modelling of two-phase flows.We present a relaxation strategy for easily dealing with both the nonlinearities associated with the pressure laws and the nonconservative termsthat are inherently present in the set of convective equations and that couple the two phases.In particular, the proposed approximate Riemann solver is given by explicit formulas, preservesthe natural phase space, and exactly captures the coupling waves between the two phases.Numerical evidences are given to corroborate the validity of our approach.

We developed a mimetic finite difference method for solving elliptic equations with tensor coefficients on polyhedral meshes. The first-order convergence estimates in a mesh-dependent H1 norm are derived.

We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows.Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes.Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e. , in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem.An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh.Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms.Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L2 norm.

We propose here a model and a numerical scheme to compute the motionof rigid particles interacting through the lubrication force. In thecase of a particle approaching a plane, we propose an algorithm andprove its convergence towards the solutions to the gluey particle modeldescribed in [B. Maury, ESAIM: Proceedings18 (2007)133–142]. We propose a multi-particle version ofthis gluey model which is based on the projection of the velocitiesonto a set of admissible velocities. Then, we describe a multi-particle algorithmfor the simulation of such systems and present numerical results.