In this paper, we develop a multiscale mortar multipoint flux mixed finite element method for second order elliptic problems. The equations in the coarse elements (or subdomains) are discretized on a fine grid scale by a multipoint flux mixed finite element method that reduces to cell-centered finite differences on irregular grids. The subdomain grids do not have to match across the interfaces. Continuity of flux between coarse elements is imposed via a mortar finite element space on a coarse grid scale. With an appropriate choice of polynomial degree of the mortar space, we derive optimal order convergence on the fine scale for both the multiscale pressure and velocity, as well as the coarse scale mortar pressure. Some superconvergence results are also derived. The algebraic system is reduced via a non-overlapping domain decomposition to a coarse scale mortar interface problem that is solved using a multiscale flux basis. Numerical experiments are presented to confirm the theory and illustrate the efficiency and flexibility of the method.

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Green’s function by Marahrens and the third author.

The numerical solution of the flow of a liquid crystal governedby a particular instance of the Ericksen–Leslie equations is considered.Convergence results for this system rely crucially upon energyestimates which involve H2 (Ω) norms of the director field. Weshow how a mixed method may be used to eliminate the need forHermite finite elements and establish convergence of the method.

A Mimetic Discretization method for the linear elasticity problemin mixed weakly symmetric form is developed. The scheme is shown toconverge linearly in the mesh size, independently of theincompressibility parameter λ, provided the discrete scalarproduct satisfies two given conditions. Finally, a family ofalgebraic scalar products which respect the above conditions isdetailed.

The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented Lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas.

In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case. In the latter situation, we prove that the discretization needs to account in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious discretization effects may appear. We provide a numerical analysis of the approach.

We study the approximation properties of some finite element subspaces ofH(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. Thiswork extends results previously obtained for quadrilateral H(div;Ω) finiteelements and for quadrilateral scalar finite element spaces. The finiteelement spaces we consider are constructed starting from a given finitedimensional space of vector fields on the reference cube, which is thentransformed to a space of vector fields on a hexahedron using the appropriatetransform (e.g., the Piola transform) associated to a trilinear isomorphism ofthe cube onto the hexahedron. After determining what vector fields are neededon the reference element to insure O(h) approximation in L 2(Ω) andin H(div;Ω) and H(curl;Ω) on the physical element, we study the properties ofthe resulting finite element spaces.

A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.

We introduce a modification of the Monge–Kantorovitch problem of exponent 2 which accommodates non balanced initial and final densities. The augmented Lagrangian numerical method introduced in [6] is adapted to this “unbalanced” problem. We illustrate the usability of this method on an idealized error estimation problem in meteorology.

We consider the Euler equations for compressible fluidsin a nozzle whose cross-section is variable and may contain discontinuities.We view these equations as a hyperbolic system in nonconservative formand investigate weak solutions in the sense of Dal Maso, LeFloch and Murat [J. Math. Pures Appl.74 (1995) 483–548].Observing that the entropy equality has a fully conservative form,we derive a minimum entropy principle satisfied by entropy solutions.We then establish the stability of a class of numerical approximations for this system.

In this paper, we study some finite volume schemes for the nonlinear hyperbolic equation ${u_t}(x,t)+\mbox{div}F(x,t,u(x,t))=0$ with the initial condition $u_{0}\in{L^\infty}(\mathbb{R}^N)$ . Passing to the limit in these schemes, we prove the existence of an entropy solution $u\in{L^infty}(\mathbb{R}^N\times\mathbb{R}_+)$ . Proving also uniqueness, we obtain the convergence of the finite volume approximation to the entropy solution in $L^p_{loc}(\mathbb{R}^N\times\mathbb{R}_+)$ , 1 ≤ p ≤ +∞. Furthermore, if ${u_0}\in {L^\infty}\cap\mbox{BV}_{loc}(\mathbb{R}^N)$ , we show that $u\in\mbox{BV}_{loc}(\mathbb{R}^N\times\mathbb{R}_+)$ , which leads to an “ $h^{\frac{1}{4}}$ ” error estimate between the approximate and the entropy solutions (where h defines the size of the mesh).

The present article is an overview of some mathematical results, whichprovide elements of rigorous basis for some multiscalecomputations in materials science. The emphasis is laid upon atomisticto continuum limits for crystalline materials. Various mathematicalapproaches are addressed. Thesetting is stationary. The relation to existing techniques used in the engineeringliterature is investigated.

We present in this paper a pressure correction scheme for the barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based a priori estimates of the continuous problem also hold for the discrete solution.The stability proof is based on two independent results for general finite volume discretizations, both interesting for their own sake: the L 2-stability of the discrete advection operator provided it is consistent, in some sense, with the mass balance and the estimate of the pressure work by means of the time derivative of the elastic potential.The proposed scheme is built in order to match these theoretical results, and combines a fractional-step time discretization of pressure-correction type with a space discretization associating low order non-conforming mixed finite elements and finite volumes.Numerical tests with an exact smooth solution show the convergence of the scheme.

In the second part of the paper, we compare the solutions producedin the framework of the conference “Mathematical and numericalaspects of low Mach number flows” organized by INRIA and MAB inPorquerolles, June 2004, to the reference solutions described inPart 1. We make some recommendations on how to produce goodquality solutions, and list a number of pitfalls to be avoided.

We analyze a force-based quasicontinuum approximation to aone-dimensional system of atoms that interact by a classicalatomistic potential. This force-based quasicontinuum approximationcan be derived as the modification of an energy-basedquasicontinuum approximation by the addition of nonconservativeforces to correct nonphysical “ghost” forces that occur in theatomistic to continuum interface during constant strain. The algorithmicsimplicity and consistency with the purely atomistic modelat constant strain has made the force-basedquasicontinuum approximation popular for large-scalequasicontinuum computations.We prove that the force-based quasicontinuum equations havea unique solution when the magnitude of the external forces satisfyexplicit bounds. For Lennard-Jones next-nearest-neighborinteractions, we show that unique solutions existfor external forces that extend the system nearly to its tensile limit.We give an analysis of the convergence of the ghost force iteration methodto solve the equilibrium equations for the force-based quasicontinuum approximation.We show that the ghost force iteration is a contraction and give an analysis for itsconvergence rate.

We design a particle interpretation of Feynman-Kac measures on path spacesbased on a backward Markovian representation combined with a traditionalmean field particle interpretation of the flow of their final timemarginals. In contrast to traditional genealogical tree based models, thesenew particle algorithms can be used to compute normalized additivefunctionals “on-the-fly” as well as theirlimiting occupation measures with a given precision degree that does notdepend on the final time horizon.We provide uniform convergence results w.r.t. the time horizon parameter aswell as functional central limit theorems and exponential concentrationestimates, yielding what seems to be the first results of this type for thisclass of models. We also illustrate these results in the context offiltering of hidden Markov models, as well as in computational physics andimaginary time Schroedinger type partial differential equations, with aspecial interest in the numerical approximation of the invariant measureassociated to h-processes.

For domains which are star-shapedw.r.t. at least one point, we give new bounds on theconstants in Jackson-inequalities in Sobolev spaces. Forconvex domains, these bounds do not depend on theeccentricity. For non-convex domains with a re-entrantcorner, the bounds are uniform w.r.t. the exteriorangle. The main tool is a new projection operator ontothe space of polynomials.

In this paper, we propose implicit and semi-implicit in time finite volume schemes forthe barotropic Euler equations (hence, as a particular case, for the shallow waterequations) and for the full Euler equations, based on staggered discretizations. Forstructured meshes, we use the MAC finite volume scheme, and, for general mixedquadrangular/hexahedral and simplicial meshes, we use the discrete unknowns of theRannacher−Turek orCrouzeix−Raviart finiteelements. We first show that a solution to each of these schemes satisfies a discretekinetic energy equation. In the barotropic case, a solution also satisfies a discreteelastic potential balance; integrating these equations over the domain readily yieldsdiscrete counterparts of the stability estimates which are known for the continuousproblem. In the case of the full Euler equations, the scheme relies on the discretizationof the internal energy balance equation, which offers two main advantages: first, we avoidthe space discretization of the total energy, which involves cell-centered andface-centered variables; second, we obtain an algorithm which boils down to a usualpressure correction scheme in the incompressible limit. Consistency (in a weak sense) withthe original total energy conservative equation is obtained thanks to corrective terms inthe internal energy balance, designed to compensate numerical dissipation terms appearingin the discrete kinetic energy inequality. It is then shown in the 1D case, that,supposing the convergence of a sequence of solutions, the limit is an entropy weaksolution of the continuous problem in the barotropic case, and a weak solution in the fullEuler case. Finally, we present numerical results which confirm this theory.

We discretize the nonlinear Schrödinger equation,with Dirichlet boundary conditions, by a linearlyimplicit two-step finite element method which conserves the L 2 norm. We prove optimal order a priori error estimates in the L 2 and H 1 norms, under mild mesh conditions for two and three space dimensions.

This paper deals with the use of wavelets in the framework of the Mortar method.We first review in an abstract framework the theory of the mortar method fornon conforming domain decomposition, and point out some basic assumptionsunder which stability and convergence of such method can be proven. We studythe application of the mortar method in the biorthogonal wavelet framework.In particular we define suitable multiplier spaces for imposing weakcontinuity. Unlike in the classical mortar method, such multiplier spaces arenot a subset of the space of traces of interior functions, but rather oftheir duals.For the resulting method, we provide with an error estimate, which is optimal in thegeometrically conforming case.