The Schwarz alternating method can be used to solveelliptic boundary value problems on domains which consist of two or moreoverlapping subdomains. The solution is approximated by an infinite sequence offunctions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. In this paper, proofs of convergence of some Schwarz alternating methods fornonlinear elliptic problems which are known to have solutions by the monotonemethod (also known as the method of subsolutions and supersolutions) aregiven. In particular, an additive Schwarz method for scalar as well as somecoupled nonlinear PDEs are shown to converge for finitely many subdomains.These results are applicable to several models in population biology.

We present here a theoretical study of eigenmodes inaxisymmetric elastic layers.The mathematical modelling allows us to bring this problem to a spectral studyof a sequence of unbounded self-adjoint operators A n , $n\in \mathbb{N}$ , in a suitableHilbert space. We show that the essential spectrum of A n is an interval oftype $[\gamma,+\infty[$ and that, under certain conditions on the coefficientsof the medium, the discrete spectrum is non empty.

We propose and analyze several finite-element schemes for solving a grade-twofluid model, with atangential boundary condition, in a two-dimensional polygon. The exactproblem is split into ageneralized Stokes problem and a transport equation, in such a way that italways has a solutionwithout restriction on the shape of the domain and on the size of the data.The first scheme usesdivergence-free discrete velocities and a centered discretization of thetransport term, whereas theother schemes use Hood-Taylor discretizations for the velocity andpressure, and either a centered or an upwinddiscretization of the transport term. One facet of our analysis isthat, without restrictions on the data,each scheme has a discrete solution and all discrete solutions convergestrongly to solutions of theexact problem. Furthermore, if the domain is convex and the data satisfycertain conditions, eachscheme satisfies error inequalities that lead to error estimates.

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.

In this paper we focus on melting and solidification processes described by phase-field models and obtain rigorous estimates for such processes. These estimates are derived in Section 2 and guarantee the convergence of solutions to non-constant equilibrium patterns. The most basic results conclude with the inequality (E2.31). The estimates in the remainder of Section 2 illustrate what obtains if the initial data is progressively more regular and may be omitted on first reading. We also present some interesting numerical simulations which demonstrate the equilibrium structures and the approach of the system to non-constant equilibrium patterns. The novel feature of these calculations is the linking of the small parameter in the system, δ, to the grid spacing, thereby producing solutions with approximate sharp interfaces. Similar ideas have been used by Caginalp and Sokolovsky [5]. A movie of these simulations may be found at http:www.math.cmu.edu/math/people/greenberg.html

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.

We propose and study semidiscrete and fully discretefinite element schemes based on appropriate relaxation models forsystems of Hyperbolic Conservation Laws.These schemes are using piecewise polynomials of arbitrary degree andtheir consistency error is of high order.The methods are combined with an adaptive strategy that yieldsfine mesh in shock regions and coarser mesh in the smooth parts of thesolution.The computational performance of these methods is demonstrated by considering scalar problems and the system of elastodynamics.

This paper presents the numerical analysis for a variational formulation of rate-independent phase transformations in elastic solids due to Mielke et al. The new model itself suggests an implicit time-discretization which is combined with thefinite element method in space. A priori error estimates are established for the quasioptimal spatial approximation of the stress fieldwithin one time-step. A posteriorierror estimates motivate an adaptive mesh-refining algorithm for efficient discretization. The proposed scheme enables numerical simulations which show that the model allows for hysteresis.

We first generalize, in an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation. In this part, all the coefficients are globally Lipchitz. The case when the nonlinearity is only locally Lipchitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation. We are not able in this case to compute a pathwise order of the approximation, we introduce the weaker notion of order in probability and generalize in that context the results of the globally Lipschitz case.

We propose a simple numerical method for capturing thesteady state solution of hyperbolic systems with geometricalsource terms. We usethe interface value, rather than the cell-averages, for the source terms that balance the nonlinear convectionat the cell interface, allowing the numerical capturing of the steadystate with a formal high order accuracy. This method applies to Godunovor Roe type upwind methods butrequires no modification of the Riemann solver. Numerical experiments on scalar conservationlaws and the one dimensional shallow water equationsshow much better resolution of the steady state than the conventionalmethod, with almost no new numerical complexity.

We present numerical evidence for the blow-up of solution for theEuler equations. Our approximate solutions are Taylor polynomials in the timevariable of an exact solution, and we believe that in terms of the exact solution,the blow-up will be rigorously proved.

In this paper, we study how solutions to elliptic problems withperiodically oscillating coefficients behave inthe neighborhood of the boundary of a domain. We extend theresults known for flat boundaries to domains with curved boundariesin the case of a layered medium. This is done by generalizing thenotion of boundary layer and by defining boundary correctors whichlead to an approximation of order ε in the energy norm.

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.

We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a"rough"coefficient function k(x). We show that the Engquist-Osher (and hence all monotone)finite difference approximations convergeto the unique entropy solution of the governing equation if, among other demands, k' is in BV, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion equations as well as new convergence results for their finite difference approximations. In the inviscid case, we also provide a rate of convergence. Our convergence proofs are based on deriving a series of a priori estimates and using a general Lp compactness criterion.

Singularly perturbed problems often yield solutions with strong directional features,e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation.To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.

In this paper, a nonlinear problem corresponding to a simplified Oldroyd-B modelwithout convective terms is considered. Assuming the domain to be a convexpolygon, existence of a solutionis proved for small relaxation times.Continuous piecewise linear finite elements together witha Galerkin Least Square (GLS) method are studied for solving this problem.Existence and a priori error estimatesare established using a Newton-chord fixed point theorem,a posteriori error estimates are also derived.An Elastic Viscous Split Stress (EVSS) scheme related to the GLS methodis introduced. Numerical results confirm the theoretical predictions.

We investigate the approximationof the evolution of compact hypersurfaces of $\mathbb{R}^N$ depending, not only on terms of curvature of the surface, but alsoon non local terms such as the measure of the set enclosedby the surface.

In this article we implement different numerical schemes to simulate the Schrödinger-Debye equations that occur in nonlinear optics. Since the existence of blow-up solutions is an open problem, we try to compute such solutions. The convergence of the methods is proved and simulations seem indeed to show that for at least small delays self-focusing solutions may exist.

We consider a general loaded arch problem with a small thickness. Toapproximate the solution of this problem, a conforming mixed finite elementmethod which takes into account an approximation of the middle line of thearch is given. But for a very small thickness such a method gives poor errorbounds. the conforming Galerkin method is then enriched with residual-freebubble functions.

In this article, we wish to investigate the behavior of a two-layer k - εturbulence model from the mathematical point of view, as this model is useful for the near-wall treatment in numerical simulations.First, we explain the difficulties inherent in themodel. Then, we present a new variable θ that enables the mathematical study. Due to a problem of definition of the turbulentviscosity on the wall boundary, we consider an alternative version of the original equation. We show that some physical aspectsof the model are preserved by the new formulation, and in particular, we show how the physicists can help us to provethe existence of a solution of our problem. Finally, we are interested in the Navier-Stokes equations coupled with the modified turbulencemodel and we show that the alternative model may be preferred to the original one, because of its good properties(existence of a solution of the coupled problems).