The convergence of a two-scale FEM for elliptic problems in divergence form with coefficients and geometries oscillating at length scale ε << 1 is analyzed. Full elliptic regularity independent of ε is shownwhen the solution is viewed as mapping from the slow into the fast scale.Two-scale FE spaces which are able to resolve the ε scale of thesolution with work independent of ε and withoutanalytical homogenization are introduced. Robustin ε error estimates for the two-scale FE spaces are proved. Numerical experiments confirm thetheoretical analysis.

This paper addresses the recovery of piecewise smooth functions from their discrete data.Reconstruction methods using both pseudo-spectral coefficients andphysical space interpolants have been discussed extensively in theliterature, and it is clear that an a priori knowledge of the jumpdiscontinuity location is essential for any reconstruction techniqueto yield spectrally accurate results with high resolution near thediscontinuities. Hence detection of the jump discontinuities iscritical for all methods. Here we formulate a new localized reconstruction method adapted from themethod developed in Gottlieb and Tadmor (1985) and recently revisited in Tadmor and Tanner (in press). Our procedure incorporates the detection of edges into the reconstruction technique. The methodis robust and highly accurate, yielding spectral accuracy up to a smallneighborhood of the jump discontinuities. Results are shown inone and two dimensions.

We propose a variational analysis for a Black and Scholes equation with stochastic volatility. This equation gives the price of a European option as a function of the time, of the price of the underlying asset and of the volatility when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The variational analysis involves weighted Sobolev spaces. It enables to prove qualitative properties of the solution, namely a maximum principle and additional regularity properties. Finally, we make numerical simulations of the solution, by finite element and finite difference methods.

Dorsey, Di Bartolo and Dolgert (Di Bartolo et al., 1996; 1997) have constructed asymptotic matched solutions at order two for the half-space Ginzburg-Landau model, in the weak-κ limit. These authors deduceda formal expansion for the superheating field in powers of $\kappa^{\frac{1}{2}}$ up to order four, extending the formula by De Gennes (De Gennes, 1966) and the two terms in Parr's formula (Parr, 1976). In this paper, we construct asymptotic matched solutions at all orders leading to a complete expansion in powers of $\kappa^{\frac{1}{2}}$ for the superheating field.

Estimates for the combined effect of boundaryapproximation and numerical integration on the approximation of(simple) eigenvalues and eigenvectors of 4th order eigenvalue problems with variable/constant coefficientsin convex domains with curved boundary by an isoparametric mixed finite element method, which,in the particular case of bending problems ofaniso-/ortho-/isotropic plates with variable/constant thickness, gives a simultaneous approximation to bending momenttensor field $\Psi= (\psi_{ij})_{1 \le i,j \le 2}$ anddisplacement field `u', have been developed.

Programming is an area at the interface between scientific computing and applied mathematics which hasbeen very active lately and so it was thought that M2AN should open its pages to it in a special issue.This is because many new tools have appeared ranging from templates in C++ to Java interface library andparallel computing tools. There has been a diffusion of computer sciences into numerical analysis and thesenew tools have made possible the implementation of very complex methods such as finite element methods ofarbitrary degree.This issue is not an overview of the field. The papers have been selected on the basis of their programmingcreativity, the quality of the final product and their relevance to numerical methods. But we have discovered onthe way that the programming community does not publish much outside conference proceedings. Furthermoreit is often difficult to pinpoint the difficulties and solutions. One must avoid tedious lists of function or subroutinedefinitions, but one must also explain in details the new programming ideas such as data driven programsor generic programming, notions which are familiar to few people only.What is new here is that the papers have been screened by reviewers who are themselves programmers andalso applied mathematicians. This successful experience leads to encourage submission of more papers of thiskind in the future as well.

We present one- and two-dimensional central-upwind schemesfor approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutionsin which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preservethis delicate balance with numerical schemes.Small perturbations of these states are also very difficultto compute. Our approach is based on extending semi-discrete central schemes forsystems of hyperbolic conservation laws to balance laws.Special attention is paid to the discretization of the sourceterm such as to preserve stationary steady-statesolutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water.This important feature allows one to compute solutions for problemsthat include dry areas.

Different effective boundary conditions or wall laws for unsteady incompressible Navier-Stokes equations over rough domains are derived in the laminar setting. First and second order unsteady wall laws are proposed using two scale asymptotic expansion techniques. The roughness elements are supposed to be periodic and the influence of the rough boundary is incorporated through constitutive constants. These constants are obtained by solving steady Stokes problems and so they are calculated only once. Numerical tests are presented to validate and compare the proposed boundary conditions.

Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi-Bellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference schemes in the stationary case with constant coefficients and in the time-dependent case with variable coefficients by using control theory and probabilistic methods. In this paper we are able to handle variable coefficients by a purely analytical method. In our opinion this way is far simpler and, for the cases we can treat, it yields a better rate of convergence than Krylov obtains in the variable coefficients case.

We present in this article two components: these components can in fact serve various goalsindependently, though we consider them here as an ensemble. The first component is a technique forthe rapid and reliable evaluation prediction of linear functional outputs of elliptic (andparabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent globalreduced–basis approximations — Galerkin projection onto a spaceW N spanned by solutions of the governing partial differentialequation at N selected points in parameter space; (ii) a posteriori error estimation — relaxations of the error–residualequation that provide inexpensive yet sharp and rigorous bounds forthe error in the outputs of interest; and (iii) off–line/on–linecomputational procedures — methods which decouple the generationand projection stages of the approximation process. This component is ideally suited — consideringthe operation count of the online stage — for the repeated and rapid evaluation required in thecontext of parameter estimation, design, optimization, andreal–time control. The second component is a framework for distributed simulations. This frameworkcomprises a library providing the necessary abstractions/concepts for distributed simulations and asmall set of tools — namely SimTeXand SimLaB— allowing an easy manipulation of thosesimulations. While the library is the backbone of the framework and is therefore general, thevarious interfaces answer specific needs. We shall describe both components and present how theyinteract.

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations.We focus on mortar finite element methods on non-matching triangulations.In particular, we discuss and analyze dual Lagrange multiplier spacesfor lowest order finite elements.These non standard Lagrange multiplier spaces yield optimal discretizationschemes and a locally supported basis for the associatedconstrained mortar spaces. As a consequence,standard efficient iterative solvers as multigrid methods or domain decomposition techniques can be easily adapted to the nonconformingsituation.Here, we introduce new dual Lagrange multiplier spaces. We concentrateon the construction of locally supported and continuous dualbasis functions.The optimality of the associated mortar method is shown. Numerical results illustrate the performance of our approach.

The stability of flat interfaces with respect to a spatial semidiscretization of a solidification model is analyzed. The considered model is the quasi-static approximation of the Stefan problem with dynamical Gibbs–Thomson law. The stability analysis bases on an argument developed by Mullins and Sekerka for the undiscretized case. The obtained stability properties differ from those with respect to the quasi-static model for certain parameter values and relatively coarse meshes. Moreover, consequences on discretization issues are discussed.

This work is concerned with the flow of a viscousplastic fluid. We choose a model of Bingham typetaking into account inhomogeneous yield limit of thefluid, which is well-adapted in the description oflandslides. After setting the generalthreedimensional problem, the blocking property isintroduced. We then focus on necessary andsufficient conditions such that blocking of the fluidoccurs.The anti-plane flow intwodimensional andonedimensional cases is considered.A variational formulation in terms of stresses isdeduced. More fine properties dealing with localstagnant regions as well as local regions where thefluid behaves like a rigid body are obtained indimension one.

In this work, we analyze hierarchic hp-finite element discretizations of the full, three-dimensional plate problem. Based on two-scale asymptotic expansion of the three-dimensional solution, we give specific mesh design principles for the hp-FEM which allow to resolve the three-dimensional boundary layer profiles at robust, exponential rate.We prove that, as the plate half-thickness ε tends to zero, the hp-discretization is consistent with the three-dimensional solution to any power of ε in the energy norm for the degree $p={\cal O}(\left|{\log \varepsilon}\right|)$ and with ${\cal O}({p^4})$ degrees of freedom.

We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ut = Δu, vt = Δv in Ω x (0,T); fully coupled by the boundary conditions $\frac{\partial u}{\partial\eta} = u^{p_{11}}v^{p_{12}}$ , $\frac{\partial v}{\partial\eta} = u^{p_{21}}v^{p_{22}}$ on ∂Ω x (0,T), where Ω is a bounded smooth domain in ${\mathbb{R}}^d$ . We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation (U,V). We prove that if U blows up in finite time then V can fail to blow up if and only if p 11 > 1 and p 21 < 2(p 11 - 1), which is the same condition as the one for non-simultaneous blow-up in the continuous problem. Moreover, we find that if the continuous problem has non-simultaneous blow-up then the same is true for the discrete one. We also prove some results about the convergence of the scheme and the convergence of the blow-up times.

In this article, we show the convergence of a class of numerical schemes for certain maximal monotone evolution systems; a by-product of this resultsis the existence of solutions in cases which had not been previouslytreated. The order of these schemes is 1/2 in general and 1 when the only non Lipschitz continuous term is the subdifferential of the indicatrix of a closed convex set. In the case of Prandtl'srheological model, our estimates in maximum norm do not dependon spatial dimension.

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.

Development of user-friendly and flexible scientific programs is a key to their usage, extension and maintenance. This paper presents an OOP (Object-Oriented Programming) approach for design of finite element analysis programs. General organization of the developed software system, called FER/SubDomain, is given which includes the solver and the pre/post processors with a friendly GUI (Graphical User Interfaces). A case study with graphical representations illustrates some functionalities of the program.

This paper is devoted to the spectral analysis of a non elliptic operator A , deriving from the study of superconducting micro-strip lines.Once a sufficient condition for the self-adjointness of operator A has been derived, we determine its continuous spectrum. Then, we show that A is unbounded from below and that it has a sequence of negative eigenvalues tending to -∞. Using the Min-Max principle, a characterization ofits positive eigenvalues is given. Thanks to this characterization, some conditions on the geometrical (large width) and physical (large dielectric permittivityin modulus) properties of the strip that ensure the existence of positive eigenvalues are derived. Finally, we analyze the asymptotic behavior of the eigenvaluesof A as the dielectric permittivity of the strip goes to -∞.

Automatic differentiation (AD) has proven its interest in many fields ofapplied mathematics, but it is still not widely used. Furthermore, existingnumerical methods have been developed under the hypotheses that computingprogram derivatives is not affordable for real size problems. Exact derivativeshave therefore been avoided, or replaced by approximations computed by divideddifferences. The hypotheses is no longer true due to the maturity of AD addedto the quick evolution of machine capacity. This encourages the development ofnew numerical methods that freely make use of program derivatives, and willrequire the definition and development of new AD strategies. AD tools mustbe extended to produce these new derivative programs, in such a modular waythat the different sub-problems can be solved independently from one another.Flexibility assures the user to be able to generate whatever specificderivative program he needs, with at the same time the possibility to generatestandard ones. This paper sketches a new model of modular, extensible andflexible AD tool that will increase tenfold the DA potential for appliedmathematics. In this model, the AD tool consists of an AD kernel namedKAD supported by a general program transformation platform.