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.

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 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.

A new Schwarz method for nonlinear systems is presented, constituting the multiplicative variant of a straightforward additive scheme. Local convergence can be guaranteed under suitable assumptions.The scheme is applied to nonlinear acoustic-structure interaction problems. Numerical examples validate the theoretical results. Further improvements are discussed by means of introducing overlapping subdomains and employing an inexact strategy for the local solvers.

In this work, we use the methods of nonequilibrium statistical mechanics in order to derive an equation which models some mechanisms of opinion formation. After proving the main mathematical properties of the model, we provide some numerical results.

In this article, we analyze the stability of various numerical schemes for differential models of viscoelastic fluids. More precisely, we consider the prototypical Oldroyd-B model, for which a free energy dissipation holds, and we show under which assumptions such a dissipation is also satisfied for the numerical scheme. Among the numerical schemes we analyze, we consider some discretizations based on the log-formulation of the Oldroyd-B system proposed by Fattal and Kupferman in [J. Non-Newtonian Fluid Mech.123 (2004) 281–285], for which solutions in some benchmark problems have been obtained beyond the limiting Weissenberg numbers for the standard scheme (see [Hulsen et al.J. Non-Newtonian Fluid Mech. 127 (2005) 27–39]). Our analysis gives some tracks to understand these numerical observations.

We analyze two numerical schemes of Euler type in time and C 0finite-element type with $\mathbb{P}_1$ -approximation in space forsolving a phase-field model of a binary alloy with thermalproperties. This model is written as a highly non-linear parabolicsystem with three unknowns: phase-field, solute concentration andtemperature, where the diffusion for the temperature and soluteconcentration may degenerate. The first scheme is nonlinear, unconditionally stableand convergent. The other scheme is linear but conditionally stableand convergent. A maximum principle is avoided in both schemes,using a truncation operator on the L 2 projection onto the $\mathbb{P}_0$ finite element for the discrete concentration. Inaddition, for the model when the heat conductivity and solutediffusion coefficients are constants, optimal error estimates forboth schemes are shown based on stability estimates.

The atomistic to continuum interface for quasicontinuum energiesexhibits nonzero forces under uniform strain that have beencalled ghost forces.In this paper,we prove for a linearization of a one-dimensional quasicontinuum energyaround a uniform strainthat the effect of the ghost forces on the displacementnearly cancels and has a small effect on the error away from the interface.We give optimal order error estimatesthat show that the quasicontinuum displacementconverges to the atomistic displacement at the rate O(h)in the discrete $\ell^\infty$ andw 1,1 norms where h is the interatomic spacing.We also give a proof that the error in the displacement gradientdecays away from the interface to O(h) at distance O(h|logh|)in the atomistic region and distance O(h) in the continuum region.Our work gives an explicit and simplified form for the decay of the effect of theatomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete $\ell^\infty$ and w1,p norms.