A simplified stochastic Hookean dumbbells model arising from viscoelastic flows is considered, the convective terms being disregarded. A finite element discretization in space is proposed. Existence of the numerical solution is proved for small data, so as a priori error estimates,using an implicit function theorem and regularity results obtained in [Bonito et al., J. Evol. Equ.6 (2006) 381–398] for the solution of the continuous problem. A posteriori error estimates are also derived.Numerical results with small time steps and a large number of realizations confirm theconvergence rate with respect to the mesh size.

We investigate unilateral contact problems with cohesive forces, leading tothe constrained minimization of a possibly nonconvex functional. Weanalyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmentedLagrangian, and sufficient conditions for the existence of a localsaddle-point are derived. Then, we derive and analyze mixed finiteelement approximations to the stationarity conditions of the three-fieldaugmented Lagrangian. The finite element spaces for the bulk displacement andthe Lagrange multiplier must satisfy a discrete inf-sup condition, whilediscontinuous finite element spaces spanned by nodal basis functions areconsidered for the unilateral contact variable so as to use collocationmethods. Two iterative algorithms are presented and analyzed, namely anUzawa-type method within a decomposition-coordination approach and anonsmooth Newton's method. Finally, numerical results illustrating thetheoretical analysis are presented.

Inspired by the growing use of non linear discretization techniques for the lineardiffusion equation in industrial codes, we construct and analyze various explicit nonlinear finite volume schemes for the heat equation in dimension one. These schemes areinspired by the Le Potier’s trick [C. R. Acad. Sci. Paris, Ser. I348 (2010) 691–695]. They preserve the maximum principle and admita finite volume formulation. We provide a original functional setting for the analysis ofconvergence of such methods. In particular we show that the fourth discrete derivative isbounded in quadratic norm. Finally we construct, analyze and test a new explicit nonlinear maximum preserving scheme with third order convergence: it is optimal on numericaltests.

This paper deals with the numerical study of a nonlinear, strongly anisotropic heatequation. The use of standard schemes in this situation leads to poor results, due to thehigh anisotropy. An Asymptotic-Preserving method is introduced in this paper, which issecond-order accurate in both, temporal and spacial variables. The discretization in timeis done using an L-stable Runge−Kutta scheme. The convergence of the method is shown to beindependent of the anisotropy parameter , andthis for fixed coarse Cartesian grids and for variable anisotropy directions. The contextof this work are magnetically confined fusion plasmas.

The magnetization of a ferromagnetic sample solves anon-convex variational problem, where its relaxation by convexifyingthe energy density resolves relevantmacroscopic information. The numerical analysis of the relaxed modelhas to deal with a constrained convexbut degenerated, nonlocal energy functional in mixed formulation formagnetic potential u and magnetization m.In [C. Carstensen and A. Prohl, Numer. Math.90(2001) 65–99], the conforming P1 - (P0)d -element in d=2,3 spatialdimensions is shown to lead toan ill-posed discrete problem in relaxed micromagnetism, and suboptimalconvergence.This observation motivated anon-conforming finite element method which leads toa well-posed discrete problem, with solutions converging atoptimal rate.In this work, we provide both an a priori and a posteriori error analysis for twostabilized conforming methods which account for inter-element jumps of thepiecewise constant magnetization.Both methods converge at optimal rate;the new approach is applied to a macroscopic nonstationary ferromagnetic model [M. Kružík and A. Prohl, Adv. Math. Sci. Appl. 14 (2004) 665–681 – M. Kružík and T. Roubíček, Z. Angew. Math. Phys. 55 (2004) 159–182 ].

Partant du principe de conservation de la masse et du principefondamental de la dynamique, on retrouvel'équation d'Euler nous permettant de décrire les modèlesasymptotiques de propagation d'ondes dans des eaux peu profondesen dimension 1. Pour décrire la propagation des ondes en dimension2, Kadomtsev et Petviashvili [ 15 (1970) 539] utilisent une perturbationlinéaire de l'équation de KdV. Mais cela ne précise pas si leséquations ainsi obtenues dérivent de l'équation d'Euler, c'est ceque montrent Ablowitz et Segur dans l'article [J. Fluid Mech.92 (1979) 691–715]. Oninsistera, de la même manière, sur le fait que les équations deKP-BBM peuvent être aussi obtenues à partir de l'équation d'Euler,et dans quelle mesure elles décrivent le modèle physique. Dans unsecond temps, on reprend la méthode introduite dans l'article deBona et al. [Lect. Appl. Math.20 (1983) 235–267] dans lequel ils comparent lessolutions d'ondes longues en dimension 1, à savoir les solutionsdes équations KdV et BBM, pour montrer ici que les solutions deséquations KP-II et KP-BBM-II sont proches sur un intervalle de tempsinversement proportionnel à l'amplitude des ondes. Du point de vue de la modélisation,il sera clair, d'après la première partie, que seul le modèledécrit par KP-BBM-II est bien posé, et comme du point de vuephysique, KP-II et KP-BBM-II décrivent les ondes longues de faibleamplitude lorsque la tension de surface est négligeable, il est intéressant de les comparer.De plus, on verra que la méthode utilisée ici reste valable pourles problèmes périodiques.

Using Burgers’ equation with mixed Neumann–Dirichlet boundary conditions, we highlight aproblem that can arise in the numerical approximation of nonlinear dynamical systems oncomputers with a finite precision floating point number system. We describe the dynamicalsystem generated by Burgers’ equation with mixed boundary conditions, summarize some ofits properties and analyze the equilibrium states for finite dimensional dynamical systemsthat are generated by numerical approximations of this system. It is important to notethat there are two fundamental differences between Burgers’ equation with mixedNeumann–Dirichlet boundary conditions and Burgers’ equation with both Dirichlet boundaryconditions. First, Burgers’ equation with homogenous mixed boundary conditions on a finiteinterval cannot be linearized by the Cole–Hopf transformation. Thus, on finite intervalsBurgers’ equation with a homogenous Neumann boundary condition is truly nonlinear. Second,the nonlinear term in Burgers’ equation with a homogenous Neumann boundary condition isnot conservative. This structure plays a key role in understanding the complex dynamicsgenerated by Burgers’ equation with a Neumann boundary condition and how this structureimpacts numerical approximations. The key point is that, regardless of the particularnumerical scheme, finite precision arithmetic will always lead to numerically generatedequilibrium states that do not correspond to equilibrium states of the Burgers’ equation.In this paper we establish the existence and stability properties of these numericalstationary solutions and employ a bifurcation analysis to provide a detailed mathematicalexplanation of why numerical schemes fail to capture the correct asymptotic dynamics. Weextend the results in [E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov,Math. Comput. Modelling 35 (2002) 1165–1195] and provethat the effect of finite precision arithmetic persists in generating a nonzero numericalfalse solution to the stationary Burgers’ problem. Thus, we show that the results obtainedin [E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov, Math. Comput.Modelling 35 (2002) 1165–1195] are not dependent on a specifictime marching scheme, but are generic to all convergent numerical approximations ofBurgers’ equation.

In this paper we propose a time discretization of a system of two parabolic equationsdescribing diffusion-driven atom rearrangement in crystalline matter. The equationsexpress the balances of microforces and microenergy; the two phase fields are the orderparameter and the chemical potential. The initial and boundary-value problem for theevolutionary system is known to be well posed. Convergence of the discrete scheme to thesolution of the continuous problem is proved by a careful development of uniformestimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, forthe difference of discrete and continuous solutions we prove an error estimate of orderone with respect to the time step.

The paper deals with some mixed finite element methods on a classof anisotropic meshes based on tetrahedra and prismatic (pentahedral)elements. Anisotropic localinterpolation error estimates are derived in some anisotropic weighted Sobolevspaces. As particularapplications, the numerical approximation by mixed methods of the Laplace equationin domainswith edges is investigated where anisotropic finiteelement meshes are appropriate. Optimal error estimates are obtained using some anisotropic regularity results of thesolutions.

We analyze a two species discrete predator-prey model inwhich the prey disperses between two patches of a heterogeneous environmentwith barriers and the mature predator disperses between thepatches with no barrier. By using the discrete dynamical system generated by a monotone, concavemaps for subcommunity of prey, we obtain the subcommunity of preyexists an equilibrium which attracts all positive solutions, andusing the stability trichotomy results on the monotone andcontinuous operator, we obtain some sufficient conditions for thepermanence of species. These results are applied to the modelswith rational growth functions and exponential growth functions.We also present numerical examples to illustrate the dynamiccomplexity of systems.

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.

Two-phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and allows for moving contact lines and contact angle hysteresis and pinning through a variational inequality. Moreover, the formulation can be extended to include non-linear contact line motion models. We prove the well-posedness of the time semi-discrete system and fully discrete method using appropriate choices of finite element spaces. A formal energy law is derived for the semi-discrete and fully discrete formulations and preliminary error estimates are also given. Simulation results are presented for a droplet in multiple configurations to illustrate the method.

We give an analysis of the stability and uniqueness of the simplylaminated microstructure for all three tetragonal to monoclinicmartensitic transformations. The energy density for tetragonal tomonoclinic transformations has four rotationally invariant wells sincethe transformation has four variants. One of these tetragonal tomonoclinic martensitic transformations corresponds to the shearing ofthe rectangular side, one corresponds to the shearing of the squarebase, and one corresponds to the shearing of the plane orthogonal to adiagonal in the square base. We show that the simply laminatedmicrostructure is stable except for a class of special materialparameters. In each case that the microstructure is stable, we deriveerror estimates for the finite element approximation.

Our aim here is to study the thermal diffusion phenomenon in a forced convective flow. A system of nonlinear parabolic equations governs the evolution of the mass fractions in multicomponent mixtures. Some existence and uniqueness results are given under suitable conditions onstate functions. Then, we present a numerical scheme based on a "mixed finite element"method adapted to a finite volume scheme, of which we give numerical analysis. In a last part, we apply an homogenization technique to the studied equations in order to obtain an efficient modelling of Soret effect and adsorption in a porous medium at a macroscopic scale.

This contribution gives an overview of current research in applying object oriented programming to scientific computing at the computational mechanics laboratory (LABMEC) at the school of civil engineering – UNICAMP. The main goal of applying object oriented programming to scientific computing is to implement increasingly complex algorithms in a structured manner and to hide the complexity behind a simple user interface. The following areas are current topics of research and documented within the paper: hp-adaptive finite elements in one-, two- and three dimensions with the development of automatic refinement strategies, multigrid methods applied to adaptively refined finite element solution spaces and parallel computing.

This article discusses the numerical approximation oftime dependent Ginzburg-Landau equations. Optimal error estimates which are robust with respectto a large Ginzburg-Landau parameter are established for asemi-discrete in time and a fully discrete approximationscheme. The proofs rely on an asymptotic expansion of the exact solution and a stability resultfor degree-one Ginzburg-Landau vortices. The error boundsprove that degree-one vortices can be approximated robustlywhile unstable higher degree vortices are critical.

The aim of this paper is to investigate the stabilityof boundary layers which appear in numerical solutionsof hyperbolic systems of conservation laws in one spacedimension on regular meshes. We prove stability under a sizecondition for Lax Friedrichs type schemes and inconditionnalstability in the scalar case. Examples of unstable boundary layersare also given.

A new system of integral equations for the exterior 2D time harmonicscattering problem is investigated. This system was first proposed by B. Després in [11]. Two new derivations of this system are given:one from elementary manipulationsof classical equations, the other based on a minimization of a quadratic functional. Numerical issues are addressed to investigate the potentialof the method.

We consider a fully practical finite element approximation of the following degenerate system $$ {\frac{\partial }{\partial t}} \rho(u)- \nabla . ( \,\alpha(u) \,\nabla u ) \ni \sigma(u)\,|\nabla\phi|^2 ,\quad \nabla . (\, \sigma(u) \,\nabla \phi ) = 0$$ subject to an initial condition on the temperature, u,and boundary conditions on both u and the electric potential, ϕ. In the above p(u) is the enthalpy incorporating the latent heat of melting, α(u) > 0 is the temperature dependent heat conductivity, and σ(u) > 0is the electricalconductivity. The latter is zero in the frozen zone, u ≤ 0,which gives rise to the degeneracy in this Stefan system.In addition to showing stability bounds, we prove (subsequence) convergence of our finite element approximation intwo and three space dimensions. The latter is non-trivial due to the degeneracy in σ(u)and the quadratic nature of the Joule heating term forcing the Stefan problem.Finally, some numerical experiments are presented in two space dimensions.

We study the time-harmonic acoustic scattering in a duct in presence of a flow and of adiscontinuous impedance boundary condition. Unlike a continuous impedance, a discontinuousone leads to still open modeling questions, as in particular the singularity of thesolution at the abrupt transition and the choice of the right unknown to formulate thescattering problem. To address these questions we propose a mathematical approach based onvariational formulations set in weighted Sobolev spaces. Considering the discontinuousimpedance as the limit of a continuous boundary condition, we prove that only the problemformulated in terms of the velocity potential converges to a well-posed problem. Moreoverwe identify the limit problem and determine some Kutta-like condition satisfied by thevelocity: its convective derivative must vanish at the ends of the impedance area. Finallywe justify why it is not possible to define limit problems for the pressure and thedisplacement. Numerical examples illustrate the convergence process.