In this paper, the convergence of a Neumann-Dirichlet algorithm to approximateCoulomb's contact problem between two elastic bodies is proved in a continuous setting. In this algorithm, the natural interface between the two bodies is retained as a decomposition zone.

We consider mathematical models describing dynamics of an elastic beam which is clamped at its left end to a vibrating support and which can move freely at its right end between two rigid obstacles. We model the contact with Signorini's complementary conditions between the displacement and the shear stress. For this infinite dimensional contact problem, we propose a family of fully discretized approximations and their convergence is proved. Moreover some examples of implementation are presented. The results obtained here are also valid in the case of a beam oscillating between two longitudinal rigid obstacles.

We consider a degenerate parabolic system which modelsthe evolution of nematic liquid crystal with variable degree of orientation.The systemis a slight modificationto that proposed in [Calderer et al., SIAM J. Math. Anal.33 (2002) 1033–1047], which is a special case of Ericksen's general continuum model in [Ericksen, Arch. Ration. Mech. Anal.113 (1991) 97–120]. We prove the global existence of weak solutions by passing to the limit in a regularized system. Moreover, we propose a practical fully discrete finite element method for this regularized system, and we establish the (subsequence) convergence of this finite element approximation to the solution of the regularized system as the mesh parameters tend to zero; andto a solution of the original degenerate parabolic system when the the mesh and regularization parameters all approach zero. Finally, numerical experiments are included which show the formation, annihilation and evolution of line singularities/defects in such models.

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.

Les méthodes sans maillage emploient une interpolation associée à un ensemble de particules : aucune information concernant la connectivité ne doit être fournie.Un des atouts de ces méthodes est que la discrétisationpeut être enrichie d'une façon très simple, soit en augmentant le nombre de particules (analogue à lastratégie de raffinement h), soit en augmentant l'ordre de consistance (analogueà la stratégie de raffinement p). Néanmoins, le coût du calcul des fonctionsd'interpolation est très élevé et ceci représente un inconvénient vis-à-visdes éléments finis. Cet article présente une interpolation mixte élémentsfinis-particules qui résulte de la généralisation de plusieurs travaux dans cedomaine. La formulation de cette interpolation mixte est valable pour n'importe quel ordre de consistance. Dans ce contexte, onénonce un estimateur d'erreur a priori dont la démonstrationse base dans les propriétés de l'interpolation mixte.Ce résultat permet d'étudier la convergence de laméthode d'enrichissement et d'établir les stratégies de raffinement del'interpolation qui permettent d'atteindre une solutionavec une précision satisfaisante.

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.

The evolution of n–dimensional graphs under a weighted curvature flow is approximated by linear finite elements. We obtain optimal error bounds for the normals and the normal velocities of the surfacesin natural norms. Furthermore we prove a global existence result for thecontinuous problem and present some examples of computed surfaces.