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L'objet de cet article consiste en une approximation d'une variante des équations de mouvement stationnaire d'un fluide incompressible de troisième grade, en dimension 2: $$\begin{gathered} - v\Delta u + rot(u - \alpha _1 \Delta u) \wedge u - (\alpha _1 + \alpha _2 )(A\Delta u + 2 div(\nabla u\nabla u^T )) \hfill \\ - \beta div(|A|^2 A) + \nabla p + \varepsilon \Delta ^2 u = f, \hfill \\ divu = 0, \hfill \\ \end{gathered}$$ qui sont une généralisation des équations de Navier-Stokes. Dans une première partie, on donne une caractérisation fondamentale de l'espaceV [Hm(Ω)]n , oùV={υ∈[D(Ω)] n ], div υ=0}. On étudie ensuite, dans une seconde partie, une approximation mixte du problème linéaire associé: $$\begin{gathered} - v\Delta u + \varepsilon \Delta ^2 u + \nabla p = f, \hfill \\ div u = 0. \hfill \\ \end{gathered}$$ Les résultats obtenus sont utilisés dans la dernière partie consacrée à une méthode d'approximation mixte de notre problème. La méthode de Taylor-Hood nous permet enfin d'obtenir des applications aux éléments finis de degré 2.

The possibility to accelerate the Picard-Lindelöf process by taking linear combinations of the iterates is discussed. The convergence is studied on infinitely long intervals.

This paper considers local convergence properties of inexact partitioned quasi-Newton algorithms for the solution of certain non-linear equations and, in particular, the optimization of partially separable objective functions. Using the bounded deterioration principle, one obtains local and linear convergence, which impliesQ-superlinear convergence under the usual conditions on the quasi-Newton updates. For the optimization case, these conditions are shown to be satisfied by any sequence of updates within the convex Broyden class, even if some Hessians are singular at the minimizer. Finally, local andQ-superlinear convergence is established for an inexact partitioned variable metric method under mild assumptions on the initial Hessian approximations.

Discrete approximations are constructed to a nonlinear evolutionary system of partial differential equations arising from modelling the dynamics of solid-state phase transitions of thermomechenical nature in the case of one space dimension. The class of problems considered includes the so-called shape memory alloys, in particular. It is shown that the obtained discrete solutions converge to the solution of the original problem, and numerical simulations for the shape memory alloy Au23Cu30Zn47 demonstrate the quality of the discrete model.

In classical numerical analysis the asymptotic convergence factor (R 1-factor) of an iterative processx m+1=Axm+b coincides with the spectral radius of then×n iteration matrixA. Thus the famous Theorem of Stein and Rosenberg can at least be partly reformulated in terms of asymptotic convergence factor. Forn×n interval matricesA with irreducible upper bound and nonnegative lower bound we compare the asymptotic convergence factor (α T ) of the total step method in interval analysis with the factorα S of the corresponding single step method. We derive a result similar to that of the Theorem of Stein and Rosenberg. Furthermore we show thatα S can be less than the spectral radius of the real single step matrix corresponding to the total step matrix |A| where |A| is the absolute value ofA. This answers an old question in interval analysis.

Two Rosenbrock-Wanner type methods for the numerical treatment of differential-algebraic equations are presented. Both methods possess a stepsize control and an index-1 monitor. The first method DAE34 is of order (3)4 and uses a full semi-implicit Rosenbrock-Wanner scheme. The second method RKF4DA is derived from the Runge-Kutta-Fehlberg 4(5)-pair, where a semi-implicit Rosenbrock-Wanner method is embedded, in order to solve the nonlinear equations. The performance of both methods is discussed in artificial test problems and in technical applications.

In this paper, we analyze the approximation of acoustic waves in a two layered media by a finite diffrences variational scheme. We examine in particular the approximation of the guided waves. We point out the existence of purely numerical parasitic phenomena and quantify the numerical dispersion relative to guided waves.