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Dans un récent article (Hairer-Wanner [1]) nous avons donné une théorie à l'aide de laquelle on peut facilement calculer les conditions d'ordre pour une méthode de Nyström. Ici nous montrons comment on peut résoudre ce système d'équations non-linéaires. Nous donnons de plus toutes les méthodes d'ordres pours=2, 3, 4 (oùs−1 indique le nombre d'évaluations de la fonction à chaque pas); des méthodes avec un paramètre d'ordres pours=5, 6 et des méthodes particulières d'ordres−1 pours=8, 9.

The sequences introduced by Carlson (1971) are variants of the Gauss arithmetic geometric sequences (which have been elegantly discussed by D. A. Cox (1984, 1985)). Given (complex)a 0,b 0 we define $$a_{n + 1} : = \tfrac{1}{2}(a_n + b_n ),b_{n + 1} : = \sqrt {(a_n a_{n + 1} )} $$ and our program is to discuss the convergence and limits of the sequences {a n }, {b n } when the determinations of the square root are made according to an assigned pattern. The original assigned pattern was all positive. Carlson's original discussion made use of an invariant integral in the case whena 0,b 0 were non-negative and all determinations were positive. The discussion of the general Gaussian case used a parametrization of the sequences by thetaconstants. Our discussion of the Carlson case will use a parameterization of the sequences by lemniscatefunctions, although it could equally be written in terms of thetafunctions (in the lemniscate case). The complex multiplication of these functions is used essentially. We made considerable use of computers and we record some sample results.

A finite element discretization of the mixed variable formulation of the biharmonic problem is considered. A multilevel algorithm for the numerical solution of the discrete equations is described. Convergence is proved under the assumption ofH 3-regularity.

A successive relaxation iterative algorithm for discrete HJB equations is proposed. Monotone convergence has been proved for the algorithm.

We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational problem. The numerical approximation of the action of the corresponding stiffness matrix consists of three steps: (1) apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space, (2) truncate each elliptic problem to a bounded domain, (3) use the finite element method for the space approximation on each truncated domain. The consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm. The results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain, the domain truncation parameter and the quadrature spacing parameter.

Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the coefficients, and grows at most as the square of the number of levels. We also characterize a class of distributions of the coefficients, called quasi-monotone, for which the weighted $L^{2}$ -projection is stable and for which we can use the standard piecewise linear functions as a coarse space. In this case, we obtain optimal methods, i.e. bounds which are independent of the number of levels and subregions. We also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas. We consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods.