A conjugate gradient method and a multigrid algorithm for Morley s finite element approximation of the biharmonic equation

Summary The numerical solution of the linear equations arising from Morley's nonconforming displacement method is studied. A suitable preconditioning for the conjugate gradient method is described. Furthermore, the nonconformity of the discretization necessitates some non-standard ingredients of multigrid algorithms.     

Approximation on Simplices with Respect to Weighted Sobolev Norms

Inequalities of Jackson and Bernstein type are derived for polynomial approximation on simplices with respect to Sobolev norms. Although we cannot use orthogonal polynomials, sharp estimates are obtained from a decomposition into orthogonal subspaces. The formulas reflect the symmetries of simplices, but analogous estimates on rectangles show that we cannot expect rotational invariance of the terms with derivatives.     

Rational approximation of Stieltjes functions by the Carathéodory-Fejér method

The Carathéodory-Fejér method provides a way for the construction of near best rational approximations. Gutknecht and Trefethen [7], [10] observed that in many cases the constructed functions yield very good estimates of the degree of rational approximation. We will Show that the correct asymptotic is obtained when the method is applied to Stieltjes functions. This fact is of interest in connection with Magnus' considerations of the 1/9-conjecture [9].     

Inelastic electron-deuteron scattering with final-state interactions

Measurements of the cross sectiond ² σ/dΩ dE′ for the inelastic electron-deuteron scattering processe+d→e+n+p have been used to determine the electromagnetic structure of the neutron. The effects on the theoretical cross section of the structure of the deuteron and of interactions between the outgoing nucleons are examined in detail. We start with the relativistic invariant expressions of the one-nucleon exchange diagram including deuteron structure to obtain the interaction Hamiltonian. For the disintegration near the quasi-elastic peak thed-n-p vertex function can be represented by nonrelativistic deuteron wave functions. The final state interaction is calculated from phenomenological nucleon-nucleon potentials. The rescattering corrections are found to lead to a decrease in the peak value of the cross section between ?6% and ?2% depending on the electron momentum transfer q² which has been varied between 2.5f ^?2 and 40f ^?2. These corrections are largest for smallq ²<5f ^?2. Furthermore the results show that the influence of our insufficient knowledge of the deuteron structure on the cross section is small.     

Eine Möglichkeit zur Konvergenzbeschleunigung bei Iterationsverfahren für bestimmte nichtlineare Probleme

Zusammenfassung Nichtlineare Gleichungssysteme, nichtlineare Approximationsaufgaben and andere nichtlineare Probleme behandelt man oft numerisch mit Iterationsverfahren. In jedem Iterationsschritt ist eine lineare Aufgabe zu lösen, die eine Näherung des vorliegenden Problems darstellt. Solche Verfahren lassen sich in einem allgemeinen Rahmen formulieren. Eine Möglichkeit zur Konvergenzbeschleunigung bietet sich dann bei Problemen, die in einem bestimmten Sinne teilweise linear sind, indem der klassische Algorithmus durch einen kleinen Optimierungsprozeß erweitert wird. Für Probleme mit Exponentialsummen erhält man these Form der Beschleunigung auch mit Hilfe von Invarianzbetrachtungen.     

Morse-Theorie für berandete Mannigfaltigkeiten

Ohne Zusammenfassung     

On the degree of approximation by spline functions with free knots

Summary A comparison theorem is derived for Chebyshev approximation by spline functions with free knots. This generalizes a result of Bernstein for approximation by polynomials.