Лекция: Semigroups of Linear Fractional Transformations on the Operator Ball
By David Shoiket.
In this talk we consider semigroups of those linear operators on Krein and Pontryagin spaces which are nonexpansive with respect to indefinite metric. These semigroups involve the consideration of semi-groups of fractional-linear transformation of the operator ball over a Hilbert space, which are nonexpansive with respect to the hyperbolic metric. We find optimal convergence for such semigroups to interior stationary and boundary sink points by a unified method. To do this, we use a special non-Euclidian «distance» which induces the original topology. In addition, we employthe notion of monotonicity with respect to the hyperbolic metric to get the order of convergence in terms of the numerical range of semigroup generators and its lower bounds.
Our approach leads to new results even in the one-dimensional case. For semigroups consisting of holomorphic self-mappings, we obtain the rather unexpected phenomenon of universal rates of convergence of exponential type. In particular, in the case of a boundary sink point we establish a continuous analog of the celebrated Julia-Wolff-Caratheodory Theorem. In addition, we discuss the so-called flow invariance (isometric) problem, and Koenigs embedding problem. [Abstracts, 13th ILAS* Conference, Amsterdam, 2006, p. 51.]
*ILAS abbrev International Linear Algebra Society
ON FEM/FV-SIMULATION OF THERMAL CONVECTIVE
FLOWS IN UPPER EATHTH'S MANTLE
M.A. Kochergina
Department of Applied Mathematics
Novosibirsk State Technical University
The physical problem of thermal convection flows in the upper Earth's mantle is described by the Navier-Stokes equations with Oberbeck-Boussinesq approximation. Moderncomputational methods allow for working with complex geometry. Therefore FEM/FV-approximations for unstructured grid are considered in the work.
The modified finite volume method (MFVM) first introduced by E.P. Shurina and T.V. Voitovich is used for discretization of transport equation. This method uses ideology and data structure of FVM to facilitate their joint application.
Inagreement with observations about FE modeling of thermal convective flows in other studies, we have isolated the violation of incompressibility constraint as the major culprit in the failure of simulation. For the purpose ofeliminating this deficiency, several modifications are considered in the investigation. These include implementation of iterative solution technique to better accommodate the dual role of pressure to both satisfy the continuity equation and balance terms in the momentum equation. With regard todiscretization of the convective terms, FLO scheme with exponential flow-oriented shape functions has been studied.