Mass and momentum conservation of the least-squares spectral collocation method for the Navier-Stokes equations

From the literature, it is known that the Least-Squares Spectral Element Method (LSSEM) for the stationary Stokes equations performs poorly with respect to mass conservation but compensates this lack by a superior conservation of momentum. Furthermore, it is known that the Least-Squares Spectral Collocation Method (LSSCM) leads to superior conservation of mass and momentum for the stationary Stokes equations. In the present paper, we consider mass and momentum conservation of the LSSCM for time-dependent Stokes and Navier-Stokes equations. We observe that the LSSCM leads to improved conservation of mass (and momentum) for these problems. Furthermore, the LSSCM leads to the well-known time-dependent profiles for the velocity and the pressure profiles. To obtain these results, we use only a few elements, each with high polynomial degree, avoid normal equations for solving the overdetermined linear systems of equations and introduce the Clenshaw-Curtis quadrature rule for imposing the average pressure to be zero. Furthermore, we combined the transformation of Gordon and Hall (transfinite mapping) with the least-squares spectral collocation scheme to discretize the internal flow problems.     

Spectral methods with sparse matrices

Summary Spectral methods employ global polynomials for approximation. Hence they give very accurate approximations for smooth solutions. Unfortunately, for Dirichlet problems the matrices involved are dense and have condition numbers growing asO(N ⁴) for polynomials of degree N in each variable. We propose a new spectral method for the Helmholtz equation with a symmetric and sparse matrix whose condition number grows only asO(N ²). Certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system. Numerical results are presented which show that we have probably found the most effective solver for spectral systems.     

Quasi-ballistic evolution of a quantum particle in a random potential with arbitrary correlation time

The evolution of the mean kinetic energy and of the mean square displacement of a quantum particle in a time-dependent random potential is studied by perturbation theory in the near ballistic regime. Convenient general formulas are derived for an arbitrary correlation time, τ, of the disorder. These formulas are studied analytically near the limit of perfect dynamic disorder (τ=0) and for static disorder (τ=∞), where detailed comparison is made with earlier results. This work is the first to relate the limits of perfect dynamic disorder and of static disorder via a unified treatment for finite τ.     

Image potential and induced dipole at inhomogeneous metal surfaces

A modified specular reflection model for the response of an inhomogeneous jellium surface discussed in a previous paper by the author is applied to the calculation of the static image potential and of the dipole moment formed by an external point charge and the induced charge distribution at the surface. Detailed results are given and compared with those of earlier treatments.     

Preparation and analysis of heart and skeletal muscle specimens with LAMMA

Summary A technique is described to obtain shockfrozen tissue specimens from a beating heart in situ using liquid propane as cooling agent. The method allows to take up to 8 samples from the ischemic area of the left ventricle of a pig heart following acute coronary artery occlusion.A freeze-drying apparatus was developed in which, under a vacuum of 10^–6 mm Hg, the specimens remain for 10 days with the temperature slowly rising from initially –100 C to room temperature. Freezing damages in the tissues samples due to the formation of ice crystals either during shock-freezing or freeze-drying are usually below the spatial resolution of the microscope system employed in the LAMMA instrument. Already after a few minutes of ischemia, a considerable decrease of the intracellular K/Na relation was measured. LAMMA spectra from skeletal muscle were taken to check for possible ion redistribution which had eventually occurred during the various steps of the preparation procedure employed.Supported by Deutsche Forschungsgemeinschaft, SFB 68, A 10     

Force between metallic films at small separation

An exact expression for the force between two dissimilar metal films in the limit of vanishing separation is derived in terms of bulk properties of the metals. This force is compared with the results of an approximate numerical treatment of Ferrante and Smith for finite separations between the films.     

Determination of lead in geological and biological materials by graphite furnace atomic absorption spectrometry

Summary Serious systematic errors inherent in the determination of lead in geological and biological samples by flameless atomic-absorption spectrometry are demonstrated. The reduced absorption of lead is due to partial interaction with alkaline, alkaline-earth and iron chlorides during the atomization stage. Incomplete dissociation of the volatile monochloride of lead in the gaseous phase reduces the absorption signal. An addition of 20% hydrogen to nitrogen (purge gas) diminishes signal suppression by removing the chlorine in form of volatile HCl. The detection limit is about 0.6 ppm in rocks and about 0.3 ppm in plant materials. Depending on the content in the samples the relative standard deviation is between 1 and 12%. The accuracy of the method was tested on 39 international standard reference samples.

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Bestimmung von Blei in geologischen und biologischen Materialien durch AAS mit der Graphitrohrküvette

Zusammenfassung Schwerwiegende systematische Fehler bei der Bestimmung von Blei in geologischen und biologischen Materialien mit der Graphitrohrküvette werden aufgezeigt. Die Reaktion von Blei mit Alkali-, Erdalkali- und Eisenchloriden während der Atomisierungsphase führt zur Bildung von Bleimonochlorid, das nur unvollständig dissoziiert und eine Signalunterdrückung zur Folge hat. Ein Zusatz von 20% Wasserstoff zu Stickstoff als Spülgas vermindert die Signalunterdrückung, indem überschüssiges Chlor als Chlorwasserstoff entfernt wird. Die Nachweisgrenze liegt bei ca. 0,6 ppm für Gesteine und bei ca. 0,3 ppm für Pflanzenmaterialien. Je nach der Konzentration beträgt die relative Standardabweichung 1–12%. Die Genauigkeit der Methode wurde an 39 internationalen Standardreferenzproben überprüft.

     

Static and Time Dependent Density Functional Theory with Internal Degrees of Freedom: Merits and Limitations Demonstrated for the Potts Model

We present an extension of the density-functional theory (DFT) formalism for lattice gases to systems with internal degrees of freedom. In order to test approximations commonly used in DFT approaches, we investigate the statics and dynamics of occupation (density) profiles in the one-dimensional Potts model. In particular, by taking the exact functional for this model we can directly evaluate the quality of the local equilibrium approximation used in time-dependent density-functional theory (TDFT). Excellent agreement is found in comparison with Monte Carlo simulations. Finally, principal limitations of TDFT are demonstrated.     

Konvergenzaussagen für Kollokationsverfahren bei elliptischen Randwertaufgaben

Summary We examine the convergence of collocation with polynomials for elliptic problems. The general theory is imbedded in theL ^p -theory from which known results of existence and regularity are adopted. Convergence results are deduced in the cases of periodic, Dirichlet and mixed boundary conditions. It becomes obvious that the order of convergence essentially depends on the smoothness of the solution.

Dieser Aufsatz enthält Ergebnisse aufbauend auf der Dissertation des Verfassers, die von Prof. Dr. K. Witsch, Universität Düsseldorf, angeregt und unterstützt wurde