A C0 interior penalty discontinuous Galerkin method for fourth‐order total variation flow I: Derivation of the method and numerical results

We consider the numerical solution of a fourth‐order total variation flow problem representing surface relaxation below the roughening temperature. Based on a regularization and scaling of the nonlinear fourth‐order parabolic equation, we perform an implicit discretization in time and a C0 Interior Penalty Discontinuous Galerkin (C0IPDG) discretization in space. The C0IPDG approximation can be derived from a mixed formulation involving numerical flux functions where an appropriate choice of the flux functions allows to eliminate the discrete dual variable. The fully discrete problem can be interpreted as a parameter dependent nonlinear system with the discrete time as a parameter. It is solved by a predictor corrector continuation strategy featuring an adaptive choice of the time step sizes. A documentation of numerical results is provided illustrating the performance of the C0IPDG method and the predictor corrector continuation strategy. The existence and uniqueness of a solution of the C0IPDG method will be shown in the second part of this paper

Spectral degeneracy and escape dynamics for intermittent maps with a hole

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron-Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds, and demonstrate $L^1$ convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.Comment: 34 page

Theoretical X-Ray Absorption Debye-Waller Factors

An approach is presented for theoretical calculations of the Debye-Waller factors in x-ray absorption spectra. These factors are represented in terms of the cumulant expansion up to third order. They account respectively for the net thermal expansion $\sigma^{(1)}(T)$, the mean-square relative displacements $\sigma^2(T)$, and the asymmetry of the pair distribution function $\sigma^{(3)}(T)$. Similarly, we obtain Debye-Waller factors for x-ray and neutron scattering in terms of the mean-square vibrational amplitudes $u^2(T)$. Our method is based on density functional theory calculations of the dynamical matrix, together with an efficient Lanczos algorithm for projected phonon spectra within the quasi-harmonic approximation. Due to anharmonicity in the interatomic forces, the results are highly sensitive to variations in the equilibrium lattice constants, and hence to the choice of exchange-correlation potential. In order to treat this sensitivity, we introduce two prescriptions: one based on the local density approximation, and a second based on a modified generalized gradient approximation. Illustrative results for the leading cumulants are presented for several materials and compared with experiment and with correlated Einstein and Debye models. We also obtain Born-von Karman parameters and corrections due to perpendicular vibrations.Comment: 11 pages, 8 figure

Nonadiabatic orientation, toroidal current, and induced magnetic field in BeO molecules

It is predicted that oriented BeO molecules would give rise to unprecedentedly strong, unidirectional electric ring current and an associated magnetic field upon excitation by a right or left circularly polarized laser pulse into the first excited degenerate singlet state. The strong toroidal electric ring current of this state is dominated by the ring current of the 1π± orbital about the molecular axis. Our predictions are based on the analysis of the orbital composition of the states involved and are substantiated by high level electronic structure calculations and wavepacket simulations of the laser-driven orientation and excitation [email protected]

Dynamical effects induced by long range activation in a nonequilibrium reaction-diffusion system

We both show experimentally and numerically that the time scales separation introduced by long range activation can induce oscillations and excitability in nonequilibrium reaction-diffusion systems that would otherwise only exhibit bistability. Namely, we show that the Chlorite-Tetrathionate reaction, where autocatalytic species diffuses faster than the substrates, the spatial bistability domain in the nonequilibrium phase diagram is extended with oscillatory and excitability domains. A simple model and a more realistic model qualitatively account for the observed behavior. The latter model provides quantitative agreement with the experiments.Comment: 19 pages + 9 figure

Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators

We discuss several adaptive mesh-refinement strategies based on (h − h/2)-error estimation. This class of adaptivemethods is particularly popular in practise since it is problem independent and requires virtually no implementational overhead. We prove that, under the saturation assumption, these adaptive algorithms are convergent. Our framework applies not only to finite element methods, but also yields a first convergence proof for adaptive boundary element schemes. For a finite element model problem, we extend the proposed adaptive scheme and prove convergence even if the saturation assumption fails to hold in general

Anisotrope Materialmodellierung für den menschlichen Unterkiefer

Im Rahmen der biomechanischen Simulation knöcherner menschlicher Organe ist die Frage nach einer befriedigenden Materialbeschreibung nach wie vor ungelöst. Computertomographische Datensätze liefern eine räumliche Verteilung der (Röntgen-) Dichte und ermöglichen damit eine gute Darstellung der individuellen Geometrie. Weiter können die verschiedenen Materialbestandteile des Knochens, Spongiosa und Kortikalis, voneinander getrennt werden. Aber die richtungsabhängige Information der Materialanisotropie ist verloren. In dieser Arbeit wird ein Ansatz für eine anisotrope Materialbeschreibung vorgestellt, die es ermöglicht, den Einfluss der individuellen knöchernen Struktur auf das makroskopische Materialverhalten abzuschätzen

Dust in Brown Dwarfs IV. Dust formation and driven turbulence on mesoscopic scales

Dust formation in brown dwarf atmospheres is studied by utilising a model for driven turbulence in the mesoscopic scale regime. We apply a pseudo-spectral method where waves are created and superimposed within a limited wavenumber interval. The turbulent kinetic energy distribution follows the Kolmogoroff spectrum which is assumed to be the most likely value. Such superimposed, stochastic waves may occur in a convectively active environment. They cause nucleation fronts and nucleation events and thereby initiate the dust formation process which continues until all condensible material is consumed. Small disturbances are found to have a large impact on the dust forming system. An initially dust-hostile region, which may originally be optically thin, becomes optically thick in a patchy way showing considerable variations in the dust properties during the formation process. The dust appears in lanes and curls as a result of the interaction with waves, i.e. turbulence, which form larger and larger structures with time. Aiming on a physical understanding of the variability of brown dwarfs, related to structure formation in substellar atmospheres, we work out first necessary criteria for small-scale closure models to be applied in macroscopic simulations of dust forming astrophysical systems.Comment: A&A accepted, 20 page