Variationelle Zeitdiskretisierungen höherer Ordnung der Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängigen Gebieten

Wir betrachten zeitabhängige Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängi- gen Gebieten, wobei die Bewegung des Gebietsrandes bekannt ist. Die zeitliche Entwicklung des Gebietes wird durch die ALE-Formulierung behandelt, die die Nachteile der klassischen Euler- und Lagrange-Betrachtungsweisen behebt. Die Position des Randes und seine Geschwindigkeit werden dabei so in das Gebietsinnere fortgesetzt, dass starke Gitterdeformationen verhindert werden. Als Zeitdiskretisierungen höherer Ordnung werden stetige Galerkin-Petrov-Verfahren (cGP) und unstetige Galerkin-Verfahren (dG) auf Probleme in zeitabhängigen Gebieten angewendet. Weiterhin werden das C 1 -stetige Galerkin-Petrov-Verfahren und das C 0 -stetige Galerkin- Verfahren vorgestellt. Deren Lösungen lassen sich auch in zeitabhängigen Gebieten durch ein einfaches einheitliches Postprocessing aus der Lösung des cGP-Problems bzw. dG-Problems erhalten. Für Problemstellungen in festen Gebieten und mit zeitlich konstanten Konvektions- und Reaktionstermen werden Stabilitätsresultate sowie optimale Fehlerabschätzungen für die nachbereiteten Lösungen der cGP-Verfahren und der dG-Verfahren angegeben. Für zeitabhängige Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängigen Gebieten präsentieren wir konservative und nicht-konservative Formulierungen, wobei eine besondere Aufmerksamkeit der Behandlung der Zeitableitung und der Gittergeschwindigkeit gilt. Stabilität und optimale Fehlerschätzungen für die in der Zeit semi-diskretisierten konservativen und nicht-konservativen Formulierungen werden vorgestellt. Abschließend wird das volldiskretisierte Problem betrachtet, wobei eine Finite-Elemente-Methode zur Ortsdiskretisierung der Konvektions-Diffusions-Reaktions-Gleichungen in zeitabhängigen Gebieten im ALE-Rahmen einbezogen wurde. Darüber hinaus wird eine lokale Projektionsstabilisierung (LPS) eingesetzt, um der Konvektionsdominanz Rechnung zu tragen. Weiterhin wird numerisch untersucht, wie sich die Approximation der Gebietsgeschwindigkeit auf die Genauigkeit der Zeitdiskretisierungsverfahren auswirkt.

Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations I: Method Formulation

In this article we consider the development of discontinuous Galerkin finite element methods for the numerical approximation of the compressible Navier-Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the interior penalty method, originally developed for the numerical approximation of linear self-adjoint second-order elliptic partial differential equations. In order to solve the resulting system of nonlinear equations, we exploit a (damped) Newton-GMRES algorithm. Numerical experiments demonstrating the practical performance of the proposed discontinuous Galerkin method with higher-order polynomials are presented.

Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations II: Goal--Oriented A Posteriori Error Estimation

In this article we consider the application of the generalization of the symmetric version of the interior penalty discontinuous Galerkin finite element method to the numerical approximation of the compressible Navier--Stokes equations. In particular, we consider the a posteriori error analysis and adaptive mesh design for the underlying discretization method. Indeed, by employing a duality argument (weighted) Type I a posteriori bounds are derived for the estimation of the error measured in terms of general target functionals of the solution; these error estimates involve the product of the finite element residuals with local weighting terms involving the solution of a certain dual problem that must be numerically approximated. This general approach leads to the design of economical finite element meshes specifically tailored to the computation of the target functional of interest, as well as providing efficient error estimation. Numerical experiments demonstrating the performance of the proposed approach will be presented.

Formulation and analysis of a diffusion-velocity particle model for transport-dispersion equations

The modelling of diffusive terms in particle methods is a delicate matter and several models were proposed in the literature to take such terms into account. The diffusion velocity method (DVM), originally designed for the diffusion of passive scalars, turns diffusive terms into convective ones by expressing them as a divergence involving a so-called diffusion velocity. In this paper, DVM is extended to the diffusion of vectorial quantities in the three-dimensional Navier–Stokes equations, in their incompressible, velocity–vorticity formulation. The integration of a large eddy simulation (LES) turbulence model is investigated and a DVM general formulation is proposed. Either with or without LES, a novel expression of the diffusion velocity is derived, which makes it easier to approximate and which highlights the analogy with the original formulation for scalar transport. From this statement, DVM is then analysed in one dimension, both analytically and numerically on test cases to point out its good behaviour.

Instantons on Calabi–Yau cones

The Hermitian Yang–Mills equations on certain vector bundles over Calabi–Yau cones can be reduced to a set of matrix equations; in fact, these are Nahm-type equations. The latter can be analysed further by generalising arguments of Donaldson and Kronheimer used in the study of the original Nahm equations. Starting from certain equivariant connections, we show that the full set of instanton equations reduce, with a unique gauge transformation, to the holomorphicity condition alone.

Beyond the Cosmological Principle

Simplifying the Einstein field equation by assuming the cosmological principle yields a set of differential equations which governs the dynamics of the universe as described in the cosmological standard model. The cosmological principle assumes the space appears the same everywhere and in every direction and moreover, the principle has earned its position as a fundamental assumption in cosmology by being compatible with the observations of the 20th century. It was not until the current century when observations in cosmological scales showed significant deviation from isotropy and homogeneity implying the violation of the principle. Among these observations are the inconsistency between local and non-local Hubble parameter evaluations, baryon acoustic features of the Lyman-α forest and the anomalies of the cosmic microwave background radiation. As a consequence, cosmological models beyond the cosmological principle have been studied vastly; after all, the principle is a hypothesis and as such should frequently be tested as any other assumption in physics. In this thesis, the effects of inhomogeneity and anisotropy, arising as a consequence of discarding the cosmological principle, is investigated. The geometry and matter content of the universe becomes more cumbersome and the resulting effects on the Einstein field equation is introduced. The cosmological standard model and its issues, both fundamental and observational are presented. Particular interest is given to the local Hubble parameter, supernova explosion, baryon acoustic oscillation, and cosmic microwave background observations and the cosmological constant problems. Explored and proposed resolutions emerging by violating the cosmological principle are reviewed. This thesis is concluded by a summary and outlook of the included research papers.

Normative reference values for the 20-meter shuttlerun test in a population-based sample of schoolchildren in Bogota, Colombia: The FUPRECOL Study

Our aim was to determine the normative reference values of cardiorespiratory fitness (CRF) and to establish the proportion of subjects with low CRF suggestive of future cardio-metabolic risk.

Charge accumulation and transport in degenerately doped semiconducting polymers with mixed ionic and electronic conductivity

This thesis is part of the fields of Material Physics and Organic Electronics and aims to determine the charge carrier density and mobility in the hydrated conducting polymer–polyelectrolyte blend PEDOT:PSS. This kind of material combines electronic semiconductor functionality with selective ionic transport, biocompatibility and electrochemical stability in water. This advantageous material properties combination makes PEDOT:PSS a unique material to build organic electrochemical transistors (OECTs), which have relevant application as amplifying transducers for bioelectronic signals. In order to measure charge carrier density and mobility, an innovative 4-wire, contact independent characterization technique was introduced, the electrolyte-gated van der Pauw (EgVDP) method, which was combined with electrochemical impedance spectroscopy. The technique was applied to macroscopic thin film samples and micro-structured PEDOT:PSS thin film devices fabricated using photolithography. The EgVDP method revealed to be effective for the measurements of holes’ mobility in hydrated PEDOT:PSS thin films, which resulted to be <μ>=(0.67±0.02) cm^2/(V*s). By comparing this result with 2-point-probe measurements, we found that contact resistance effects led to a mobility overestimation in the latter. Ion accumulation at the drain contact creates a gate-dependent potential barrier and is discussed as a probable reason for the overestimation in 2-point-probe measurements. The measured charge transport properties of PEDOT:PSS were analyzed in the framework of an extended drift-diffusion model. The extended model fits well also to the non-linear response in the transport characterization and results suggest a Gaussian DOS for PEDOT:PSS. The PEDOT:PSS-electrolyte interface capacitance resulted to be voltage-independent, confirming the hypothesis of its morphological origin, related to the separation between the electronic (PEDOT) and ionic (PSS) phases in the blend.

Small polarons in spin-orbit coupled osmates

Small polarons (SP) have been thoroughly investigated in 3d transition metal oxides and they have been found to play a crucial role in physical phenomena such as charge transport, colossal magnetoresistance and surface reactivity. However, our knowledge about these quasi-particles in 5d systems remains very limited, since the more delocalised nature of the 5d orbitals reduces the strength of the Electronic Correlation (EC), making SP formation in these compounds rather unexpected. Nevertheless, the Spin-Orbit coupled Dirac-Mott insulator Ba2NaOsO6 (BNOO) represents a good candidate for enabling polaron formation in a relativistic background, due to the relatively large EC (U ∼ 3 eV) and Jahn-Teller activity. Moreover, anomalous peaks in Nuclear Magnetic Resonance (NMR) spectroscopy experiments suggest the presence of thermally activated SP dynamics when BNOO is doped with Ca atoms. We investigate SP formation in BNOO both from an electronic and structural point of view by means of fully relativistic first principles calculations. Our numerical simulations predict a stable SP ground state and agree on the value of 810 K for the dynamical process peak found by NMR experiments.

Defect states in MAPbBr3 by photo-induced current transient spectroscopy

In the past decade, perovskites have been under the spotlight as promising semicon- ductors with unique properties. Hybrid halide perovskites show excellent characteristic properties suitable for optoelectronic applications as tunable band gap, high absorption coefficient, large mobility and long carrier recombination lifetime. However, a complete understanding of environmental instability and the nature of defects in these materials is still lacking, hindering the development of perovskite-based technologies. In this work we studied MAPbBr3 single crystals, fabricated with Inverse Temperature Crystallization (ITC) technique, with Photo-Induced Current Transient Spectroscopy (PICTS). PICTS is a transient photocurrent measurement rarely employed for studying perovskites mate- rials, that allows for the defects characterization in high resistivity materials. We studied the samples under different conditions, such as negative and positive voltage biases, bias stress, different contact geometries and different illumination wavelengths, in order to study their effect on the material physical properties and to evaluate the trap activation energies and their behavior under different working conditions.