Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the $L^2$-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier--Stokes equations. The novel error estimators only take the $\mathrm{curl}$ of the right-hand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressure-dominant situations. This is also confirmed by some numerical examples with the novel pressure-robust modifications of the Taylor--Hood and mini finite element methods

A Variational Approach to the Spinless Relativistic Coulomb Problem

By application of a straightforward variational procedure we derive a simple, analytic upper bound on the ground-state energy eigenvalue of a semirelativistic Hamiltonian for (one or two) spinless particles which experience some Coulomb-type interaction.Comment: 7 pages, HEPHY-PUB 606/9

High-resolution transmission electron microscopy investigation of diffusion in metallic glass multilayer films

Lack of plasticity is one of the main disadvantages of metallic glasses. One of the solutions to this problem can be composite materials. Diffusion bonding is promising for composite fabrication. In the present work the diffusion process in glassy multilayer films was investigated. A combination of advanced transmission electron microscopy (TEM)methods and precision sputtering techniques allows visualization and study of diffusion in amorphous metallic layers with high resolution. Multilayered films were obtained by radio frequency sputter deposition of Zr-Cu and Zr-Pd. The multilayers were annealed under a high vacuum (10 −5 Pa)for 1 and 5 h at 400 °C, that is, well below the crystallization temperatures but very close to the glass-transition temperatures of both types of the glassy layer. The structural evolution in the deposited films was investigated by high-resolution transmission electron microscopy. It was observed that, despite the big differences in the atomic mass and size, Pd and Cu have similar diffusion coefficients. Surprisingly, 1 h of annealing results in formation of metastable copper nanocrystals in the Zr-Cu layers which, however, disappear after 5 h of annealing. This effect may be connected with nanovoid formation under a complex stress state evolving upon annealing, and is related to the exceptionally slow relaxation of the glassy layers sealed with a Ta overlayer.The authors acknowledge the financial support through the European Research Council under the ERC Advanced Grants INTELHYB (grant ERC-2013-ADG-340025) and ExtendGlass (grant ERC-2015-AdG-695487), the German Science Foundation (DFG) under the grant SO 1518/1-1, and the Ministry of Education and Science of the Russian Federation in the framework of the ‘Increase Competitiveness’ program of NUST ‘MISiS’ (№ К2-2014-013 and К2-2017-089)

Variational Estimation of the Wave Function at Origin for Heavy Quarkonium

The wave function at the origin (WFO) is an important quantity in studying many physical problems concerning heavy quarkonia. However, when one used the variational method with fewer parameters, in general, the deviation of resultant WFO from the "accurate" solution was not well estimated. In this paper, we discuss this issue by employing several potential forms and trial wave functions in detail and study the relation between WFO and the reduced mass.Comment: 17 pages, .zip file of the LATEX2

Instantaneous Bethe-Salpeter equation: utmost analytic approach

The Bethe-Salpeter formalism in the instantaneous approximation for the interaction kernel entering into the Bethe-Salpeter equation represents a reasonable framework for the description of bound states within relativistic quantum field theory. In contrast to its further simplifications (like, for instance, the so-called reduced Salpeter equation), it allows also the consideration of bound states composed of "light" constituents. Every eigenvalue equation with solutions in some linear space may be (approximately) solved by conversion into an equivalent matrix eigenvalue problem. We demonstrate that the matrices arising in these representations of the instantaneous Bethe-Salpeter equation may be found, at least for a wide class of interactions, in an entirely algebraic manner. The advantages of having the involved matrices explicitly, i.e., not "contaminated" by errors induced by numerical computations, at one's disposal are obvious: problems like, for instance, questions of the stability of eigenvalues may be analyzed more rigorously; furthermore, for small matrix sizes the eigenvalues may even be calculated analytically.Comment: LaTeX, 23 pages, 2 figures, version to appear in Phys. Rev.

Pion Generalized Dipole Polarizabilities by Virtual Compton Scattering πe→πeγ\pi e \to \pi e\gamma

We present a calculation of the cross section and the event generator of the reaction $\pi e\to \pi e \gamma$. This reaction is sensitive to the pion generalized dipole polarizabilities, namely, the longitudinal electric $\alpha_L(q^2)$, the transverse electric $\alpha_T(q^2)$, and the magnetic $\beta(q^2)$ which, in the real-photon limit, reduce to the ordinary electric and magnetic polarizabilities $\bar{\alpha}$ and $\bar{\beta}$, respectively. The calculation of the cross section is done in the framework of chiral perturbation theory at ${\cal O}(p^4)$. A pion VCS event generator has been written which is ready for implementation in GEANT simulation codes or for independent use.Comment: 33 pages, Revtex, 15 figure

Finite element simulation of three-dimensional free-surface flow problems

An adaptive finite element algorithm is described for the stable solution of three-dimensional free-surface-flow problems based primarily on the use of node movement. The algorithm also includes a discrete remeshing procedure which enhances its accuracy and robustness. The spatial discretisation allows an isoparametric piecewise-quadratic approximation of the domain geometry for accurate resolution of the curved free surface. The technique is illustrated through an implementation for surface-tension-dominated viscous flows modelled in terms of the Stokes equations with suitable boundary conditions on the deforming free surface. Two three-dimensional test problems are used to demonstrate the performance of the method: a liquid bridge problem and the formation of a fluid droplet

Meson exchange and nucleon polarizabilities in the quark model

Modifications to the nucleon electric polarizability induced by pion and sigma exchange in the q-q potentials are studied by means of sum rule techniques within a non-relativistic quark model. Contributions from meson exchange interactions are found to be small and in general reduce the quark core polarizability for a number of hybrid and one-boson-exchange q-q models. These results can be explained by the constraints that the baryonic spectrum impose on the short range behavior of the mesonic interactions.Comment: 11 pages, 1 figure added, expanded discussio