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

Quality of Variational Trial States

Besides perturbation theory (which clearly requires the knowledge of the exact unperturbed solution), variational techniques represent the main tool for any investigation of the eigenvalue problem of some semibounded operator H in quantum theory. For a reasonable choice of the employed trial subspace of the domain of H, the lowest eigenvalues of H usually can be located with acceptable precision whereas the trial-subspace vectors corresponding to these eigenvalues approximate, in general, the exact eigenstates of H with much less accuracy. Accordingly, various measures for the accuracy of the approximate eigenstates derived by variational techniques are scrutinized. In particular, the matrix elements of the commutator of the operator H and (suitably chosen) different operators with respect to degenerate approximate eigenstates of H obtained by variational methods are proposed as new criteria for the accuracy of variational eigenstates. These considerations are applied to precisely that Hamiltonian for which the eigenvalue problem defines the well-known spinless Salpeter equation. This bound-state wave equation may be regarded as (the most straightforward) relativistic generalization of the usual nonrelativistic Schroedinger formalism, and is frequently used to describe, e.g., spin-averaged mass spectra of bound states of quarks.Comment: LaTeX, 7 pages, version to appear in Physical Review

Semi-Relativistic Hamiltonians of Apparently Nonrelativistic Form

We construct effective Hamiltonians which despite their apparently nonrelativistic form incorporate relativistic effects by involving parameters which depend on the relevant momentum. For some potentials the corresponding energy eigenvalues may be determined analytically. Applied to two-particle bound states, it turns out that in this way a nonrelativistic treatment may indeed be able to simulate relativistic effects. Within the framework of hadron spectroscopy, this lucky circumstance may be an explanation for the sometimes extremely good predictions of nonrelativistic potential models even in relativistic regions.Comment: 20 pages, LaTeX, no 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

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

TMEM16B, a novel protein with calcium-dependent chloride channel activity, associates with a presynaptic protein complex in photoreceptor terminals

Photoreceptor ribbon synapses release glutamate in response to graded changes in membrane potential evoked by vast, logarithmically scalable light intensities. Neurotransmitter release is modulated by intracellular calcium levels. Large Ca2+-dependent chloride currents are important regulators of synaptic transmission from photoreceptors to second-order neurons; the molecular basis underlying these currents is unclear. We cloned human and mouse TMEM16B, a member of the TMEM16 family of transmembrane proteins, and show that it is abundantly present in the photoreceptor synaptic terminals in mouse retina. TMEM16B colocalizes with adaptor proteins PSD95, VELI3, and MPP4 at the ribbon synapses and contains a consensus PDZ class I binding motif capable of interacting with PDZ domains of PSD95. Furthermore, TMEM16B is lost from photoreceptor membranes of MPP4-deficient mice. This suggests that TMEM16B is a novel component of a presynaptic protein complex recruited to specialized plasma membrane domains of photoreceptors. TMEM16B confers Ca2+-dependent chloride currents when overexpressed in mammalian cells as measured by halide sensitive fluorescent protein assays and whole-cell patch-clamp recordings. The compartmentalized localization and the electrophysiological properties suggest TMEM16B to be a strong candidate for the long sought-after Ca2+-dependent chloride channel in the photoreceptor synapse

Multigrid algorithms for hp-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes

In this paper we analyze the convergence properties of two-level and W-cycle multigrid solvers for the numerical solution of the linear system of equations arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polygonal/polyhedral meshes. We prove that the two-level method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large. An analogous result is obtained for the W-cycle multigrid algorithm, which is proved to be uniformly convergent with respect to the mesh size, the polynomial approximation degree, and the number of levels, provided the latter remains bounded and the number of smoothing steps is chosen sufficiently large. Numerical experiments are presented which underpin the theoretical predictions; moreover, the proposed multilevel solvers are shown to be convergent in practice, even when some of the theoretical assumptions are not fully satisfied