Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: Strongly monotone quasi-Newtonian flows

In this article, we develop the a priori and a posteriori error analysis of hp-version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain Ω ⊂ ℝd, d = 2, 3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm, which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp-adaptive refinement algorithm.

hp-DGFEM for second-order mixed elliptic problems polyhedra

We prove exponential rates of convergence of hp-version discontinuous Galerkin (dG) interior penalty finite element methods for second-order elliptic problems with mixed Dirichlet-Neumann boundary conditions in axiparallel polyhedra. The dG discretizations are based on axiparallel, σ-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of μ-bounded variation. We consider piecewise analytic solutions which belong to a larger analytic class than those for the pure Dirichlet problem considered in [11, 12]. For such solutions, we establish the exponential convergence of a nonconforming dG interpolant given by local L 2 -projections on elements away from corners and edges, and by suitable local low-order quasi-interpolants on elements at corners and edges. Due to the appearance of non-homogeneous, weighted norms in the analytic regularity class, new arguments are introduced to bound the dG consistency errors in elements abutting on Neumann edges. The non-homogeneous norms also entail some crucial modifications of the stability and quasi-optimality proofs, as well as of the analysis for the anisotropic interpolation operators. The exponential convergence bounds for the dG interpolant constructed in this paper generalize the results of [11, 12] for the pure Dirichlet case.

On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators

We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.

Measurement of flow harmonics with multi-particle cumulants in Pb+Pb collisions at √sNN=2.76 TeV with the ATLAS detector

ATLAS measurements of the azimuthal anisotropy in lead–lead collisions at √sNN = 2.76 TeV are shown using a dataset of approximately 7μb−1 collected at the LHC in 2010. The measurements are performed for charged particles with transversemomenta 0.5 < pT < 20 GeV and in the pseudorapidity range |η| < 2.5. The anisotropy is characterized by the Fourier coefficients, vn, of the charged-particle azimuthal angle distribution for n = 2–4. The Fourier coefficients are evaluated using multi-particle cumulants calculated with the generating function method. Results on the transverse momentum, pseudorapidity and centrality dependence of the vn coefficients are presented. The elliptic flow, v2, is obtained from the two-, four-, six- and eight-particle cumulants while higher-order coefficients, v3 and v4, are determined with two- and four-particle cumulants. Flow harmonics vn measured with four-particle cumulants are significantly reduced compared to the measurement involving two-particle cumulants. A comparison to vn measurements obtained using different analysis methods and previously reported by the LHC experiments is also shown. Results of measurements of flow fluctuations evaluated with multiparticle cumulants are shown as a function of transverse momentum and the collision centrality. Models of the initial spatial geometry and its fluctuations fail to describe the flow fluctuations measurements.

Response profiles of murine spiral ganglion neurons on multi-electrode arrays.

OBJECTIVE Cochlear implants (CIs) have become the gold standard treatment for deafness. These neuroprosthetic devices feature a linear electrode array, surgically inserted into the cochlea, and function by directly stimulating the auditory neurons located within the spiral ganglion, bypassing lost or not-functioning hair cells. Despite their success, some limitations still remain, including poor frequency resolution and high-energy consumption. In both cases, the anatomical gap between the electrode array and the spiral ganglion neurons (SGNs) is believed to be an important limiting factor. The final goal of the study is to characterize response profiles of SGNs growing in intimate contact with an electrode array, in view of designing novel CI devices and stimulation protocols, featuring a gapless interface with auditory neurons. APPROACH We have characterized SGN responses to extracellular stimulation using multi-electrode arrays (MEAs). This setup allows, in our view, to optimize in vitro many of the limiting interface aspects between CIs and SGNs. MAIN RESULTS Early postnatal mouse SGN explants were analyzed after 6-18 days in culture. Different stimulation protocols were compared with the aim to lower the stimulation threshold and the energy needed to elicit a response. In the best case, a four-fold reduction of the energy was obtained by lengthening the biphasic stimulus from 40 μs to 160 μs. Similarly, quasi monophasic pulses were more effective than biphasic pulses and the insertion of an interphase gap moderately improved efficiency. Finally, the stimulation with an external electrode mounted on a micromanipulator showed that the energy needed to elicit a response could be reduced by a factor of five with decreasing its distance from 40 μm to 0 μm from the auditory neurons. SIGNIFICANCE This study is the first to show electrical activity of SGNs on MEAs. Our findings may help to improve stimulation by and to reduce energy consumption of CIs and thereby contribute to the development of fully implantable devices with better auditory resolution in the future.