hp-dGFEM for Second-Order Elliptic Problems in Plyhedra I: Stability and Quasioptimality on Geometric Meshes

We introduce and analyze hp-version discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems in three-dimensional polyhedral domains. To resolve possible corner-, edge- and corner-edge singularities, we consider hexahedral meshes that are geometrically and anisotropically refined toward the corresponding neighborhoods. Similarly, the local polynomial degrees are increased linearly and possibly anisotropically away from singularities. We design interior penalty hp-dG methods and prove that they are well-defined for problems with singular solutions and stable under the proposed hp-refinements. We establish (abstract) error bounds that will allow us to prove exponential rates of convergence in the second part of this work.

Geometric properties of the lung parenchyma after postnatal glucocorticoid treatment in rats

While glucocorticoid (GC) administration appears to be beneficial during the acute phase of treatment of neonates at risk of developing chronic lung disease, it is still not clear whether steroid application has an adverse long-term effect on the lung maturation. Thus, the goal of the present work was to analyze GC effects on the pulmonary structure in a rat model where dosage and timing of drug administration were adapted to the therapeutic situation in human neonatology. The animals received daily a maximum of 0.1 mg dexamethasone phosphate per kilogram body weight during the first 4 postnatal days. Investigations were performed at the light microscopic level by means of a digital image analysis system. While there were no differences in the lung architecture between experimental animals and controls on day 4, the earliest time point of observation, we found a widening of airspaces with a concomitant decrease in the alveolar surface area density, representing a loss of parenchymal complexity, on days 10 and 21 in treated rats. On days 36 and 60, however, no alterations in the pulmonary parenchyma could be detected in experimental animals. We conclude from these findings that the GC-induced initial inhibition of development (days 10 and 21) was completely reversed, so that a normal parenchymal architecture and also a normal alveolar surface area density were found in adult rats (days 36 and 60). From the results obtained using the regimen of GC administration described, mimicking more closely the steroid treatment in human neonatology, we conclude that the observed short-term adverse effects on lung development can be fully compensated until adult age.

Genetic Control of Plant Development by Overriding a Geometric Division Rule

Formative cell divisions are critical for multicellular patterning. In the early plant embryo, such divisions follow from orienting the division plane. A major unanswered question is how division plane orientation is genetically controlled, and in particular whether this relates to cell geometry. We have generated a complete 4D map of early Arabidopsis embryogenesis and used computational analysis to demonstrate that several divisions follow a rule that uses the smallest wall area going through the center of the cell. In other cases, however, cell division clearly deviates from this rule, which invariably leads to asymmetric cell division. By analyzing mutant embryos and through targeted genetic perturbation, we show that response to the hormone auxin triggers a deviation from the ``shortest wall'' rule. Our work demonstrates that a simple default rule couples division orientation to cell geometry in the embryo and that genetic regulation can create patterns by overriding the default rule.