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.

hp-dGFEM for Second-Order Elliptic Problems in Plyhedra II: Exponential Convergence

The goal of this paper is to establish exponential convergence of $hp$-version interior penalty (IP) discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems with homogeneous Dirichlet boundary conditions and piecewise analytic data in three-dimensional polyhedral domains. More precisely, we shall analyze the convergence of the $hp$-IP dG methods considered in [D. Schötzau, C. Schwab, T. P. Wihler, SIAM J. Numer. Anal., 51 (2013), pp. 1610--1633] based on axiparallel $\sigma$-geometric anisotropic meshes and $\bm{s}$-linear anisotropic polynomial degree distributions.

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.

Sharing decisions in breast cancer care: Development of the Decision Analysis System for Oncology (DAS-O) to identify shared decision making during treatment consultations

Shared Decision Making (SDM) is widely accepted as the preferred method for reaching treatment decisions in the oncology setting including those about clinical trial participation: however, there is some disagreement between researchers over the components of SDM. Specific standardized coding systems are needed to help overcome this difficulty.

Strategies for the prevention of postoperative recurrence in Crohn's disease: results of a decision analysis

Nearly 70% of patients with Crohn's disease (CD) undergo surgical resection, with one-quarter subsequently developing clinical recurrence within 12 months. Several options exist for the prevention of postoperative recurrence in CD, but the comparative cost effectiveness of these competing strategies has not been previously analyzed.

Usefulness of preemptive anticoagulation in patients with suspected pulmonary embolism: a decision analysis

The diagnostic workup of pulmonary embolism (PE) may take several hours. The usefulness of anticoagulant treatment while awaiting the results of diagnostic tests has not been assessed. The objective of this study was to compare the risks and benefits of bid low-molecular-weight heparin vs no treatment in patients with suspected PE.

Sex Differences in Semantic Processing: Event-Related Brain Potentials Distinguish between Lower and Higher Order Semantic Analysis during Word Reading

Behavioral studies suggest that women and men differ in the strategic elaboration of verbally encoded information especially in the absence of external task demand. However, measuring such covert processing requires other than behavioral data. The present study used event-related potentials to compare sexes in lower and higher order semantic processing during the passive reading of semantically related and unrelated word pairs. Women and men showed the same early context effect in the P1-N1 transition period. This finding indicates that the initial lexical-semantic access is similar in men and women. In contrast, sexes differed in higher order semantic processing. Women showed an earlier and longer lasting context effect in the N400 accompanied by larger signal strength in temporal networks similarly recruited by men and women. The results suggest that women spontaneously conduct a deeper semantic analysis. This leads to faster processing of related words in the active neural networks as reflected in a shorter stability of the N400 map in women. Taken together, the findings demonstrate that there is a selective sex difference in the controlled semantic analysis during passive word reading that is not reflected in different functional organization but in the depth of processing.

Relative Predicativity and dependent recursion in second-order set theory and higher-orders theories

This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditional ω. Namely, the equivalence between normal transfinite recursion scheme and new dependent transfinite recursion scheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universe V of sets is treated as the given totality (nor in the contexts of those of n+3-th order number or set theories, where the class of all n+2-th order objects is treated as the given totality).