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.

Three-dimensional surface reconstruction within noncontact diffuse optical tomography using structured light

A main field in biomedical optics research is diffuse optical tomography, where intensity variations of the transmitted light traversing through tissue are detected. Mathematical models and reconstruction algorithms based on finite element methods and Monte Carlo simulations describe the light transport inside the tissue and determine differences in absorption and scattering coefficients. Precise knowledge of the sample's surface shape and orientation is required to provide boundary conditions for these techniques. We propose an integrated method based on structured light three-dimensional (3-D) scanning that provides detailed surface information of the object, which is usable for volume mesh creation and allows the normalization of the intensity dispersion between surface and camera. The experimental setup is complemented by polarization difference imaging to avoid overlaying byproducts caused by inter-reflections and multiple scattering in semitransparent tissue.

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.

Rising serum sodium levels are associated with a concurrent development of metabolic alkalosis in critically ill patients

PURPOSE Changes in electrolyte homeostasis are important causes of acid-base disorders. While the effects of chloride are well studied, only little is known of the potential contributions of sodium to metabolic acid-base state. Thus, we investigated the effects of intensive care unit (ICU)-acquired hypernatremia on acid-base state. METHODS We included critically ill patients who developed hypernatremia, defined as a serum sodium concentration exceeding 149 mmol/L, after ICU admission in this retrospective study. Data on electrolyte and acid-base state in all included patients were gathered in order to analyze the effects of hypernatremia on metabolic acid-base state by use of the physical-chemical approach. RESULTS A total of 51 patients were included in the study. The time of rising serum sodium and hypernatremia was accompanied by metabolic alkalosis. A transient increase in total base excess (standard base excess from 0.1 to 5.5 mmol/L) paralleled by a transient increase in the base excess due to sodium (base excess sodium from 0.7 to 4.1 mmol/L) could be observed. The other determinants of metabolic acid-base state remained stable. The increase in base excess was accompanied by a slight increase in overall pH (from 7.392 to 7.429, standard base excess from 0.1 to 5.5 mmol/L). CONCLUSIONS Hypernatremia is accompanied by metabolic alkalosis and an increase in pH. Given the high prevalence of hypernatremia, especially in critically ill patients, hypernatremic alkalosis should be part of the differential diagnosis of metabolic acid-base disorders.

Cognitive and emotional effects of carotid stenosis

PRINCIPLES Patients with carotid artery stenosis (CAS) are at risk of ipsilateral stroke and chronic compromise of cerebral blood flow. It is under debate whether the hypo-perfusion or embolism in CAS is directly related to cognitive impairment. Alternatively, CAS may be a marker for underlying risk factors, which themselves influence cognition. We aimed to determine cognitive performance level and the emotional state of patients with CAS. We hypothesised that patients with high grade stenosis, bilateral stenosis, symptomatic patients and/or those with relevant risk factors would suffer impairment of their cognitive performance and emotional state. METHODS A total of 68 patients with CAS of ≥70% were included in a prospective exploratory study design. All patients underwent structured assessment of executive functions, language, verbal and visual memory, motor speed, anxiety and depression. RESULTS Significantly more patients with CAS showed cognitive impairments (executive functions, word production, verbal and visual memory, motor speed) and anxiety than expected in a normative sample. Bilateral and symptomatic stenosis was associated with slower processing speed. Cognitive performance and anxiety level were not influenced by the side and the degree of stenosis or the presence of collaterals. Factors associated with less cognitive impairment included higher education level, female gender, ambidexterity and treated hypercholesterolemia. CONCLUSIONS Cognitive impairment and increased level of anxiety are frequent in patients with carotid stenosis. The lack of a correlation between cognitive functioning and degree of stenosis or the presence of collaterals, challenges the view that CAS per se leads to cognitive impairment.

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.

The Utilization of Classical Spin Monte Carlo Methods to Simulate the Magnetic Behavior of Extended Three-Dimensional Cubic Networks Incorporating M(II) Ions with an S = 5/2 Ground State Spin

The numerical simulations of the magnetic properties of extended three-dimensional networks containing M(II) ions with an S = 5/2 ground-state spin have been carried out within the framework of the isotropic Heisenberg model. Analytical expressions fitting the numerical simulations for the primitive cubic, diamond, together with (10−3) cubic networks have all been derived. With these empirical formulas in hands, we can now extract the interaction between the magnetic ions from the experimental data for these networks. In the case of the primitive cubic network, these expressions are directly compared with those from the high-temperature expansions of the partition function. A fit of the experimental data for three complexes, namely [(N(CH3)4][Mn(N3)] 1, [Mn(CN4)]n 2, and [FeII(bipy)3][MnII2(ox)3] 3, has been carried out. The best fits were those obtained using the following parameters, J = −3.5 cm-1, g = 2.01 (1); J = −8.3 cm-1, g = 1.95 (2); and J = −2.0 cm-1, g = 1.95 (3).