Fully Adaptive Newton--Galerkin Methods for Semilinear Elliptic Partial Differential Equations

In this paper we develop an adaptive procedure for the numerical solution of general, semilinear elliptic problems with possible singular perturbations. Our approach combines both prediction-type adaptive Newton methods and a linear adaptive finite element discretization (based on a robust a posteriori error analysis), thereby leading to a fully adaptive Newton–Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples

Essential spectrum of systems of singular differential equations

In this paper we develop a new method to determine the essential spectrum of coupled systems of singular differential equations. Applications to problems from magnetohydrodynamics and astrophysics are given.

Population pharmacokinetics of sevoflurane in conjunction with the AnaConDa: toward target-controlled infusion of volatiles into the breathing system

BACKGROUND: The Anesthetic Conserving Device (AnaConDa) uncouples delivery of a volatile anesthetic (VA) from fresh gas flow (FGF) using a continuous infusion of liquid volatile into a modified heat-moisture exchanger capable of adsorbing VA during expiration and releasing adsorbed VA during inspiration. It combines the simplicity and responsiveness of high FGF with low agent expenditures. We performed in vitro characterization of the device before developing a population pharmacokinetic model for sevoflurane administration with the AnaConDa, and retrospectively testing its performance (internal validation). MATERIALS AND METHODS: Eighteen females and 20 males, aged 31-87, BMI 20-38, were included. The end-tidal concentrations were varied and recorded together with the VA infusion rates into the device, ventilation and demographic data. The concentration-time course of sevoflurane was described using linear differential equations, and the most suitable structural model and typical parameter values were identified. The individual pharmacokinetic parameters were obtained and tested for covariate relationships. Prediction errors were calculated. RESULTS: In vitro studies assessed the contribution of the device to the pharmacokinetic model. In vivo, the sevoflurane concentration-time courses on the patient side of the AnaConDa were adequately described with a two-compartment model. The population median absolute prediction error was 27% (interquartile range 13-45%). CONCLUSION: The predictive performance of the two-compartment model was similar to that of models accepted for TCI administration of intravenous anesthetics, supporting open-loop administration of sevoflurane with the AnaConDa. Further studies will focus on prospective testing and external validation of the model implemented in a target-controlled infusion device.

Spectral bounds and basis results for non-self-adjoint pencils, with an application to Hagen-Poiseuille flow

We obtain eigenvalue enclosures and basisness results for eigen- and associated functions of a non-self-adjoint unbounded linear operator pencil A−λBA−λB in which BB is uniformly positive and the essential spectrum of the pencil is empty. Both Riesz basisness and Bari basisness results are obtained. The results are applied to a system of singular differential equations arising in the study of Hagen–Poiseuille flow with non-axisymmetric disturbances.