Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

Abstract. We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods. 1

The weighted and shifted seven-step BDF method for parabolic equations

Stability of the BDF methods of order up to five for parabolic equations can be established by the energy technique via Nevanlinna--Odeh multipliers. The nonexistence of Nevanlinna--Odeh multipliers makes the six-step BDF method special; however, the energy technique was recently extended by the authors in [Akrivis et al., SIAM J. Numer. Anal. \textbf{59} (2021) 2449--2472] and covers all six stable BDF methods. The seven-step BDF method is unstable for parabolic equations, since it is not even zero-stable. In this work, we construct and analyze a stable linear combination of two non zero-stable schemes, the seven-step BDF method and its shifted counterpart, referred to as WSBDF7 method. The stability regions of the WSBDF$q, q\leqslant 7$, with a weight $\vartheta\geqslant1$, increase as $\vartheta$ increases, are larger than the stability regions of the classical BDF$q,$ corresponding to $\vartheta=1$. We determine novel and suitable multipliers for the WSBDF7 method and establish stability for parabolic equations by the energy technique. The proposed approach is applicable for mean curvature flow, gradient flows, fractional equations and nonlinear equations.Comment: 23 page

Quadratic constraint consistency in the projection-free approximation of harmonic maps and bending isometries

We devise a projection-free iterative scheme for the approximation of harmonic maps that provides a second-order accuracy of the constraint violation and is unconditionally energy stable. A corresponding error estimate is valid under a mild but necessary discrete regularity condition. The method is based on the application of a BDF2 scheme and the considered problem serves as a model for partial differential equations with holonomic constraint. The performance of the method is illustrated via the computation of stationary harmonic maps and bending isometries

ENhancinG vAGinal dElivery in Greece through educational and behavioral interventions among maternity care providers regarding labor management: the ENGAGE stepped-wedge randomized prospective trial protocol

Background: There is an emerging need to systematically investigate the causes for the increased cesarean section rates in Greece and undertake interventions so as to substantially reduce its rates. To this end, the ability of the participating Greek obstetricians to follow evidence-based guidelines and respond to other educational and behavioral interventions while managing labor will be explored, along with barriers and enablers. Herein discussed is the protocol of a stepped-wedge designed intervention trial in Greek maternity units with the aforementioned goals in mind, named ENGAGE (ENhancinG vAGinal dElivery in Greece). Methods: Twenty-two selected maternity units in Greece will participate in a multicenter stepped-wedge randomized prospective trial involving 20,000 to 25,000 births, with two of them entering the intervention period of the study each month (stepped randomization). The maternity care units entering the study will apply the suggested interventions for a period of 8–18 months depending on the time they enter the intervention stage of the study. There will also be an initial phase of the study lasting from 8 to 18 months including observation and recording of the routine practice (cesarean section, vaginal birth, and maternal and perinatal morbidity and mortality) in the participating units. The second phase, the intervention period, will include such interventions as the application of the HSOG (the Hellenic Society of Obstetrics and Gynecology) Guidelines on labor management, training on the correct interpretation of cardiotocography, and dealing with emergencies in vaginal deliveries, while the steering committee members will be available to discuss and implement organizational and behavioral changes, answer questions, clarify relevant issues, and provide practical instructions to the participating healthcare professionals during regular visits or video conferences. Furthermore, during the study, the results will be available for the participating units in order for them to monitor their own performance while also receiving feedback regarding their rates. Τhe final 2-month phase of the study will be devoted to completing follow-up questionnaires with data concerning maternal and neonatal morbidities that occurred after the completion of the intervention period. The total duration of the study is estimated at 28 months. The primary outcome assessed will be the cesarean section rate change and the secondary outcomes will be maternal and neonatal morbidity and mortality. Discussion: The study is expected to yield new information on the effects, advantages, possibilities, and challenges of consistent clinical engagement and implementation of behavioral, educational, and organizational interventions described in detail in the protocol on cesarean section practice in Greece. The results may lead to new insights into means of improving the quality of maternal and neonatal care, particularly since this represents a shared effort to reduce the high cesarean section rates in Greece and, moreover, points the way to their reduction in other countries. Trial registration: NCT 04504500 (ClinicalTrials.gov). The trial was prospectively registered. Ethics Reference No: 320/23.6.2020, Bioethics and Conduct Committee, School of Medicine, National and Kapodistrian University of Athens, Athens, Greec