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In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.

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Objectives: Hyperoxia could lead to a worse outcome after cardiac arrest. Few studies have investigated the impact of oxygenation status on patient outcomes following extracorporeal cardiopulmonary resuscitation. We sought to delineate the association between oxygenation status and neurologic outcomes in patients receiving extracorporeal cardiopulmonary resuscitation. Design: Retrospective analysis of a prospective extracorporeal cardiopulmonary resuscitation registry database. Setting: An academic tertiary care hospital. Patients: Patients receiving extracorporeal cardiopulmonary resuscitation between 2000 and 2014. Interventions: None. Measurements and Main Results: A total of 291 patients were included, and 80.1% were male. Their mean age was 56.0 years. The arterial blood gas data employed in the primary analysis were recorded from the first sample over the first 24 hours in the ICUs after return of spontaneous circulation. The mean Pao 2 after initiation of venoarterial extracorporeal membrane oxygenation was 178.0 mm Hg, and the mean Pao 2/Fio 2 ratio was 322.0. Only 88 patients (30.2%) demonstrated favorable neurologic status at hospital discharge. Multivariate logistic regression analysis indicated that Pao 2 between 77 and 220 mm Hg (odds ratio, 2.29; 95% CI, 1.01–5.22; p = 0.05) and Pao 2/Fio 2 ratio between 314 and 788 (odds ratio, 5.09; 95% CI, 2.13–12.14; p < 0.001) were both positively associated with favorable neurologic outcomes. Conclusions: Oxygenation status during extracorporeal membrane oxygenation affects neurologic outcomes in patients receiving extracorporeal cardiopulmonary resuscitation. The Pao 2 range of 77 to 220 mm Hg, which is slightly narrower than previously defined, seems optimal. The Pao 2/Fio 2 ratio was also associated with outcomes in our analysis, indicating that both Pao 2 and the Pao 2/Fio 2 ratio should be closely monitored during the early postcardiac arrest phase for postextracorporeal cardiopulmonary resuscitation patients.

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