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This paper develops divergence-free mixed finite element methods for the Stokes equation. Using H(div)-conforming velocities and discontinuous pressures ensures the inf-sup condition for the velocity--pressure pair and yields pointwise divergence-free velocities. However, this choice makes the vector Laplacian difficult to discretize. Inspired by mass-conserving mixed formulations with stresses, tangential--normal continuous traceless tensor elements are introduced to discretize the vector Laplacian. An inf-sup condition for the weak div operator between the stress and velocity spaces is then proved. Two key properties characterize the scheme. First, the stress--velocity inf-sup stability gives a stable discretization of the vector Laplacian without additional stabilization, unlike discontinuous Galerkin or virtual element methods. Second, the scheme has the property that if a stress field is distributionally divergence-free against the discrete divergence-free velocity space, then it is also distributionally divergence-free against the continuous divergence-free velocity space. This property decouples the stress and velocity errors and leads to superconvergence. As a result, optimal-order error estimates are obtained for the stress, while the velocity and pressure converge at rates higher than the approximation orders of the chosen spaces. Numerical experiments confirm the theoretical results. 

The interaction between K atoms and oxygen molecules on solid surfaces is of topical interest to oxidation–reduction processes in K–O2 batteries. Alkali metals have one ns electron in their valence shell, making them highly chemically reactive toward oxidizing reactants. Mechanistic information on the oxygen reduction by K at the atomic level is scarce despite its key role in defining the alkali metal–O2 battery performance. Here, we use scanning tunneling microscopy and density functional theory to investigate the reduction of a single oxygen molecule by K atoms codeposited on the Ag(111) surface. Our study provides fundamental chemical information on the binary and collective interactions between the O2 and K atoms on metal surfaces. 

This paper addresses the challenge of constructing finite element curl div complexes in three dimensions. Tangential-normal continuity is introduced in order to develop distributional finite element curl div complexes. The spaces constructed are applied to discretize the quad curl problem, demonstrating optimal order of convergence. Furthermore, a hybridization technique is proposed, demonstrating its equivalence to nonconforming finite elements and weak Galerkin methods. 

Abstract Although the existing causal inference literature focuses on the forward-looking perspective by estimating effects of causes, the backward-looking perspective can provide insights into causes of effects. In backward-looking causal inference, the probability of necessity measures the probability that a certain event is caused by the treatment given the observed treatment and outcome. Most existing results focus on binary outcomes. Motivated by applications with ordinal outcomes, we propose a general definition of the probability of necessity. However, identifying the probability of necessity is challenging because it involves the joint distribution of the potential outcomes. We propose a novel assumption of monotonic incremental treatment effect to identify the probability of necessity with ordinal outcomes. We also discuss the testable implications of this key identification assumption. When it fails, we derive explicit formulas of the sharp large-sample bounds on the probability of necessity. 

Spring sea ice cover in the Barents-Kara Seas (BKS) exhibits notable interannual to decadal variability with spatial patterns distinct from winter. While previous studies suggest that internal climate variability dominates, the specific drivers remain to be identified. Using observations and model simulations, we reveal that atmospheric teleconnection from the western North Pacific accounts for ~26 to 40% of the interannual to decadal variability of the BKS sea ice during February-March-April. A poleward-propagating Rossby wave train, forced by divergent winds associated with the Western Pacific (WP) pattern, links WP to BKS sea ice variability by inducing lower-atmospheric circulation anomalies over the North Atlantic that enhance poleward heat and moisture transport. Furthermore, decadal shifts in the WP pattern, potentially originating from the Pacific Decadal Oscillation, have modulated the sea ice trend over the BKS, driving accelerated retreat in the 2000s and a slowdown after 2015. A projected transition of WP to its negative phase may accelerate BKS sea ice loss in the coming decade.