We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continuous Galerkin, non-conforming and a new, wide class of hybridizable discontinuous Galerkin methods. The distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements. Since the associated matrix is sparse, symmetric and positive definite, these methods can be efficiently implemented. Moreover, the framework allows, in a single implementation, the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled. Finally, the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques, as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom.

COVID-19 pandemic caused by SARS-CoV-2 infection is a public health emergency. COVID-19 typically exhibits respiratory illness. Unexpectedly, emerging clinical reports indicate that neurological symptoms continue to rise, suggesting detrimental effects of SARS-CoV-2 on the central nervous system (CNS). Here, we show that a Düsseldorf isolate of SARS-CoV-2 enters 3D human brain organoids within 2 days of exposure. We identified that SARS-CoV-2 preferably targets neurons of brain organoids. Imaging neurons of organoids reveal that SARS-CoV-2 exposure is associated with altered distribution of Tau from axons to soma, hyperphosphorylation, and apparent neuronal death. Our studies, therefore, provide initial insights into the potential neurotoxic effect of SARS-CoV-2 and emphasize that brain organoids could model CNS pathologies of COVID-19.

We lay out a program for constructing discontinuous Petrov-Galerkin (DPG) schemes having test function spaces that are automatically computable to guarantee stability. Given a trial space, a DPG discretization using its optimal test space counterpart inherits stability from the well posedness of the undiscretized problem. Although the question of stable test space choice had attracted the attention of many previous authors, the novelty in our approach lies in the fact we identify a discontinuous Galerkin (DG) framework wherein test functions, arbitrarily close to the optimal ones, can be locally computed. The idea is presented abstractly and its feasibility illustrated through several theoretical and numerical examples.

A mutation in the centrosomal‐P4.1‐associated protein (CPAP) causes Seckel syndrome with microcephaly, which is suggested to arise from a decline in neural progenitor cells (NPCs) during development. However, mechanisms of NPCs maintenance remain unclear. Here, we report an unexpected role for the cilium in NPCs maintenance and identify CPAP as a negative regulator of ciliary length independent of its role in centrosome biogenesis. At the onset of cilium disassembly, CPAP provides a scaffold for the cilium disassembly complex (CDC), which includes Nde1, Aurora A, and OFD1, recruited to the ciliary base for timely cilium disassembly. In contrast, mutated CPAP fails to localize at the ciliary base associated with inefficient CDC recruitment, long cilia, retarded cilium disassembly, and delayed cell cycle re‐entry leading to premature differentiation of patient iPS‐derived NPCs. Aberrant CDC function also promotes premature differentiation of NPCs in Seckel iPS‐derived organoids. Thus, our results suggest a role for cilia in microcephaly and its involvement during neurogenesis and brain size control.